Explanation:
Okay, so I need to find the measure of angle STR in this diagram. Let me try to visualize it first. There's a diagram with points P, Q, R, S, and T. The Asymptote code shows that Q is at (0,0), R is at (1,0), P is at (1.1, 0.5), S is 0.6*P, and T is R + 0.6*(P - R). Then they draw lines connecting T to S, S to P, P to R, R to Q, and Q to S.

First, let me sketch this mentally. Points Q and R are on the x-axis, with Q at the origin and R at (1,0). Point P is above them, a bit to the right of R. Then S is 0.6 times P, so it's somewhere between the origin and P? Wait, no. If S is 0.6*P, then since P is at (1.1, 0.5), multiplying by 0.6 would give S at (0.66, 0.3). But then T is R + 0.6*(P - R). Let me compute that. P - R is (1.1 - 1, 0.5 - 0) = (0.1, 0.5). Then 0.6*(0.1, 0.5) is (0.06, 0.3). Adding that to R (1,0) gives T at (1.06, 0.3). So S is at (0.66, 0.3) and T is at (1.06, 0.3). So S and T are both on the line y=0.3? Wait, but then ST is horizontal? Because both have the same y-coordinate. And QR is from (0,0) to (1,0), which is also horizontal. So ST is parallel to QR, as given.

So the diagram has ST parallel to QR. Also, angles at P and Q are given: angle P is 40 degrees, angle Q is 35 degrees. Need to find angle STR.

Let me try to label the diagram. Points Q and R are on the base, P is above, S is somewhere between Q and P? Wait, S is 0.6*P, so if P is at (1.1, 0.5), then S is 0.6 of the way from the origin to P? Wait, no. If S is 0.6*P, then it's scaling the coordinates of P by 0.6. So if P is (1.1, 0.5), then S is (0.66, 0.3). But Q is at (0,0), so S is somewhere between Q and P? Maybe. Similarly, T is R + 0.6*(P - R). So starting at R, moving 0.6 of the way towards P. Since R is (1,0) and P is (1.1, 0.5), then T is (1 + 0.6*(0.1), 0 + 0.6*(0.5)) = (1.06, 0.3). So T is above R, but not as high as P.

So the lines drawn are T to S, S to P, P to R, R to Q, and Q to S. So triangle QSP? And triangle PTR? Maybe. Also, ST is parallel to QR.

Given that ST is parallel to QR, which are both horizontal in this coordinate system. So maybe we can use some properties of parallel lines and transversals. Also, angles at P and Q are given. Let me think.

First, angle P is 40 degrees. In triangle PQR? Wait, but where is point S and T? Maybe triangle PQS or something else? Wait, the diagram is a bit complex. Let me try to figure out which triangles we have.

From the Asymptote code, the figure is drawn as T--SS--P--R--Q--SS. So connecting T to S, S to P, P to R, R to Q, Q to S. So the figure includes triangle QSP, triangle PRT? Maybe quadrilateral QSPR? Not sure.

But since ST is parallel to QR, maybe we can use similar triangles or something. Also, angles at P and Q are given. Let me think about triangle PQR. Wait, but is P connected to Q? In the diagram, P is connected to R and S, and Q is connected to S and R. So maybe triangle PQR is not directly drawn, but maybe we can consider it.

Alternatively, maybe triangle PQS and triangle PRT? Wait, not sure. Let me try to find some relationships.

Since ST is parallel to QR, then maybe there are some corresponding angles or alternate interior angles. Let me think. If I consider ST parallel to QR, and maybe some transversal cutting them. For example, if we consider line SP as a transversal, then angle at S and angle at Q? Wait, not sure.

Alternatively, maybe triangle SPQ and triangle...? Wait, maybe triangle SPT? Not sure.

Alternatively, since we have angles at P and Q given, maybe we can find some other angles in the diagram.

Let me note that angle P is 40 degrees. In the diagram, angle at P is between which lines? Since P is connected to S and R. So angle at P is between SP and PR. Similarly, angle Q is 35 degrees, which is at Q, between RQ and QS.

So maybe we can find some angles in triangle PQS or triangle PQR?

Wait, if I can find some angles in triangle PQR, maybe. Let me try.

But first, let me check if triangle PQR is a triangle. Points P, Q, R. Q is connected to R, R is connected to P? Yes, in the diagram, P is connected to R. So triangle PQR exists.

In triangle PQR, we know angle at Q is 35 degrees, angle at P is 40 degrees? Wait, but angle at P in triangle PQR would be angle QPR? Wait, but in the diagram, angle P is given as 40 degrees. Is that angle QPR?

Wait, the problem says "angle P = 40°", so that's angle at point P. In triangle PQR, angle at P is between sides PQ and PR. But in the diagram, point S is connected to P, so maybe angle P is between SP and PR? Wait, the problem says "angle P = 40°", so maybe that's angle SPQ? Wait, no. The Asymptote code draws P connected to S and R. So angle at P is between SP and PR. So angle SP R? Wait, but the problem says angle P is 40°, so that's angle at P between SP and PR.

Similarly, angle Q is 35°, which is at Q between RQ and QS.

So maybe we can find some angles in triangle PQS or triangle PQR?

Alternatively, since ST is parallel to QR, then maybe triangle SPT is similar to triangle QRT? Or something like that?

Alternatively, since ST is parallel to QR, then the corresponding angles would be equal. For example, if we consider line SP as a transversal cutting ST and QR, then angle at S and angle at Q would be equal? Wait, but ST is parallel to QR, so if SP is a transversal, then angle between SP and ST would be equal to angle between SP and QR?

Wait, maybe I need to find some angles. Let me try to find angle at S or T.

Alternatively, maybe using the fact that ST is parallel to QR, so the triangles involved might be similar.

Alternatively, since we have coordinates given in the Asymptote code, maybe I can compute coordinates and then compute the angle? But since the problem is general (not specific to the coordinates given in Asymptote code, since it's just an example), but maybe the Asymptote code is just for illustration. Wait, but the problem says "in the diagram below", so maybe the diagram is specific. However, since the Asymptote code is given, maybe I can use coordinates to calculate?

But maybe the problem is intended to be solved with pure geometry, not coordinates. Let me try that first.

So, given that ST is parallel to QR. Let me consider triangle PQR. If I can find some angles in triangle PQR, then maybe I can find other angles.

But angle at P is 40°, angle at Q is 35°, so maybe angle at R can be found?

Wait, in triangle PQR, angles at P, Q, R. Wait, but is triangle PQR a triangle? If points P, Q, R are connected, then yes. But angle at P is 40°, angle at Q is 35°, so angle at R would be 180 - 40 - 35 = 105°.

Wait, but is that correct? Wait, if triangle PQR has angles at P, Q, R, then yes. But is angle at P in triangle PQR the same as angle P given in the problem?

Wait, the problem says "angle P = 40°", which is angle at P. If in triangle PQR, angle at P is 40°, angle at Q is 35°, then angle at R is 105°, as above.

But then, how does ST come into play? Since ST is parallel to QR.

So maybe triangle STR? Wait, we need angle STR. So point S, T, R.

Alternatively, maybe triangle STR is related to triangle Q... something.

Alternatively, since ST is parallel to QR, then maybe triangle SPT is similar to triangle QPR?

Wait, let me think. If ST is parallel to QR, then the corresponding angles would be equal. For example, angle at S would correspond to angle at Q, and angle at T would correspond to angle at R?

But I need to figure out which triangles are similar.

Alternatively, since ST is parallel to QR, then the line SP is cutting the sides of triangle PQR? Wait, if ST is parallel to QR, then maybe S is on PQ and T is on PR? But in the Asymptote code, S is 0.6*P, which is not on PQ. Wait, maybe not.

Wait, in the Asymptote code, S is defined as 0.6*P, so if P is (1.1, 0.5), then S is (0.66, 0.3). But Q is (0,0), R is (1,0). So S is not on PQ or PR. Wait, maybe S is on PS? Wait, no.

Alternatively, maybe S is on PQ? Let me check coordinates. If S is (0.66, 0.3), and Q is (0,0), P is (1.1, 0.5). So the line from Q to P is from (0,0) to (1.1, 0.5). Let me see if S lies on that line.

Parametric equations: from Q(0,0) to P(1.1, 0.5). So any point on that line can be written as (1.1t, 0.5t) where t is between 0 and 1.

If S is (0.66, 0.3), then 1.1t = 0.66 => t = 0.66 / 1.1 = 0.6. Similarly, 0.5t = 0.3 => t = 0.6. So yes! S is on QP, at t=0.6. Similarly, T is defined as R + 0.6*(P - R). So R is (1,0), P - R is (0.1, 0.5). So 0.6*(0.1, 0.5) is (0.06, 0.3). Adding to R gives T as (1.06, 0.3). Let me check if T is on PR.

PR is from P(1.1, 0.5) to R(1, 0). Let me parametrize PR. Let me write parametric equations. Let parameter be s from 0 to 1.

x = 1.1 - 0.1s

y = 0.5 - 0.5s

So when s=0, we are at P(1.1, 0.5); when s=1, we are at R(1, 0).

Now, T is at (1.06, 0.3). Let me see if there's an s such that:

1.1 - 0.1s = 1.06 => 0.1s = 1.1 - 1.06 = 0.04 => s = 0.4

Similarly, y-coordinate: 0.5 - 0.5s = 0.3 => 0.5s = 0.2 => s = 0.4.

Yes! So T is on PR at s=0.4.

Therefore, S is on QP at t=0.6, and T is on PR at s=0.4.

Therefore, ST is a line connecting a point on QP to a point on PR, and it's parallel to QR.

So given that ST is parallel to QR, and S divides QP in ratio t=0.6, and T divides PR in ratio s=0.4.

But since ST is parallel to QR, by the converse of the basic proportionality theorem (Thales' theorem), if a line divides two sides of a triangle proportionally, then it is parallel to the third side.

But here, ST is parallel to QR, so it should divide the sides QP and PR proportionally.

But in this case, QS / QP = 0.6, and RT / PR = ?

Wait, let me check.

QS is the segment from Q to S. Since S divides QP into QS and SP, with QS / QP = 0.6. Similarly, RT is the segment from R to T. Since T divides PR into PT and TR. Since T is at s=0.4 from P to R, then PT / PR = 0.4, so TR / PR = 1 - 0.4 = 0.6.

Therefore, QS / QP = 0.6 and TR / PR = 0.6. Therefore, QS / QP = TR / PR, which means that ST is parallel to QR by the converse of Thales' theorem.

So that's consistent with the diagram.

Therefore, in triangle PQR, ST is a line segment connecting points S on QP and T on PR such that QS / QP = TR / PR = 0.6.

Therefore, maybe we can use this to find some relations.

But how does this help us find angle STR?

Angle STR is the angle at point T between points S, T, R. So angle at T between S and R.

To find angle STR, maybe we can consider triangle STR. If we can find some angles or sides in triangle STR, we can compute it.

Alternatively, since we know some angles in triangle PQR, maybe we can find angles in triangle STR.

First, let me recap. In triangle PQR, angles at P, Q, R are 40°, 35°, and 105°, as calculated earlier.

But wait, is that correct? Wait, angle at P is 40°, angle at Q is 35°, so angle at R is 180 - 40 - 35 = 105°, yes.

But how does that relate to triangle STR?

Alternatively, maybe we can find coordinates of all points and compute the angle. Since the Asymptote code gives coordinates, even though it's just an example, maybe we can use those coordinates to calculate the angle.

Let me try that approach.

Given coordinates:

Q = (0, 0)

R = (1, 0)

P = (1.1, 0.5)

S = 0.6 * P = (0.66, 0.3)

T = R + 0.6*(P - R) = (1, 0) + 0.6*(0.1, 0.5) = (1 + 0.06, 0 + 0.3) = (1.06, 0.3)

So coordinates:

S = (0.66, 0.3)

T = (1.06, 0.3)

R = (1, 0)

So angle STR is the angle at T between points S, T, R.

So to find angle at T between S and R.

To compute this angle, we can use vectors or coordinate geometry.

First, find vectors TS and TR, then compute the angle between them.

Vector TS = S - T = (0.66 - 1.06, 0.3 - 0.3) = (-0.4, 0)

Vector TR = R - T = (1 - 1.06, 0 - 0.3) = (-0.06, -0.3)

Then, the angle between vectors TS and TR can be found using the dot product formula:

cos(theta) = (TS • TR) / (|TS| |TR|)

Compute TS • TR = (-0.4)(-0.06) + (0)(-0.3) = 0.024 + 0 = 0.024

|TS| = sqrt((-0.4)^2 + 0^2) = sqrt(0.16) = 0.4

|TR| = sqrt((-0.06)^2 + (-0.3)^2) = sqrt(0.0036 + 0.09) = sqrt(0.0936) ≈ 0.306

Therefore, cos(theta) ≈ 0.024 / (0.4 * 0.306) ≈ 0.024 / 0.1224 ≈ 0.196

Then theta ≈ arccos(0.196) ≈ 78.7 degrees.

Wait, but the problem is likely expecting an exact value, not an approximate decimal. So maybe my approach using coordinates is not the right way, since the coordinates are specific to the Asymptote code, which might not be to scale or might be just an example.

Alternatively, maybe there's a geometric relationship that gives an exact value.

Alternatively, maybe my calculation is correct, and the answer is approximately 78.7°, but since angles in the problem are given as whole numbers (40°, 35°), maybe the exact value is 75° or something else.

Wait, but let me check my calculation again.

First, coordinates:

S = (0.66, 0.3)

T = (1.06, 0.3)

R = (1, 0)

So vector TS is S - T = (0.66 - 1.06, 0.3 - 0.3) = (-0.4, 0). Correct.

Vector TR is R - T = (1 - 1.06, 0 - 0.3) = (-0.06, -0.3). Correct.

Dot product: (-0.4)(-0.06) + (0)(-0.3) = 0.024 + 0 = 0.024. Correct.

|TS| = sqrt( (-0.4)^2 + 0^2 ) = 0.4. Correct.

|TR| = sqrt( (-0.06)^2 + (-0.3)^2 ) = sqrt(0.0036 + 0.09) = sqrt(0.0936). Let me compute sqrt(0.0936).

Note that 0.3^2 = 0.09, 0.306^2 = 0.0936. Because 0.3^2 = 0.09, 0.306^2 = (0.3 + 0.006)^2 = 0.09 + 2*0.3*0.006 + 0.006^2 = 0.09 + 0.0036 + 0.000036 = 0.093636. So sqrt(0.0936) ≈ 0.306. So yes, |TR| ≈ 0.306.

Therefore, cos(theta) ≈ 0.024 / (0.4 * 0.306) ≈ 0.024 / 0.1224 ≈ 0.196.

Now, arccos(0.196). Let me check what angle has cosine ≈ 0.196.

cos(80°) ≈ 0.1736

cos(75°) ≈ 0.2588

So 0.196 is between 75° and 80°, closer to 80°. Let me compute it more accurately.

Let me use calculator-like approach. Let me recall that cos(78°) ≈ 0.2079

cos(79°) ≈ 0.1908

So 0.196 is between 78° and 79°, closer to 79°.

Let me compute the difference:

At 78°, cos ≈ 0.2079

At 79°, cos ≈ 0.1908

Difference between 78° and 79° is 1°, and the difference in cos values is 0.2079 - 0.1908 = 0.0171

Our value is 0.196, which is 0.2079 - 0.196 = 0.0119 less than cos(78°).

So fraction = 0.0119 / 0.0171 ≈ 0.696. So approximately 78° + 0.696° ≈ 78.696°, so approximately 78.7°, as before.

But the problem is likely expecting an exact value. Therefore, maybe my coordinate approach is not the right way, or maybe there's an exact value here.

Alternatively, maybe I made a wrong assumption in considering triangle PQR. Let me think again.

Alternatively, maybe using the Law of Sines or Law of Cosines in some triangle.

Let me try to find some relations.

Since ST is parallel to QR, then angle between ST and SP is equal to angle between QR and QP? Wait, maybe.

Alternatively, since ST || QR, then angle SPT is equal to angle PQR?

Wait, let me think. If ST is parallel to QR, then the transversal PQ would create corresponding angles.

Wait, if we consider line PQ intersecting ST and QR. But S is on PQ, and T is on PR. Maybe not straightforward.

Alternatively, since ST || QR, then angle between SP and ST is equal to angle between SP and QR? Not sure.

Alternatively, since ST || QR, then triangle STR is similar to triangle Q... something?

Alternatively, since S divides QP in ratio 0.6 and T divides PR in ratio 0.4, as we saw earlier.

Wait, in triangle PQR, with ST || QR, then by the basic proportionality theorem, QS / QP = RT / PR = 0.6.

But since QS / QP = 0.6, then QS = 0.6 QP, so SP = QP - QS = 0.4 QP. Similarly, RT = 0.6 PR, so PT = PR - RT = 0.4 PR.

Therefore, SP / QP = 0.4 and PT / PR = 0.4.

Therefore, SP / QP = PT / PR = 0.4.

Therefore, triangles SPQ and PTR? Wait, not sure.

Alternatively, since SP / QP = PT / PR, and angle at P is common? Wait, if we consider triangles SPQ and PTR?

Wait, angle at P is common? No, because SP is part of PQ, and PT is part of PR.

Wait, maybe triangles SPT and QPR?

Wait, not sure.

Alternatively, since ST || QR, then angle PST = angle PQR = 35°, because they are corresponding angles.

Wait, if ST || QR, and PQ is a transversal, then angle between PQ and ST (which is angle at S) is equal to angle between PQ and QR (which is angle at Q).

Wait, angle at Q is 35°, so angle between PQ and QR is 35°, so angle between PQ and ST is also 35°, which is angle at S.

Therefore, angle at S (angle QSP) is 35°.

Similarly, since ST || QR, and PR is a transversal, then angle between PR and ST (which is angle at T) is equal to angle between PR and QR (which is angle at R).

Angle at R in triangle PQR is 105°, so angle between PR and QR is 105°, so angle between PR and ST is also 105°, which is angle at T (angle PTR).

Wait, but angle PTR is at T between P and R. But we need angle STR, which is at T between S and R.

Wait, so if angle PTR is 105°, and angle at T between S and R is angle STR, then maybe we can find angle STR by considering triangle STR?

Wait, but we need more information.

Alternatively, in triangle STR, if we can find two angles, we can find the third.

But we need to find angle STR.

Alternatively, let me consider triangle SPT.

Wait, maybe not.

Alternatively, since we know angle at S is 35°, angle at T is...?

Wait, if angle between PR and ST is 105°, which is angle PTR. But angle PTR is at T between P and R. So angle PTR = 105°, and angle STR is adjacent to it?

Wait, if we look at point T, connected to S, P, and R. So at point T, there are angles: angle STR (which we need), angle PTR (which is 105°), and angle STP.

Wait, but unless we know something about the other angles, it's hard to say.

Alternatively, since we know angle at S is 35°, and maybe we can find angle at S in triangle STR?

Wait, in triangle STR, angle at S is angle SRT? No, angle at S is angle STR? No, angle at S is angle S in triangle STR. Wait, triangle STR has vertices S, T, R. So angles at S, T, R.

We need angle at T, which is angle STR.

If we can find angles at S and R in triangle STR, then we can find angle at T.

But how?

Alternatively, let me consider triangle SPQ.

In triangle SPQ, we know angle at Q is 35°, angle at S is 35°, as we said earlier? Wait, no. Wait, angle at S in triangle SPQ is angle QSP, which we said is 35°, because ST || QR and PQ is a transversal.

Wait, if angle QSP is 35°, and angle at Q is 35°, then triangle SPQ has two angles of 35°, making it an isosceles triangle?

Wait, if angle at Q is 35°, angle at S is 35°, then angle at P in triangle SPQ would be 180 - 35 - 35 = 110°.

But angle at P in triangle SPQ is angle SPQ. However, in the original problem, angle at P is given as 40°, which is angle between SP and PR.

Wait, so angle SPQ is part of angle at P.

Wait, angle at P in the diagram is between SP and PR, which is 40°, but angle SPQ is between SP and PQ.

So unless PQ and PR are related...

Wait, maybe I need to find angle SPQ.

If in triangle SPQ, angles at Q and S are 35°, then angle at P (angle SPQ) is 110°, as above.

But angle at P in the diagram is angle between SP and PR, which is given as 40°.

Therefore, angle between SP and PR is 40°, and angle between SP and PQ is 110°, so the angle between PQ and PR is 110° - 40° = 70°?

Wait, but angle between PQ and PR is angle QPR, which is part of triangle PQR.

Wait, in triangle PQR, angle at P is angle QPR, which we previously thought was 40°, but according to this, it's 70°?

This seems conflicting.

Wait, there must be a mistake here.

Let me clarify.

Earlier, I assumed that angle at P in triangle PQR is 40°, but maybe that's not the case.

Wait, the problem says "angle P = 40°", which is angle at P. In the diagram, angle at P is between SP and PR. However, in triangle PQR, angle at P is between PQ and PR.

Therefore, angle between SP and PR is 40°, and angle between PQ and PR is something else.

Therefore, my previous assumption that angle at P in triangle PQR is 40° is incorrect.

Therefore, I need to correct that.

So, let me start over.

Given that angle P = 40°, which is the angle between SP and PR.

Angle Q = 35°, which is the angle between RQ and QS.

We need to find angle STR.

Since ST is parallel to QR, we can use that to find some relationships.

Let me consider triangle SPQ.

Since ST || QR, and PQ is a transversal, then angle between PQ and ST (which is angle QSP) is equal to angle between PQ and QR (which is angle PQR = 35°). Therefore, angle QSP = 35°.

Similarly, since ST || QR, and PR is a transversal, then angle between PR and ST (which is angle PTR) is equal to angle between PR and QR (which is angle PRQ).

But angle PRQ is angle at R in triangle PQR.

But we don't know angle at R yet.

But let me denote some variables.

Let me denote angle at P (between SP and PR) as 40°, which is given.

Let me denote angle at Q (between RQ and QS) as 35°, given.

Let me denote angle QSP = 35°, as established.

In triangle SPQ, we have angle at Q = 35°, angle at S = 35°, so angle at P (angle SPQ) is 180 - 35 - 35 = 110°.

But angle SPQ is the angle at P between SP and PQ.

But the given angle at P is between SP and PR, which is 40°.

Therefore, the angle between PQ and PR is angle QPR = angle SPQ - angle between SP and PR? Wait, no.

Wait, angle between SP and PQ is 110°, and angle between SP and PR is 40°, so the angle between PQ and PR is angle QPR = angle SPQ - angle SPR? Wait, need to visualize.

At point P, there are two lines: SP and PR. The angle between them is 40°. Also, there is line PQ, which forms angle SPQ = 110° with SP.

Therefore, the angle between PQ and PR is angle QPR = angle SPQ - angle SPR? Wait, if SP is between PQ and PR, then angle QPR = angle SPQ - angle SPR?

Wait, if you imagine at point P, with SP going off in one direction, PQ going off in another, and PR going off in another.

If angle between SP and PQ is 110°, and angle between SP and PR is 40°, then the angle between PQ and PR is 110° - 40° = 70°.

Yes, because if SP is between PQ and PR, then the angle between PQ and PR is the difference between the angles SPQ and SPR.

Therefore, angle QPR = 70°.

Therefore, in triangle PQR, we have angle at Q = 35°, angle at P = 70°, so angle at R is 180 - 35 - 70 = 75°.

Ah! So angle at R is 75°, not 105° as I previously thought.

This was my mistake earlier, assuming angle at P in triangle PQR was 40°, but actually, angle at P in triangle PQR is 70°, because angle between PQ and PR is 70°, while the given angle at P (between SP and PR) is 40°.

Therefore, in triangle PQR, angles are:

At Q: 35°, at P: 70°, at R: 75°.

Now, going back to the earlier thought: since ST || QR, and PR is a transversal, then angle between PR and ST (which is angle PTR) is equal to angle between PR and QR (which is angle PRQ = 75°).

Therefore, angle PTR = 75°.

Now, looking at point T, which is connected to S, P, and R.

At point T, we have angles:

- angle PTR = 75° (between PT and TR)

- angle STR (which we need, between ST and TR)

- angle STP (between ST and TP)

These three angles at point T should add up to 360°, but since we are dealing with a triangle, maybe not. Wait, point T is connected to S, P, and R. So the angles around point T are:

- angle STR (between S and R)

- angle PTR (between P and R)

- angle STP (between S and P)

But these three angles are around point T, so they should add up to 360°. However, if we consider triangle STR and triangle PTR, it's more complex.

Alternatively, consider triangle STR. To find angle STR, we need more information.

But we know angle PTR = 75°, which is part of triangle PTR.

Alternatively, consider triangle STR and triangle... something.

Alternatively, since we know angle at R in triangle PQR is 75°, and ST || QR, maybe there's a relationship.

Alternatively, consider triangle STR. If we can find two angles, we can find the third.

We know angle at R in triangle STR is...?

Wait, angle at R in triangle STR is angle SRT.

But angle SRT is part of angle PRQ (which is 75°).

Since T is on PR, angle SRT is the same as angle PRQ? No, because S is not on PR.

Wait, point T is on PR, so angle SRT is the angle at R between SR and RT.

But RT is part of PR, so angle SRT is the angle between SR and PR.

But we might need to find that angle.

Alternatively, consider triangle SRT.

Wait, maybe using the Law of Sines in some triangles.

Let me consider triangle SPQ, where we know angles:

- angle at Q: 35°

- angle at S: 35°

- angle at P: 110°

Therefore, triangle SPQ is isosceles with SP = SQ.

Wait, if two angles are equal, then sides opposite are equal. So side opposite angle at Q (35°) is SP, and side opposite angle at S (35°) is PQ. Wait, no.

Wait, in triangle SPQ, angle at Q is 35°, angle at S is 35°, so sides opposite are equal.

Side opposite angle at Q is SP, and side opposite angle at S is PQ. Therefore, SP = PQ.

Wait, so SP = PQ.

Similarly, in triangle PQR, we can find some relations.

But I need to relate this to other parts.

Alternatively, since SP = PQ, and we know some angles, maybe we can find lengths.

But without specific lengths, it's hard.

Alternatively, since ST || QR, and we know ratios from the coordinates.

Earlier, in the coordinate system, we found angle STR ≈ 78.7°, but with the corrected angles, maybe we can find an exact value.

Wait, let me think again.

In triangle PQR, angles are 35°, 70°, 75°.

Since ST || QR, and S divides QP in ratio 0.6, T divides PR in ratio 0.4.

Let me consider triangle STR.

To find angle STR, maybe we can use the Law of Sines in triangle STR and some other triangle.

Alternatively, since we know angle at T is 75° (angle PTR), and if we can find angle at S or R in triangle STR, we can find angle STR.

Wait, angle at S in triangle STR is angle SRT? No, angle at S in triangle STR is angle STR? No, angle at S is angle STR? No, triangle STR has vertices S, T, R. So angles are:

- At S: angle STR? No, angle at S is angle TSR.

Wait, this is getting confusing. Let me clarify:

In triangle STR:

- Vertex S: angle between ST and SR

- Vertex T: angle between ST and TR (which is angle STR, the one we need)

- Vertex R: angle between SR and TR

So to find angle at T (angle STR), we need to know other angles.

If we can find angle at S or angle at R in triangle STR, we can find angle STR.

Let me try to find angle at R in triangle STR, which is angle SRT.

Angle SRT is the angle at R between SR and RT.

But RT is part of PR, and SR is a line from R to S.

Wait, in triangle PQR, angle at R is 75°, which is angle between PR and QR.

But SR is a line from R to S, which is somewhere inside the triangle.

Alternatively, since we know SP = PQ (from triangle SPQ being isosceles), and S divides QP in ratio 0.6, maybe we can find some relations.

Wait, SP = PQ.

But S is on QP, so SP = PQ?

Wait, if SP = PQ, and S is on QP, then SP = PQ implies that S is located such that SP = PQ.

But if QP is a segment, and S is between Q and P, then SP = PQ would mean that S is located such that from S to P is equal to the entire length of QP.

But that would mean that S is beyond P, which contradicts S being between Q and P.

Wait, this suggests a mistake in my earlier conclusion that SP = PQ.

Wait, in triangle SPQ, angles at Q and S are both 35°, so sides opposite are equal.

Side opposite angle at Q (35°) is SP.

Side opposite angle at S (35°) is PQ.

Therefore, SP = PQ.

But if S is on QP, then SP is a part of QP.

So SP = PQ implies that SP = PQ, but since S is between Q and P, SP is less than QP.

This is a contradiction. Therefore, my earlier conclusion must be wrong.

Wait, what's the problem here?

In triangle SPQ, angle at Q is 35°, angle at S is 35°, so sides opposite are equal.

Side opposite angle Q is SP.

Side opposite angle S is PQ.

Therefore, SP = PQ.

But if S is on QP, then SP is a segment from S to P, and PQ is the entire segment from P to Q.

So SP = PQ would mean that the length from S to P is equal to the length from P to Q.

Therefore, S must be located such that SP = PQ, which would place S outside of segment QP.

But according to the Asymptote code, S is on QP, between Q and P.

This contradiction suggests that my earlier assumption that angle QSP = 35° is incorrect.

Wait, let me re-examine that.

Earlier, I said that since ST || QR, and PQ is a transversal, then angle QSP = angle PQR = 35°.

But is that correct?

If ST || QR, and PQ is a transversal, then the corresponding angles would be equal.

But angle between PQ and ST is angle QSP, and angle between PQ and QR is angle PQR.

But are these corresponding angles?

Let me visualize: PQ is a line, ST is parallel to QR.

At point Q, the line QR is going to the right (since Q is at (0,0) and R is at (1,0)), and PQ is going up to P.

At point S, which is on PQ, the line ST is going to the right (since ST is parallel to QR).

Therefore, the angle between PQ and ST at S is equal to the angle between PQ and QR at Q, because they are corresponding angles.

Yes, that should be correct. Therefore, angle QSP = angle PQR = 35°.

But then in triangle SPQ, angles at Q and S are both 35°, leading to SP = PQ, which seems contradictory.

But according to the coordinates, S is on QP, and SP is shorter than PQ.

Wait, let me calculate lengths using coordinates.

In the Asymptote code, Q is (0,0), P is (1.1, 0.5), S is (0.66, 0.3).

Length PQ: distance from Q(0,0) to P(1.1, 0.5) is sqrt(1.1² + 0.5²) = sqrt(1.21 + 0.25) = sqrt(1.46) ≈ 1.208.

Length SP: distance from S(0.66, 0.3) to P(1.1, 0.5) is sqrt((1.1 - 0.66)² + (0.5 - 0.3)²) = sqrt(0.44² + 0.2²) = sqrt(0.1936 + 0.04) = sqrt(0.2336) ≈ 0.483.

So SP ≈ 0.483, PQ ≈ 1.208, so SP ≠ PQ.

Therefore, my earlier conclusion that SP = PQ is wrong.

Where is the mistake?

In triangle SPQ, if angles at Q and S are both 35°, then sides opposite should be equal.

Side opposite angle Q is SP, side opposite angle S is PQ.

But according to coordinates, SP ≠ PQ, so this suggests that angles at Q and S are not both 35°.

Therefore, my earlier assumption that angle QSP = 35° is incorrect.

Wait, why did I think that?

Because I thought that since ST || QR, and PQ is a transversal, then angle QSP = angle PQR.

But maybe it's not corresponding angles.

Let me think again about the transversal.

If ST is parallel to QR, and PQ is a transversal, then the angles formed at Q and S should be corresponding angles.

But angle at Q is between PQ and QR, and angle at S is between PQ and ST.

Since ST || QR, these should be corresponding angles, hence equal.

But according to coordinates, angle QSP is not 35°.

Let me calculate angle QSP using coordinates.

Point Q(0,0), S(0.66, 0.3), P(1.1, 0.5).

Angle at S between Q and P.

Wait, angle QSP is the angle at S between Q and P.

Wait, no. Angle QSP is the angle at S between Q and P? No, angle QSP is the angle at S between Q and P? Wait, no.

Wait, angle QSP is the angle at S between Q and P? No, it's the angle at S between Q and P? Wait, no.

Wait, angle QSP is the angle at S between Q and P? No, it's the angle at S between Q and P? Wait, no.

Actually, angle QSP is the angle at S between Q and P? No, it's the angle at S between Q and P? Wait, no.

Wait, angle QSP is the angle at S between Q and P? No, it's the angle at S between Q and P? Wait, no.

Actually, angle QSP is the angle at S between Q and P? No, it's the angle at S between Q and P? Wait, no.

Wait, angle QSP is the angle at S between Q and P? No, it's the angle at S between Q and P? No, it's the angle at S between Q and P? No, it's the angle at S between Q and P?

Wait, I think I'm confused. Let me clarify.

Angle QSP is the angle at point S, between points Q, S, P. So it's the angle at S between SQ and SP.

Similarly, angle PQR is the angle at Q between P, Q, R.

If ST || QR, and PQ is a transversal, then the corresponding angles would be angle QSP (at S) and angle PQR (at Q).

But according to coordinates, let's calculate angle QSP.

Vector SQ = Q - S = (0 - 0.66, 0 - 0.3) = (-0.66, -0.3)

Vector SP = P - S = (1.1 - 0.66, 0.5 - 0.3) = (0.44, 0.2)

Angle between vectors SQ and SP at point S.

Using the dot product:

cos(theta) = (SQ • SP) / (|SQ| |SP|)

SQ • SP = (-0.66)(0.44) + (-0.3)(0.2) = -0.2904 - 0.06 = -0.3504

|SQ| = sqrt((-0.66)^2 + (-0.3)^2) = sqrt(0.4356 + 0.09) = sqrt(0.5256) ≈ 0.725

|SP| = sqrt(0.44² + 0.2²) ≈ sqrt(0.1936 + 0.04) = sqrt(0.2336) ≈ 0.483

So cos(theta) ≈ -0.3504 / (0.725 * 0.483) ≈ -0.3504 / 0.349 ≈ -1.004

Wait, this is approximately -1, which would mean the angle is 180°, which is impossible.

Wait, this suggests a calculation error.

Wait, let me recalculate SQ • SP:

SQ is (-0.66, -0.3)

SP is (0.44, 0.2)

Dot product: (-0.66)(0.44) + (-0.3)(0.2) = -0.2904 - 0.06 = -0.3504. Correct.

|SQ|: sqrt(0.66² + 0.3²) = sqrt(0.4356 + 0.09) = sqrt(0.5256) ≈ 0.725. Correct.

|SP|: sqrt(0.44² + 0.2²) = sqrt(0.1936 + 0.04) = sqrt(0.2336) ≈ 0.483. Correct.

So cos(theta) ≈ -0.3504 / (0.725 * 0.483) ≈ -0.3504 / 0.349 ≈ -1.004.

This is approximately -1, which would mean the angle is 180°, which is impossible.

This suggests that vectors SQ and SP are pointing in nearly opposite directions, which would mean that points Q, S, P are colinear? But S is on QP, so they are colinear.

Wait, if S is on QP, then vectors SQ and SP are in opposite directions, hence the angle between them is 180°, which is correct.

But this contradicts the earlier assumption that angle QSP is 35°.

Wait, what's going on here?

Ah! I think I made a mistake in identifying the angles.

If ST is parallel to QR, and PQ is a transversal, then the corresponding angles would be angle between PQ and ST at S, and angle between PQ and QR at Q.

But angle between PQ and ST at S is not angle QSP, but angle between PQ and ST at S.

Since ST is parallel to QR, and PQ is a transversal, the corresponding angle at S would be the angle between PQ and ST, which is angle between PQ and ST at S.

But since S is on PQ, the line PQ passes through S, so the angle between PQ and ST at S is the angle between PQ (which is the line itself) and ST.

Since ST is parallel to QR, and PQ is a transversal, the corresponding angle would be the angle between PQ and QR at Q, which is angle PQR = 35°.

But since ST is parallel to QR, the angle between PQ and ST at S is equal to angle PQR = 35°.

But how is this angle measured?

At point S, the line ST is going in some direction, and PQ is passing through S. The angle between PQ and ST at S is 35°.

But since S is on PQ, the line PQ is straight, so the angle between PQ and ST at S is the angle between the line PQ (which is straight) and the line ST.

This angle is 35°, so if we consider the direction of ST, it makes 35° with PQ.

But since ST is parallel to QR, and QR is horizontal in the diagram, then the angle between PQ and ST is 35°, which would mean that ST is inclined at 35° from PQ.

But since PQ is going from Q(0,0) to P(1.1, 0.5), its slope is (0.5 - 0)/(1.1 - 0) = 0.5 / 1.1 ≈ 0.4545, so angle of PQ with x-axis is arctan(0.4545) ≈ 24.5°.

If ST is making 35° with PQ, then the angle of ST with x-axis is 24.5° + 35° = 59.5°, or 24.5° - 35° = -10.5°.

But in the diagram, ST is horizontal (since S and T have the same y-coordinate), so it's parallel to x-axis, which is 0°.

But if ST is horizontal, then the angle between PQ and ST is the angle between PQ (≈24.5°) and horizontal (0°), which is 24.5°, not 35°.

This contradicts the earlier assumption that it should be 35°.

Therefore, my earlier reasoning about corresponding angles was incorrect.

This suggests that my initial approach was wrong.

Let me try to find the correct relationship.

Since ST is parallel to QR, and QR is horizontal, then ST is horizontal.

Therefore, the line ST is horizontal.

In the diagram, PQ is a line from Q(0,0) to P(1.1, 0.5), which has a slope of 0.5 / 1.1 ≈ 0.4545, as before.

At point S, which is on PQ, the line ST is horizontal. Therefore, the angle between PQ and ST at S is the angle between PQ and the horizontal line ST.

This angle can be calculated as the angle of PQ with the x-axis, which is arctan(0.5 / 1.1) ≈ 24.5°, as before.

But the problem states that angle Q is 35°, which is the angle at Q between RQ and QS.

Since RQ is along the x-axis from Q(0,0) to R(1,0), and QS is from Q(0,0) to S(0.66, 0.3), the angle at Q is between RQ (along x-axis) and QS.

Let me calculate this angle.

Vector QR is (1, 0) - (0, 0) = (1, 0)

Vector QS is (0.66, 0.3) - (0, 0) = (0.66, 0.3)

The angle at Q is between QR and QS, which can be calculated using the dot product:

cos(theta) = (QR • QS) / (|QR| |QS|)

QR • QS = (1)(0.66) + (0)(0.3) = 0.66

|QR| = 1

|QS| = sqrt(0.66² + 0.3²) ≈ sqrt(0.4356 + 0.09) ≈ sqrt(0.5256) ≈ 0.725

So cos(theta) ≈ 0.66 / 0.725 ≈ 0.909

Therefore, theta ≈ arccos(0.909) ≈ 24.5°, which matches the angle of PQ with x-axis.

But the problem states that angle Q is 35°, but according to this calculation, it's approximately 24.5°.

This is a contradiction.

Wait, this suggests that the Asymptote code's coordinates do not match the problem's given angles.

The problem says angle Q is 35°, but in the Asymptote code's coordinates, it's approximately 24.5°.

This means that the diagram is just an example, and the actual problem has different proportions.

Therefore, I cannot rely on the coordinates to calculate the angle, as they are not to scale.

Therefore, I need to solve it using pure geometry, without relying on specific coordinates.

Let me start over, using the given angles and the parallel lines.

Given: ST || QR, angle at P is 40°, angle at Q is 35°. Find angle STR.

Let me draw triangle PQR, with ST parallel to QR, S on PQ, T on PR.

Given angle at P (between SP and PR) is 40°, angle at Q (between RQ and QS) is 35°.

Since ST || QR, let's use the properties of parallel lines and transversals.

Let me consider the transversal PQ intersecting ST and QR. Since ST || QR, the corresponding angles are equal.

The angle between PQ and ST at S is equal to the angle between PQ and QR at Q.

Let me denote this angle as x.

So, angle between PQ and ST at S is x, and angle between PQ and QR at Q is x.

But angle at Q is given as 35°, which is the angle between RQ and QS.

Wait, if the angle between PQ and QR at Q is x, and angle at Q is 35°, which is between RQ and QS, then x is part of that 35°?

Wait, this is getting too vague. Let me try to label the diagram with variables.

Let me denote:

At point Q, angle between RQ and QS is 35°.

Since ST || QR, and PQ is a transversal, the corresponding angle at S (between PQ and ST) is equal to the angle at Q (between RQ and QS).

Wait, if ST || QR, then the corresponding angle at S (between PQ and ST) is equal to the angle at Q (between PQ and QR).

Wait, but angle at Q between PQ and QR is part of the given angle at Q.

Wait, let me clarify.

At point Q, there are two lines: QR and QS. The angle between them is 35°.

If we consider PQ as a transversal, and ST || QR, then the corresponding angle at S (between PQ and ST) is equal to the angle at Q between PQ and QR.

Let me denote the angle between PQ and QR at Q as y. Then, the angle between PQ and ST at S is also y.

But the angle at Q between RQ and QS is 35°, which is the angle between QR and QS.

If I can relate y to this 35°, I can find y.

Let me consider triangle QRS? No, S is on PQ.

Wait, at point Q, the lines are QR, QS, and PQ.

The angle between QR and QS is 35°, and the angle between PQ and QR is y.

Therefore, the angle between PQ and QS is 35° - y? Or y - 35°? Depends on the configuration.

This is getting too confusing without a diagram.

Let me try to use the fact that ST || QR and apply the basic proportionality theorem.

Since ST || QR, and S is on PQ, T is on PR, then QS/QP = RT/PR.

Let me denote QS/QP = RT/PR = k.

Let me denote k as the ratio.

Let me also consider the angles.

At point P, angle between SP and PR is 40°.

At point Q, angle between RQ and QS is 35°.

Let me consider triangle PQS and triangle PRT.

Wait, not sure.

Alternatively, let me use the Law of Sines in triangle PQS and triangle PRT.

But I need more information.

Alternatively, let me consider the angles in triangle PQR.

Let me denote angle at P as 40°, but wait, no.

Wait, the given angle at P is between SP and PR, which is 40°.

Let me denote angle QPR as α, which is the angle at P between PQ and PR.

Then, the given angle at P (between SP and PR) is 40°, and since S is on PQ, the angle between SP and PQ is α - 40°?

Wait, if angle between SP and PR is 40°, and angle between PQ and PR is α, then the angle between SP and PQ is α - 40°.

But I need to relate this to other parts.

This is getting too abstract. Maybe I should use the fact that ST || QR and apply the Law of Sines in triangles involving ST and QR.

Let me consider triangle PQR and the line ST parallel to QR.

Since ST || QR, the triangles PST and PQR are similar? No, because S is on PQ and T is on PR, but ST is parallel to QR, so by the basic proportionality theorem, the triangles PST and PQR are similar?

Wait, if a line divides two sides of a triangle proportionally, then it is parallel to the third side. Conversely, if a line is parallel to the third side, it divides the other two sides proportionally.

So, since ST || QR, then triangles PST and PQR are similar? No, because ST is not necessarily connecting to the third side.

Wait, if ST is parallel to QR, then triangles PST and PQR are similar? No, because S is on PQ and T is on PR, so the line ST is cutting PQ and PR, and is parallel to QR, so triangles PST and PQR are similar.

Wait, yes! Because if a line parallel to one side of a triangle intersects the other two sides, then it creates a smaller similar triangle.

Wait, in this case, ST is parallel to QR, and S is on PQ, T is on PR, so triangle PST is similar to triangle PQR.

Wait, but ST is parallel to QR, so triangle PST ~ triangle PQR by AA similarity.

Because angle at P is common, and angle PST = angle PQR (since ST || QR).

Yes!

Therefore, triangle PST ~ triangle PQR.

Therefore, corresponding angles are equal.

So, angle PST = angle PQR = 35°, and angle PTS = angle PRQ.

Wait, let me confirm.

In triangle PST and PQR:

- Angle at P is common.

- Angle PST = angle PQR (since ST || QR, corresponding angles).

Therefore, triangles PST ~ PQR by AA similarity.

Therefore, the ratio of sides is equal.

Let me denote the ratio of similarity as k = PS / PQ = PT / PR = ST / QR.

Now, let me use this to find some relations.

Let me denote angle at P as 40°, which is angle between SP and PR.

Wait, in triangle PQR, angle at P is angle QPR, which is the angle between PQ and PR.

In triangle PST, angle at P is the same, angle QPR.

But in triangle PST, angle at P is also part of the triangle.

Wait, but we know angle at P in triangle PST is the same as in triangle PQR.

But we also know that in triangle PST, angle PST = 35°, and angle PTS = angle PRQ.

Let me denote angle PRQ as β.

In triangle PQR, angles are:

- At Q: 35°

- At P: let's call it γ

- At R: β

So, γ + 35° + β = 180° => γ + β = 145°

In triangle PST, angles are:

- At P: γ

- At S: 35°

- At T: β

Therefore, γ + 35° + β = 180°, which is the same as above.

But this doesn't give new information.

However, we know that in triangle PST, angle at P is γ, and in the problem, the angle at P (between SP and PR) is 40°.

Wait, the angle at P in triangle PST is the same as angle QPR in triangle PQR, which is γ.

But the problem states that the angle at P (between SP and PR) is 40°, which is the same as angle QPR, which is γ.

Therefore, γ = 40°.

Therefore, in triangle PQR, γ = 40°, angle at Q is 35°, so angle at R is β = 180° - 40° - 35° = 105°.

Wait, this is different from earlier, but now it makes sense.

So, in triangle PQR, angles are:

- At P: 40°

- At Q: 35°

- At R: 105°

Now, since triangles PST ~ PQR, the ratio of sides is k = PS / PQ = PT / PR = ST / QR.

Let me denote this ratio as k.

Now, we need to find angle STR.

Let me consider triangle STR.

Point T is on PR, and we need angle at T between S and R.

Let me consider triangle STR. To find angle STR, I can use the Law of Sines or Law of Cosines if I know some sides or other angles.

Alternatively, since we know some angles in triangle PQR and the similarity, maybe we can find relations.

Let me consider triangle STR.

At point T, we have angle STR, which is what we need.

Let me consider triangle STR and triangle TPR or something.

Alternatively, since ST || QR, and we know angle at R is 105°, maybe angle STR is related.

Wait, let me think about the angles at point T.

In triangle PQR, angle at R is 105°, which is angle PRQ.

Since ST || QR, the angle between PR and ST at T is equal to angle PRQ = 105°.

Wait, this is the corresponding angle.

So, at point T, the angle between PR and ST is 105°, which is angle PTR.

But angle PTR is part of triangle PTR.

Now, in triangle STR, we have angle at T: angle STR, and angle at R: angle SRT, and angle at S: angle TSR.

But how to relate these?

Wait, if I can find angle SRT, then I can find angle STR.

Angle SRT is the angle at R between SR and RT.

But RT is part of PR, and SR is a line from R to S.

In triangle PQR, angle at R is 105°, which is between PR and QR.

Since ST || QR, and S is on PQ, the line SR is a transversal.

Maybe we can find angle SRT using some relations.

Alternatively, since triangles PST ~ PQR, the ratio of sides is k = PS / PQ.

Let me denote PQ = a, PR = b, QR = c.

Then, PS = k * PQ = k * a, PT = k * PR = k * b, and ST = k * QR = k * c.

But without knowing the actual lengths, it's hard to proceed.

However, we can use the Law of Sines in triangle PQR.

In triangle PQR:

- angle at P: 40°

- angle at Q: 35°

- angle at R: 105°

By the Law of Sines:

PQ / sin(105°) = PR / sin(35°) = QR / sin(40°)

Let me denote PQ = a, PR = b, QR = c.

So:

a / sin(105°) = b / sin(35°) = c / sin(40°)

Let me compute the ratios.

Let me choose a value for one side to make calculations easier. Let me set PQ = a = 1.

Then, PR = b = sin(35°) / sin(105°) * a ≈ (0.5736) / (0.9659) ≈ 0.5937

QR = c = sin(40°) / sin(105°) * a ≈ (0.6428) / (0.9659) ≈ 0.665

Now, since triangles PST ~ PQR with ratio k, then PS = k * PQ = k * 1 = k, PT = k * PR = k * 0.5937, and ST = k * QR = k * 0.665.

But also, since S is on PQ, PS + SQ = PQ = 1, so SQ = 1 - k.

Similarly, PT + TR = PR = 0.5937, so TR = 0.5937 - k * 0.5937 = 0.5937(1 - k).

Now, let me consider triangle STR. To find angle STR, I need to know some sides or other angles.

Let me try to use the Law of Sines in triangle STR.

But I need to know some sides.

Alternatively, let me consider triangle STR and triangle TPR.

Wait, at point T, we have angle PTR = 105° (from earlier), and angle STR is what we need.

If I can find the other angles in triangle STR, I can find angle STR.

Let me consider triangle STR.

At point S, angle is angle TSR.

At point R, angle is angle SRT.

At point T, angle is angle STR.

If I can find angle TSR or angle SRT, I can find angle STR.

Let me try to find angle SRT.

Angle SRT is the angle at R between SR and RT.

In triangle PQR, angle at R is 105°, which is between PR and QR.

Since ST || QR, and SR is a line from R to S, which is on PQ.

Let me consider the line SR.

In triangle PQR, we can find the coordinates of S if we know the ratio k.

But since I set PQ = 1, and PS = k, then S divides PQ in the ratio k : (1 - k).

But without knowing k, it's hard to proceed.

Wait, but from the similarity of triangles PST and PQR, the ratio k is the same for all sides.

But how to find k?

Wait, we don't have enough information to find k.

Wait, but maybe we can relate it to the given angles.

Wait, in triangle PST, we know angle at S is 35°, angle at P is 40°, so angle at T is 180 - 35 - 40 = 105°.

Wait, in triangle PST, angles are:

- At P: 40°

- At S: 35°

- At T: 105°

Therefore, angle at T in triangle PST is 105°, which is angle PTS.

But angle PTS is the same as angle PTR, which we previously said is 105°, which matches.

Now, in triangle STR, we need angle STR.

Let me consider triangle STR and triangle PTS.

Wait, angle at T in triangle PTS is 105°, which is angle PTR.

If I can find some relationship between triangles STR and PTS, maybe.

Alternatively, let me consider the straight line at point T.

At point T, the angles around it are:

- angle STR (which we need)

- angle PTR (105°)

- angle STP (which is part of triangle PST)

But since ST is a straight line, the sum of angles around T should be 360°, but I need to consider the specific angles.

Actually, at point T, there are two angles: angle STR (between S and R) and angle PTR (between P and R), and angle STP (between S and P).

These three angles should add up to 180°, since they are on a straight line? No, because T is a point with multiple lines.

Actually, at point T, the lines are ST, TR, and TP.

So the angles at T are:

- Between ST and TR: angle STR (what we need)

- Between TR and TP: angle PTR = 105°

- Between TP and ST: angle STP

These three angles should add up to 360°, but since they are around point T, but if we consider the straight lines, it's more complex.

Actually, if you look at the lines at point T: ST, TR, and TP.

If you consider the angles between ST and TR (angle STR), between TR and TP (angle PTR = 105°), and between TP and ST (angle STP), then these three angles should add up to 360°.

But in reality, since ST and TP are two lines from T, and TR is another line, it's a bit complex.

Alternatively, if we consider the straight line from T to P and T to R, and T to S, it's better to use triangle relationships.

Let me consider triangle STR and triangle STP.

In triangle STP, we know all angles: 40°, 35°, 105°.

In triangle STR, we need angle STR.

If I can find the lengths of sides in these triangles, I can use the Law of Sines or Cosines.

Let me assume some lengths.

Let me set PQ = 1, as before.

From earlier, PR ≈ 0.5937, QR ≈ 0.665.

From the similarity, k = PS / PQ = PS / 1 = PS.

But we need to find k.

Wait, but how?

Wait, in triangle PST, we know the angles, and if we can relate it to the sides, maybe we can find k.

Let me use the Law of Sines in triangle PST.

In triangle PST:

- angle at P: 40°

- angle at S: 35°

- angle at T: 105°

Let me denote PS = k, PT = k * PR = k * 0.5937, and ST = k * QR = k * 0.665.

By the Law of Sines in triangle PST:

PS / sin(angle at T) = PT / sin(angle at S) = ST / sin(angle at P)

So:

k / sin(105°) = (k * 0.5937) / sin(35°) = (k * 0.665) / sin(40°)

Let me check the first two ratios:

k / sin(105°) = (k * 0.5937) / sin(35°)

Cancel k:

1 / sin(105°) = 0.5937 / sin(35°)

Calculate sin(105°) ≈ sin(60° + 45°) = sin60cos45 + cos60sin45 ≈ (0.8660 * 0.7071) + (0.5 * 0.7071) ≈ 0.6124 + 0.3536 ≈ 0.9660

sin(35°) ≈ 0.5736

So 0.5937 / 0.5736 ≈ 1.035

1 / 0.9660 ≈ 1.035

Yes, this checks out.

Similarly, check the first and third ratios:

k / sin(105°) = (k * 0.665) / sin(40°)

Cancel k:

1 / sin(105°) = 0.665 / sin(40°)

Calculate sin(40°) ≈ 0.6428

0.665 / 0.6428 ≈ 1.035

1 / 0.9660 ≈ 1.035

Also checks out.

Therefore, the ratio is consistent.

Now, to find angle STR, let's consider triangle STR.

Let me use the Law of Sines in triangle STR.

But I need to know some sides or angles.

Let me consider triangle STR and triangle PTR.

Wait, in triangle PTR, we know angle at T is 105°, and we can find other angles if we know sides.

Alternatively, let me consider triangle STR and use the Law of Sines.

Let me denote:

In triangle STR:

- angle at T: angle STR = x (what we need)

- angle at R: angle SRT = y

- angle at S: angle TSR = z

We know that x + y + z = 180°

If I can find y or z, I can find x.

Let me try to find angle at R, y = angle SRT.

Angle SRT is the angle at R between SR and RT.

In triangle PQR, angle at R is 105°, which is between PR and QR.

Since ST || QR, and SR is a line from R to S, which is on PQ, we can find angle SRT.

Let me consider triangle SRT and triangle QRT?

Wait, not sure.

Alternatively, let me use the Law of Sines in triangle SRT and triangle SQR.

Wait, this is getting too complex.

Let me try to use coordinates with the given angles.

Let me set up a coordinate system where Q is at (0, 0), R is at (1, 0), and P is somewhere in the plane.

Given angle at Q is 35°, angle at P is 40°, and ST || QR.

Let me try to find coordinates of P.

Let me assume QR is from (0, 0) to (1, 0).

Let me let Q be at (0, 0), R at (1, 0).

Let me let P be at some point (x, y).

Given angle at Q is 35°, which is the angle between RQ (from Q to R, along x-axis) and QS (from Q to S).

But S is on PQ.

Also, angle at P is 40°, which is the angle between SP and PR.

This is complex, but let me try.

Let me assume that S divides PQ in some ratio, say k:1-k, and T divides PR in the same ratio, since ST || QR.

But without knowing k, it's difficult.

Alternatively, let me use the Law of Sines in triangle PQR.

In triangle PQR, angles are 40°, 35°, 105°.

Let me set QR = 1.

Then, by the Law of Sines:

PQ / sin(105°) = PR / sin(35°) = QR / sin(40°) = 1 / sin(40°)

So:

PQ = sin(105°) / sin(40°) ≈ 0.9659 / 0.6428 ≈ 1.503

PR = sin(35°) / sin(40°) ≈ 0.5736 / 0.6428 ≈ 0.892

Now, since ST || QR, and S is on PQ, T is on PR, with PS / PQ = PT / PR = k.

Let me choose k = 0.6, as in the Asymptote code, but this is just an example.

But since the actual value of k is not given, I need to find it.

Wait, but how?

Wait, in the problem, there is no additional information, so the angle STR should be determined solely by the given angles and the parallel lines.

This suggests that the angle STR is constant, regardless of the position of S and T, as long as ST || QR and the given angles at P and Q.

This is possible, and it might be a fixed value.

Let me try to find it using the Law of Sines in triangle STR.

Let me consider the following:

In triangle PQR, with angles 40°, 35°, 105°, and ST || QR.

Let me consider the line ST, which is parallel to QR, and let me find the angles in triangle STR.

At point T, angle PTR = 105° (as established earlier).

In triangle STR, angle at T is angle STR, and angle at R is angle SRT.

Let me consider the line SR.

In triangle PQR, let me find the angle at S.

Wait, in triangle PQR, we can find the lengths and then use the Law of Sines to find other angles.

But this is getting too involved.

Wait, let me consider the following:

Since ST || QR, the angle between ST and PR is equal to the angle between QR and PR, which is 105°.

This is angle PTR = 105°.

Now, in triangle STR, we have angle at T = 105°, and we need to find angle STR.

Wait, no, angle at T in triangle STR is not 105°, it's angle STR.

Wait, I think I'm getting confused again.

Let me try to use the fact that in triangle PST, we have angles 40°, 35°, 105°, and in triangle STR, we need to find angle STR.

If I can find the relationship between these triangles.

Let me consider that in triangle PST, angle at T is 105°, and in triangle STR, angle at T is what we need.

If I can find the relationship between these two angles.

At point T, the angle between ST and PR is 105° (angle PTR), and the angle between ST and TR is angle STR.

If I can find the angle between PR and TR, which is zero, since T is on PR.

Wait, no.

At point T, the line TR is part of PR, and ST is a line from T to S.

So, the angle between ST and PR at T is 105°, and the angle between ST and TR is the same as the angle between ST and PR, which is 105°.

Wait, but TR is part of PR, so the angle between ST and TR is the same as the angle between ST and PR, which is 105°.

But this would mean that angle STR is 105°, but this can't be, because in the coordinate example, it was approximately 78.7°.

This suggests a mistake in reasoning.

Wait, if at point T, the angle between ST and PR is 105°, and TR is part of PR, then the angle between ST and TR is 105°, which would mean angle STR = 105°.

But this contradicts the coordinate calculation.

However, in the coordinate example, with the given angles, the calculation gave approximately 78.7°, which is different from 105°.

This suggests that my assumption is wrong.

Let me re-examine.

If ST || QR, and PR is a transversal, then the corresponding angle at T (between ST and PR) is equal to the corresponding angle at R (between QR and PR).

But the angle at R between QR and PR is angle PRQ, which is 105°.

Therefore, the angle between ST and PR at T is 105°, which is angle PTR.

But angle PTR is at T between P and R, while angle STR is at T between S and R.

These are different angles.

At point T, the lines are ST, TR, and TP.

The angle between ST and TR is angle STR, and the angle between TR and TP is angle PTR = 105°.

The angle between ST and TP is angle STP.

These three angles at T add up to 180°, because they are on a straight line? No, because TP and TR are on the same line PR, so the angles at T are:

- Between ST and TR: angle STR

- Between TR and TP: angle PTR = 105°

- Between TP and ST: angle STP

But since TP and TR are on the same line, the angle between TP and TR is 180°, but angle PTR is part of it.

Actually, at point T, the line PR passes through T, so the angles on one side of PR are angle STR and angle STP, and on the other side is angle PTR.

Wait, this is getting too confusing.

Let me use the fact that in triangle PST, angle at T is 105°, and in triangle STR, angle at T is what we need.

If I can find the relationship between these two angles.

At point T, the line ST is common to both triangles PST and STR.

The angle at T in triangle PST is 105°, which is between ST and PT.

The angle at T in triangle STR is between ST and RT.

Since PT and RT are on the same line PR, the angle between ST and PT (105°) and the angle between ST and RT (angle STR) are supplementary.

Wait, if PT and RT are on the same line, then the angle between ST and PT (105°) and the angle between ST and RT (angle STR) add up to 180°.

Yes! Because PT and RT are in opposite directions along PR.

Therefore, angle between ST and PT (105°) + angle between ST and RT (angle STR) = 180°.

Therefore, angle STR = 180° - 105° = 75°.

Ah! This makes sense!

So, angle STR is 75°.

This matches with the earlier calculation in the coordinate example, where it was approximately 78.7°, but that was due to the specific coordinates not matching the given angles.

But with the correct angles, it should be 75°.

Let me verify this logic.

At point T, the line PR passes through T, so the angles on either side of PR are supplementary.

The angle between ST and PT is 105° (from triangle PST), and the angle between ST and RT is angle STR.

Since PT and RT are in opposite directions, the sum of these two angles is 180°.

Therefore, angle STR = 180° - 105° = 75°.

Yes, this makes sense.

Therefore, the measure of angle STR is 75 degrees.

**Final Answer**
\boxed{75}

To determine the measure of angle $ \angle STR $, we are given the following:

- $ \overline{ST} \parallel \overline{QR} $
- $ \angle P = 40^\circ $
- $ \angle Q = 35^\circ $

We are to find $ \angle STR $, the angle at point $ T $ between segments $ \overline{ST} $ and $ \overline{TR} $.

---

### Step 1: Understand the Geometry

We are given that $ \overline{ST} \parallel \overline{QR} $, and that points $ S $ and $ T $ lie on segments $ \overline{PQ} $ and $ \overline{PR} $, respectively. This setup implies that triangle $ \triangle PST $ is similar to triangle $ \triangle PQR $ by the **Basic Proportionality Theorem** (also known as Thales' Theorem), since a line parallel to one side of a triangle intersects the other two sides proportionally.

---

### Step 2: Use Triangle Angle Sum to Find $ \angle R $

In triangle $ \triangle PQR $, we are given:

- $ \angle Q = 35^\circ $
- $ \angle P = 40^\circ $

Using the triangle angle sum property:

$$
\angle R = 180^\circ - \angle P - \angle Q = 180^\circ - 40^\circ - 35^\circ = 105^\circ
$$

So, $ \angle PRQ = 105^\circ $.

---

### Step 3: Use Parallel Lines to Find $ \angle PTR $

Since $ \overline{ST} \parallel \overline{QR} $, and $ \overline{PR} $ is a transversal, the corresponding angles are equal. Therefore:

$$
\angle PTR = \angle PRQ = 105^\circ
$$

This is the angle at point $ T $ between segments $ \overline{PT} $ and $ \overline{TR} $.

---

### Step 4: Use Straight Line Angle Sum at Point $ T $

At point $ T $, the lines $ \overline{ST} $, $ \overline{TR} $, and $ \overline{TP} $ meet. Since $ \overline{PT} $ and $ \overline{TR} $ are collinear (they lie on the same line $ \overline{PR} $), the angles formed at point $ T $ by $ \overline{ST} $ with $ \overline{PT} $ and $ \overline{TR} $ are supplementary.

That is:

$$
\angle STR + \angle PTR = 180^\circ
$$

Substitute the known value of $ \angle PTR $:

$$
\angle STR = 180^\circ - 105^\circ = 75^\circ
$$

---

### Final Answer

$$
\boxed{75}
$$

[Assistant]
Explanation:
Okay, so I need to find the measure of angle STR in this diagram. Let me try to visualize it first. There's a diagram with points P, Q, R, S, and T. The Asymptote code shows that Q is at (0,0), R is at (1,0), P is at (1.1, 0.5), S is 0.6*P, and T is R + 0.6*(P - R). Then they draw lines connecting T to S, S to P, P to R, R to Q, and Q to S.

First, let me sketch this mentally. Points Q and R are on the x-axis, with Q at the origin and R at (1,0). Point P is above them, a bit to the right of R. Then S is 0.6 times P, so it's somewhere between the origin and P? Wait, no. If S is 0.6*P, then since P is at (1.1, 0.5), multiplying by 0.6 would give S at (0.66, 0.3). But then T is R + 0.6*(P - R). Let me compute that. P - R is (1.1 - 1, 0.5 - 0) = (0.1, 0.5). Then 0.6*(0.1, 0.5) is (0.06, 0.3). Adding that to R (1,0) gives T at (1.06, 0.3). So S is at (0.66, 0.3) and T is at (1.06, 0.3). So S and T are both on the line y=0.3? Wait, but then ST is horizontal? Because both have the same y-coordinate. And QR is from (0,0) to (1,0), which is also horizontal. So ST is parallel to QR, as given.

So the diagram has ST parallel to QR. Also, angles at P and Q are given: angle P is 40 degrees, angle Q is 35 degrees. Need to find angle STR.

Let me try to label the diagram. Points Q and R are on the base, P is above, S is somewhere between Q and P? Wait, S is 0.6*P, so if P is at (1.1, 0.5), then S is 0.6 of the way from the origin to P? Wait, no. If S is 0.6*P, then it's scaling the coordinates of P by 0.6. So if P is (1.1, 0.5), then S is (0.66, 0.3). But Q is at (0,0), so S is somewhere between Q and P? Maybe. Similarly, T is R + 0.6*(P - R). So starting at R, moving 0.6 of the way towards P. Since R is (1,0) and P is (1.1, 0.5), then T is (1 + 0.6*(0.1), 0 + 0.6*(0.5)) = (1.06, 0.3). So T is above R, but not as high as P.

So the lines drawn are T to S, S to P, P to R, R to Q, and Q to S. So triangle QSP? And triangle PTR? Maybe. Also, ST is parallel to QR.

Given that ST is parallel to QR, which are both horizontal in this coordinate system. So maybe we can use some properties of parallel lines and transversals. Also, angles at P and Q are given. Let me think.

First, angle P is 40 degrees. In triangle PQR? Wait, but where is point S and T? Maybe triangle PQS or something else? Wait, the diagram is a bit complex. Let me try to figure out which triangles we have.

From the Asymptote code, the figure is drawn as T--SS--P--R--Q--SS. So connecting T to S, S to P, P to R, R to Q, Q to S. So the figure includes triangle QSP, triangle PRT? Maybe quadrilateral QSPR? Not sure.

But since ST is parallel to QR, maybe we can use similar triangles or something. Also, angles at P and Q are given. Let me think about triangle PQR. Wait, but is P connected to Q? In the diagram, P is connected to R and S, and Q is connected to S and R. So maybe triangle PQR is not directly drawn, but maybe we can consider it.

Alternatively, maybe triangle PQS and triangle PRT? Wait, not sure. Let me try to find some relationships.

Since ST is parallel to QR, then maybe there are some corresponding angles or alternate interior angles. Let me think. If I consider ST parallel to QR, and maybe some transversal cutting them. For example, if we consider line SP as a transversal, then angle at S and angle at Q? Wait, not sure.

Alternatively, maybe triangle SPQ and triangle...? Wait, maybe triangle SPT? Not sure.

Alternatively, since we have angles at P and Q given, maybe we can find some other angles in the diagram.

Let me note that angle P is 40 degrees. In the diagram, angle at P is between which lines? Since P is connected to S and R. So angle at P is between SP and PR. Similarly, angle Q is 35 degrees, which is at Q, between RQ and QS.

So maybe we can find some angles in triangle PQS or triangle PQR?

Wait, if I can find some angles in triangle PQR, maybe. Let me try.

But first, let me check if triangle PQR is a triangle. Points P, Q, R. Q is connected to R, R is connected to P? Yes, in the diagram, P is connected to R. So triangle PQR exists.

In triangle PQR, we know angle at Q is 35 degrees, angle at P is 40 degrees? Wait, but angle at P in triangle PQR would be angle QPR? Wait, but in the diagram, angle P is given as 40 degrees. Is that angle QPR?

Wait, the problem says "angle P = 40°", so that's angle at point P. In triangle PQR, angle at P is between sides PQ and PR. But in the diagram, point S is connected to P, so maybe angle P is between SP and PR? Wait, the problem says "angle P = 40°", so maybe that's angle SPQ? Wait, no. The Asymptote code draws P connected to S and R. So angle at P is between SP and PR. So angle SP R? Wait, but the problem says angle P is 40°, so that's angle at P between SP and PR.

Similarly, angle Q is 35°, which is at Q between RQ and QS.

So maybe we can find some angles in triangle PQS or triangle PQR?

Alternatively, since ST is parallel to QR, then maybe triangle SPT is similar to triangle QRT? Or something like that?

Alternatively, since ST is parallel to QR, then the corresponding angles would be equal. For example, if we consider line SP as a transversal cutting ST and QR, then angle at S and angle at Q would be equal? Wait, but ST is parallel to QR, so if SP is a transversal, then angle between SP and ST would be equal to angle between SP and QR?

Wait, maybe I need to find some angles. Let me try to find angle at S or T.

Alternatively, maybe using the fact that ST is parallel to QR, so the triangles involved might be similar.

Alternatively, since we have coordinates given in the Asymptote code, maybe I can compute coordinates and then compute the angle? But since the problem is general (not specific to the coordinates given in Asymptote code, since it's just an example), but maybe the Asymptote code is just for illustration. Wait, but the problem says "in the diagram below", so maybe the diagram is specific. However, since the Asymptote code is given, maybe I can use coordinates to calculate?

But maybe the problem is intended to be solved with pure geometry, not coordinates. Let me try that first.

So, given that ST is parallel to QR. Let me consider triangle PQR. If I can find some angles in triangle PQR, then maybe I can find other angles.

But angle at P is 40°, angle at Q is 35°, so maybe angle at R can be found?

Wait, in triangle PQR, angles at P, Q, R. Wait, but is triangle PQR a triangle? If points P, Q, R are connected, then yes. But angle at P is 40°, angle at Q is 35°, so angle at R would be 180 - 40 - 35 = 105°.

Wait, but is that correct? Wait, if triangle PQR has angles at P, Q, R, then yes. But is angle at P in triangle PQR the same as angle P given in the problem?

Wait, the problem says "angle P = 40°", which is angle at P. If in triangle PQR, angle at P is 40°, angle at Q is 35°, then angle at R is 105°, as above.

But then, how does ST come into play? Since ST is parallel to QR.

So maybe triangle STR? Wait, we need angle STR. So point S, T, R.

Alternatively, maybe triangle STR is related to triangle Q... something.

Alternatively, since ST is parallel to QR, then maybe triangle SPT is similar to triangle QPR?

Wait, let me think. If ST is parallel to QR, then the corresponding angles would be equal. For example, angle at S would correspond to angle at Q, and angle at T would correspond to angle at R?

But I need to figure out which triangles are similar.

Alternatively, since ST is parallel to QR, then the line SP is cutting the sides of triangle PQR? Wait, if ST is parallel to QR, then maybe S is on PQ and T is on PR? But in the Asymptote code, S is 0.6*P, which is not on PQ. Wait, maybe not.

Wait, in the Asymptote code, S is defined as 0.6*P, so if P is (1.1, 0.5), then S is (0.66, 0.3). But Q is (0,0), R is (1,0). So S is not on PQ or PR. Wait, maybe S is on PS? Wait, no.

Alternatively, maybe S is on PQ? Let me check coordinates. If S is (0.66, 0.3), and Q is (0,0), P is (1.1, 0.5). So the line from Q to P is from (0,0) to (1.1, 0.5). Let me see if S lies on that line.

Parametric equations: from Q(0,0) to P(1.1, 0.5). So any point on that line can be written as (1.1t, 0.5t) where t is between 0 and 1.

If S is (0.66, 0.3), then 1.1t = 0.66 => t = 0.66 / 1.1 = 0.6. Similarly, 0.5t = 0.3 => t = 0.6. So yes! S is on QP, at t=0.6. Similarly, T is defined as R + 0.6*(P - R). So R is (1,0), P - R is (0.1, 0.5). So 0.6*(0.1, 0.5) is (0.06, 0.3). Adding to R gives T as (1.06, 0.3). Let me check if T is on PR.

PR is from P(1.1, 0.5) to R(1, 0). Let me parametrize PR. Let me write parametric equations. Let parameter be s from 0 to 1.

x = 1.1 - 0.1s

y = 0.5 - 0.5s

So when s=0, we are at P(1.1, 0.5); when s=1, we are at R(1, 0).

Now, T is at (1.06, 0.3). Let me see if there's an s such that:

1.1 - 0.1s = 1.06 => 0.1s = 1.1 - 1.06 = 0.04 => s = 0.4

Similarly, y-coordinate: 0.5 - 0.5s = 0.3 => 0.5s = 0.2 => s = 0.4.

Yes! So T is on PR at s=0.4.

Therefore, S is on QP at t=0.6, and T is on PR at s=0.4.

Therefore, ST is a line connecting a point on QP to a point on PR, and it's parallel to QR.

So given that ST is parallel to QR, and S divides QP in ratio t=0.6, and T divides PR in ratio s=0.4.

But since ST is parallel to QR, by the converse of the basic proportionality theorem (Thales' theorem), if a line divides two sides of a triangle proportionally, then it is parallel to the third side.

But here, ST is parallel to QR, so it should divide the sides QP and PR proportionally.

But in this case, QS / QP = 0.6, and RT / PR = ?

Wait, let me check.

QS is the segment from Q to S. Since S divides QP into QS and SP, with QS / QP = 0.6. Similarly, RT is the segment from R to T. Since T divides PR into PT and TR. Since T is at s=0.4 from P to R, then PT / PR = 0.4, so TR / PR = 1 - 0.4 = 0.6.

Therefore, QS / QP = 0.6 and TR / PR = 0.6. Therefore, QS / QP = TR / PR, which means that ST is parallel to QR by the converse of Thales' theorem.

So that's consistent with the diagram.

Therefore, in triangle PQR, ST is a line segment connecting points S on QP and T on PR such that QS / QP = TR / PR = 0.6.

Therefore, maybe we can use this to find some relations.

But how does this help us find angle STR?

Angle STR is the angle at point T between points S, T, R. So angle at T between S and R.

To find angle STR, maybe we can consider triangle STR. If we can find some angles or sides in triangle STR, we can compute it.

Alternatively, since we know some angles in triangle PQR, maybe we can find angles in triangle STR.

First, let me recap. In triangle PQR, angles at P, Q, R are 40°, 35°, and 105°, as calculated earlier.

But wait, is that correct? Wait, angle at P is 40°, angle at Q is 35°, so angle at R is 180 - 40 - 35 = 105°, yes.

But how does that relate to triangle STR?

Alternatively, maybe we can find coordinates of all points and compute the angle. Since the Asymptote code gives coordinates, even though it's just an example, maybe we can use those coordinates to calculate the angle.

Let me try that approach.

Given coordinates:

Q = (0, 0)

R = (1, 0)

P = (1.1, 0.5)

S = 0.6 * P = (0.66, 0.3)

T = R + 0.6*(P - R) = (1, 0) + 0.6*(0.1, 0.5) = (1 + 0.06, 0 + 0.3) = (1.06, 0.3)

So coordinates:

S = (0.66, 0.3)

T = (1.06, 0.3)

R = (1, 0)

So angle STR is the angle at T between points S, T, R.

So to find angle at T between S and R.

To compute this angle, we can use vectors or coordinate geometry.

First, find vectors TS and TR, then compute the angle between them.

Vector TS = S - T = (0.66 - 1.06, 0.3 - 0.3) = (-0.4, 0)

Vector TR = R - T = (1 - 1.06, 0 - 0.3) = (-0.06, -0.3)

Then, the angle between vectors TS and TR can be found using the dot product formula:

cos(theta) = (TS • TR) / (|TS| |TR|)

Compute TS • TR = (-0.4)(-0.06) + (0)(-0.3) = 0.024 + 0 = 0.024

|TS| = sqrt((-0.4)^2 + 0^2) = sqrt(0.16) = 0.4

|TR| = sqrt((-0.06)^2 + (-0.3)^2) = sqrt(0.0036 + 0.09) = sqrt(0.0936) ≈ 0.306

Therefore, cos(theta) ≈ 0.024 / (0.4 * 0.306) ≈ 0.024 / 0.1224 ≈ 0.196

Then theta ≈ arccos(0.196) ≈ 78.7 degrees.

Wait, but the problem is likely expecting an exact value, not an approximate decimal. So maybe my approach using coordinates is not the right way, since the coordinates are specific to the Asymptote code, which might not be to scale or might be just an example.

Alternatively, maybe there's a geometric relationship that gives an exact value.

Alternatively, maybe my calculation is correct, and the answer is approximately 78.7°, but since angles in the problem are given as whole numbers (40°, 35°), maybe the exact value is 75° or something else.

Wait, but let me check my calculation again.

First, coordinates:

S = (0.66, 0.3)

T = (1.06, 0.3)

R = (1, 0)

So vector TS is S - T = (0.66 - 1.06, 0.3 - 0.3) = (-0.4, 0). Correct.

Vector TR is R - T = (1 - 1.06, 0 - 0.3) = (-0.06, -0.3). Correct.

Dot product: (-0.4)(-0.06) + (0)(-0.3) = 0.024 + 0 = 0.024. Correct.

|TS| = sqrt( (-0.4)^2 + 0^2 ) = 0.4. Correct.

|TR| = sqrt( (-0.06)^2 + (-0.3)^2 ) = sqrt(0.0036 + 0.09) = sqrt(0.0936). Let me compute sqrt(0.0936).

Note that 0.3^2 = 0.09, 0.306^2 = 0.0936. Because 0.3^2 = 0.09, 0.306^2 = (0.3 + 0.006)^2 = 0.09 + 2*0.3*0.006 + 0.006^2 = 0.09 + 0.0036 + 0.000036 = 0.093636. So sqrt(0.0936) ≈ 0.306. So yes, |TR| ≈ 0.306.

Therefore, cos(theta) ≈ 0.024 / (0.4 * 0.306) ≈ 0.024 / 0.1224 ≈ 0.196.

Now, arccos(0.196). Let me check what angle has cosine ≈ 0.196.

cos(80°) ≈ 0.1736

cos(75°) ≈ 0.2588

So 0.196 is between 75° and 80°, closer to 80°. Let me compute it more accurately.

Let me use calculator-like approach. Let me recall that cos(78°) ≈ 0.2079

cos(79°) ≈ 0.1908

So 0.196 is between 78° and 79°, closer to 79°.

Let me compute the difference:

At 78°, cos ≈ 0.2079

At 79°, cos ≈ 0.1908

Difference between 78° and 79° is 1°, and the difference in cos values is 0.2079 - 0.1908 = 0.0171

Our value is 0.196, which is 0.2079 - 0.196 = 0.0119 less than cos(78°).

So fraction = 0.0119 / 0.0171 ≈ 0.696. So approximately 78° + 0.696° ≈ 78.696°, so approximately 78.7°, as before.

But the problem is likely expecting an exact value. Therefore, maybe my coordinate approach is not the right way, or maybe there's an exact value here.

Alternatively, maybe I made a wrong assumption in considering triangle PQR. Let me think again.

Alternatively, maybe using the Law of Sines or Law of Cosines in some triangle.

Let me try to find some relations.

Since ST is parallel to QR, then angle between ST and SP is equal to angle between QR and QP? Wait, maybe.

Alternatively, since ST || QR, then angle SPT is equal to angle PQR?

Wait, let me think. If ST is parallel to QR, then the transversal PQ would create corresponding angles.

Wait, if we consider line PQ intersecting ST and QR. But S is on PQ, and T is on PR. Maybe not straightforward.

Alternatively, since ST || QR, then angle between SP and ST is equal to angle between SP and QR? Not sure.

Alternatively, since ST || QR, then triangle STR is similar to triangle Q... something?

Alternatively, since S divides QP in ratio 0.6 and T divides PR in ratio 0.4, as we saw earlier.

Wait, in triangle PQR, with ST || QR, then by the basic proportionality theorem, QS / QP = RT / PR = 0.6.

But since QS / QP = 0.6, then QS = 0.6 QP, so SP = QP - QS = 0.4 QP. Similarly, RT = 0.6 PR, so PT = PR - RT = 0.4 PR.

Therefore, SP / QP = 0.4 and PT / PR = 0.4.

Therefore, SP / QP = PT / PR = 0.4.

Therefore, triangles SPQ and PTR? Wait, not sure.

Alternatively, since SP / QP = PT / PR, and angle at P is common? Wait, if we consider triangles SPQ and PTR?

Wait, angle at P is common? No, because SP is part of PQ, and PT is part of PR.

Wait, maybe triangles SPT and QPR?

Wait, not sure.

Alternatively, since ST || QR, then angle PST = angle PQR = 35°, because they are corresponding angles.

Wait, if ST || QR, and PQ is a transversal, then angle between PQ and ST (which is angle at S) is equal to angle between PQ and QR (which is angle at Q).

Wait, angle at Q is 35°, so angle between PQ and QR is 35°, so angle between PQ and ST is also 35°, which is angle at S.

Therefore, angle at S (angle QSP) is 35°.

Similarly, since ST || QR, and PR is a transversal, then angle between PR and ST (which is angle at T) is equal to angle between PR and QR (which is angle at R).

Angle at R in triangle PQR is 105°, so angle between PR and QR is 105°, so angle between PR and ST is also 105°, which is angle at T (angle PTR).

Wait, but angle PTR is at T between P and R. But we need angle STR, which is at T between S and R.

Wait, so if angle PTR is 105°, and angle at T between S and R is angle STR, then maybe we can find angle STR by considering triangle STR?

Wait, but we need more information.

Alternatively, in triangle STR, if we can find two angles, we can find the third.

But we need to find angle STR.

Alternatively, let me consider triangle SPT.

Wait, maybe not.

Alternatively, since we know angle at S is 35°, angle at T is...?

Wait, if angle between PR and ST is 105°, which is angle PTR. But angle PTR is at T between P and R. So angle PTR = 105°, and angle STR is adjacent to it?

Wait, if we look at point T, connected to S, P, and R. So at point T, there are angles: angle STR (which we need), angle PTR (which is 105°), and angle STP.

Wait, but unless we know something about the other angles, it's hard to say.

Alternatively, since we know angle at S is 35°, and maybe we can find angle at S in triangle STR?

Wait, in triangle STR, angle at S is angle SRT? No, angle at S is angle STR? No, angle at S is angle S in triangle STR. Wait, triangle STR has vertices S, T, R. So angles at S, T, R.

We need angle at T, which is angle STR.

If we can find angles at S and R in triangle STR, then we can find angle at T.

But how?

Alternatively, let me consider triangle SPQ.

In triangle SPQ, we know angle at Q is 35°, angle at S is 35°, as we said earlier? Wait, no. Wait, angle at S in triangle SPQ is angle QSP, which we said is 35°, because ST || QR and PQ is a transversal.

Wait, if angle QSP is 35°, and angle at Q is 35°, then triangle SPQ has two angles of 35°, making it an isosceles triangle?

Wait, if angle at Q is 35°, angle at S is 35°, then angle at P in triangle SPQ would be 180 - 35 - 35 = 110°.

But angle at P in triangle SPQ is angle SPQ. However, in the original problem, angle at P is given as 40°, which is angle between SP and PR.

Wait, so angle SPQ is part of angle at P.

Wait, angle at P in the diagram is between SP and PR, which is 40°, but angle SPQ is between SP and PQ.

So unless PQ and PR are related...

Wait, maybe I need to find angle SPQ.

If in triangle SPQ, angles at Q and S are 35°, then angle at P (angle SPQ) is 110°, as above.

But angle at P in the diagram is angle between SP and PR, which is given as 40°.

Therefore, angle between SP and PR is 40°, and angle between SP and PQ is 110°, so the angle between PQ and PR is 110° - 40° = 70°?

Wait, but angle between PQ and PR is angle QPR, which is part of triangle PQR.

Wait, in triangle PQR, angle at P is angle QPR, which we previously thought was 40°, but according to this, it's 70°?

This seems conflicting.

Wait, there must be a mistake here.

Let me clarify.

Earlier, I assumed that angle at P in triangle PQR is 40°, but maybe that's not the case.

Wait, the problem says "angle P = 40°", which is angle at P. In the diagram, angle at P is between SP and PR. However, in triangle PQR, angle at P is between PQ and PR.

Therefore, angle between SP and PR is 40°, and angle between PQ and PR is something else.

Therefore, my previous assumption that angle at P in triangle PQR is 40° is incorrect.

Therefore, I need to correct that.

So, let me start over.

Given that angle P = 40°, which is the angle between SP and PR.

Angle Q = 35°, which is the angle between RQ and QS.

We need to find angle STR.

Since ST is parallel to QR, we can use that to find some relationships.

Let me consider triangle SPQ.

Since ST || QR, and PQ is a transversal, then angle between PQ and ST (which is angle QSP) is equal to angle between PQ and QR (which is angle PQR = 35°). Therefore, angle QSP = 35°.

Similarly, since ST || QR, and PR is a transversal, then angle between PR and ST (which is angle PTR) is equal to angle between PR and QR (which is angle PRQ).

But angle PRQ is angle at R in triangle PQR.

But we don't know angle at R yet.

But let me denote some variables.

Let me denote angle at P (between SP and PR) as 40°, which is given.

Let me denote angle at Q (between RQ and QS) as 35°, given.

Let me denote angle QSP = 35°, as established.

In triangle SPQ, we have angle at Q = 35°, angle at S = 35°, so angle at P (angle SPQ) is 180 - 35 - 35 = 110°.

But angle SPQ is the angle at P between SP and PQ.

But the given angle at P is between SP and PR, which is 40°.

Therefore, the angle between PQ and PR is angle QPR = angle SPQ - angle between SP and PR? Wait, no.

Wait, angle between SP and PQ is 110°, and angle between SP and PR is 40°, so the angle between PQ and PR is angle QPR = angle SPQ - angle SPR? Wait, need to visualize.

At point P, there are two lines: SP and PR. The angle between them is 40°. Also, there is line PQ, which forms angle SPQ = 110° with SP.

Therefore, the angle between PQ and PR is angle QPR = angle SPQ - angle SPR? Wait, if SP is between PQ and PR, then angle QPR = angle SPQ - angle SPR?

Wait, if you imagine at point P, with SP going off in one direction, PQ going off in another, and PR going off in another.

If angle between SP and PQ is 110°, and angle between SP and PR is 40°, then the angle between PQ and PR is 110° - 40° = 70°.

Yes, because if SP is between PQ and PR, then the angle between PQ and PR is the difference between the angles SPQ and SPR.

Therefore, angle QPR = 70°.

Therefore, in triangle PQR, we have angle at Q = 35°, angle at P = 70°, so angle at R is 180 - 35 - 70 = 75°.

Ah! So angle at R is 75°, not 105° as I previously thought.

This was my mistake earlier, assuming angle at P in triangle PQR was 40°, but actually, angle at P in triangle PQR is 70°, because angle between PQ and PR is 70°, while the given angle at P (between SP and PR) is 40°.

Therefore, in triangle PQR, angles are:

At Q: 35°, at P: 70°, at R: 75°.

Now, going back to the earlier thought: since ST || QR, and PR is a transversal, then angle between PR and ST (which is angle PTR) is equal to angle between PR and QR (which is angle PRQ = 75°).

Therefore, angle PTR = 75°.

Now, looking at point T, which is connected to S, P, and R.

At point T, we have angles:

- angle PTR = 75° (between PT and TR)

- angle STR (which we need, between ST and TR)

- angle STP (between ST and TP)

These three angles at point T should add up to 360°, but since we are dealing with a triangle, maybe not. Wait, point T is connected to S, P, and R. So the angles around point T are:

- angle STR (between S and R)

- angle PTR (between P and R)

- angle STP (between S and P)

But these three angles are around point T, so they should add up to 360°. However, if we consider triangle STR and triangle PTR, it's more complex.

Alternatively, consider triangle STR. To find angle STR, we need more information.

But we know angle PTR = 75°, which is part of triangle PTR.

Alternatively, consider triangle STR and triangle... something.

Alternatively, since we know angle at R in triangle PQR is 75°, and ST || QR, maybe there's a relationship.

Alternatively, consider triangle STR. If we can find two angles, we can find the third.

We know angle at R in triangle STR is...?

Wait, angle at R in triangle STR is angle SRT.

But angle SRT is part of angle PRQ (which is 75°).

Since T is on PR, angle SRT is the same as angle PRQ? No, because S is not on PR.

Wait, point T is on PR, so angle SRT is the angle at R between SR and RT.

But RT is part of PR, so angle SRT is the angle between SR and PR.

But we might need to find that angle.

Alternatively, consider triangle SRT.

Wait, maybe using the Law of Sines in some triangles.

Let me consider triangle SPQ, where we know angles:

- angle at Q: 35°

- angle at S: 35°

- angle at P: 110°

Therefore, triangle SPQ is isosceles with SP = SQ.

Wait, if two angles are equal, then sides opposite are equal. So side opposite angle at Q (35°) is SP, and side opposite angle at S (35°) is PQ. Wait, no.

Wait, in triangle SPQ, angle at Q is 35°, angle at S is 35°, so sides opposite are equal.

Side opposite angle at Q is SP, and side opposite angle at S is PQ. Therefore, SP = PQ.

Wait, so SP = PQ.

Similarly, in triangle PQR, we can find some relations.

But I need to relate this to other parts.

Alternatively, since SP = PQ, and we know some angles, maybe we can find lengths.

But without specific lengths, it's hard.

Alternatively, since ST || QR, and we know ratios from the coordinates.

Earlier, in the coordinate system, we found angle STR ≈ 78.7°, but with the corrected angles, maybe we can find an exact value.

Wait, let me think again.

In triangle PQR, angles are 35°, 70°, 75°.

Since ST || QR, and S divides QP in ratio 0.6, T divides PR in ratio 0.4.

Let me consider triangle STR.

To find angle STR, maybe we can use the Law of Sines in triangle STR and some other triangle.

Alternatively, since we know angle at T is 75° (angle PTR), and if we can find angle at S or R in triangle STR, we can find angle STR.

Wait, angle at S in triangle STR is angle SRT? No, angle at S in triangle STR is angle STR? No, angle at S is angle STR? No, triangle STR has vertices S, T, R. So angles are:

- At S: angle STR? No, angle at S is angle TSR.

Wait, this is getting confusing. Let me clarify:

In triangle STR:

- Vertex S: angle between ST and SR

- Vertex T: angle between ST and TR (which is angle STR, the one we need)

- Vertex R: angle between SR and TR

So to find angle at T (angle STR), we need to know other angles.

If we can find angle at S or angle at R in triangle STR, we can find angle STR.

Let me try to find angle at R in triangle STR, which is angle SRT.

Angle SRT is the angle at R between SR and RT.

But RT is part of PR, and SR is a line from R to S.

Wait, in triangle PQR, angle at R is 75°, which is angle between PR and QR.

But SR is a line from R to S, which is somewhere inside the triangle.

Alternatively, since we know SP = PQ (from triangle SPQ being isosceles), and S divides QP in ratio 0.6, maybe we can find some relations.

Wait, SP = PQ.

But S is on QP, so SP = PQ?

Wait, if SP = PQ, and S is on QP, then SP = PQ implies that S is located such that SP = PQ.

But if QP is a segment, and S is between Q and P, then SP = PQ would mean that S is located such that from S to P is equal to the entire length of QP.

But that would mean that S is beyond P, which contradicts S being between Q and P.

Wait, this suggests a mistake in my earlier conclusion that SP = PQ.

Wait, in triangle SPQ, angles at Q and S are both 35°, so sides opposite are equal.

Side opposite angle at Q (35°) is SP.

Side opposite angle at S (35°) is PQ.

Therefore, SP = PQ.

But if S is on QP, then SP is a part of QP.

So SP = PQ implies that SP = PQ, but since S is between Q and P, SP is less than QP.

This is a contradiction. Therefore, my earlier conclusion must be wrong.

Wait, what's the problem here?

In triangle SPQ, angle at Q is 35°, angle at S is 35°, so sides opposite are equal.

Side opposite angle Q is SP.

Side opposite angle S is PQ.

Therefore, SP = PQ.

But if S is on QP, then SP is a segment from S to P, and PQ is the entire segment from P to Q.

So SP = PQ would mean that the length from S to P is equal to the length from P to Q.

Therefore, S must be located such that SP = PQ, which would place S outside of segment QP.

But according to the Asymptote code, S is on QP, between Q and P.

This contradiction suggests that my earlier assumption that angle QSP = 35° is incorrect.

Wait, let me re-examine that.

Earlier, I said that since ST || QR, and PQ is a transversal, then angle QSP = angle PQR = 35°.

But is that correct?

If ST || QR, and PQ is a transversal, then the corresponding angles would be equal.

But angle between PQ and ST is angle QSP, and angle between PQ and QR is angle PQR.

But are these corresponding angles?

Let me visualize: PQ is a line, ST is parallel to QR.

At point Q, the line QR is going to the right (since Q is at (0,0) and R is at (1,0)), and PQ is going up to P.

At point S, which is on PQ, the line ST is going to the right (since ST is parallel to QR).

Therefore, the angle between PQ and ST at S is equal to the angle between PQ and QR at Q, because they are corresponding angles.

Yes, that should be correct. Therefore, angle QSP = angle PQR = 35°.

But then in triangle SPQ, angles at Q and S are both 35°, leading to SP = PQ, which seems contradictory.

But according to the coordinates, S is on QP, and SP is shorter than PQ.

Wait, let me calculate lengths using coordinates.

In the Asymptote code, Q is (0,0), P is (1.1, 0.5), S is (0.66, 0.3).

Length PQ: distance from Q(0,0) to P(1.1, 0.5) is sqrt(1.1² + 0.5²) = sqrt(1.21 + 0.25) = sqrt(1.46) ≈ 1.208.

Length SP: distance from S(0.66, 0.3) to P(1.1, 0.5) is sqrt((1.1 - 0.66)² + (0.5 - 0.3)²) = sqrt(0.44² + 0.2²) = sqrt(0.1936 + 0.04) = sqrt(0.2336) ≈ 0.483.

So SP ≈ 0.483, PQ ≈ 1.208, so SP ≠ PQ.

Therefore, my earlier conclusion that SP = PQ is wrong.

Where is the mistake?

In triangle SPQ, if angles at Q and S are both 35°, then sides opposite should be equal.

Side opposite angle Q is SP, side opposite angle S is PQ.

But according to coordinates, SP ≠ PQ, so this suggests that angles at Q and S are not both 35°.

Therefore, my earlier assumption that angle QSP = 35° is incorrect.

Wait, why did I think that?

Because I thought that since ST || QR, and PQ is a transversal, then angle QSP = angle PQR.

But maybe it's not corresponding angles.

Let me think again about the transversal.

If ST is parallel to QR, and PQ is a transversal, then the angles formed at Q and S should be corresponding angles.

But angle at Q is between PQ and QR, and angle at S is between PQ and ST.

Since ST || QR, these should be corresponding angles, hence equal.

But according to coordinates, angle QSP is not 35°.

Let me calculate angle QSP using coordinates.

Point Q(0,0), S(0.66, 0.3), P(1.1, 0.5).

Angle at S between Q and P.

Wait, angle QSP is the angle at S between Q and P.

Wait, no. Angle QSP is the angle at S between Q and P? No, angle QSP is the angle at S between Q and P? Wait, no.

Wait, angle QSP is the angle at S between Q and P? No, it's the angle at S between Q and P? Wait, no.

Wait, angle QSP is the angle at S between Q and P? No, it's the angle at S between Q and P? Wait, no.

Actually, angle QSP is the angle at S between Q and P? No, it's the angle at S between Q and P? Wait, no.

Wait, angle QSP is the angle at S between Q and P? No, it's the angle at S between Q and P? Wait, no.

Actually, angle QSP is the angle at S between Q and P? No, it's the angle at S between Q and P? Wait, no.

Wait, angle QSP is the angle at S between Q and P? No, it's the angle at S between Q and P? No, it's the angle at S between Q and P? No, it's the angle at S between Q and P?

Wait, I think I'm confused. Let me clarify.

Angle QSP is the angle at point S, between points Q, S, P. So it's the angle at S between SQ and SP.

Similarly, angle PQR is the angle at Q between P, Q, R.

If ST || QR, and PQ is a transversal, then the corresponding angles would be angle QSP (at S) and angle PQR (at Q).

But according to coordinates, let's calculate angle QSP.

Vector SQ = Q - S = (0 - 0.66, 0 - 0.3) = (-0.66, -0.3)

Vector SP = P - S = (1.1 - 0.66, 0.5 - 0.3) = (0.44, 0.2)

Angle between vectors SQ and SP at point S.

Using the dot product:

cos(theta) = (SQ • SP) / (|SQ| |SP|)

SQ • SP = (-0.66)(0.44) + (-0.3)(0.2) = -0.2904 - 0.06 = -0.3504

|SQ| = sqrt((-0.66)^2 + (-0.3)^2) = sqrt(0.4356 + 0.09) = sqrt(0.5256) ≈ 0.725

|SP| = sqrt(0.44² + 0.2²) ≈ sqrt(0.1936 + 0.04) = sqrt(0.2336) ≈ 0.483

So cos(theta) ≈ -0.3504 / (0.725 * 0.483) ≈ -0.3504 / 0.349 ≈ -1.004

Wait, this is approximately -1, which would mean the angle is 180°, which is impossible.

Wait, this suggests a calculation error.

Wait, let me recalculate SQ • SP:

SQ is (-0.66, -0.3)

SP is (0.44, 0.2)

Dot product: (-0.66)(0.44) + (-0.3)(0.2) = -0.2904 - 0.06 = -0.3504. Correct.

|SQ|: sqrt(0.66² + 0.3²) = sqrt(0.4356 + 0.09) = sqrt(0.5256) ≈ 0.725. Correct.

|SP|: sqrt(0.44² + 0.2²) = sqrt(0.1936 + 0.04) = sqrt(0.2336) ≈ 0.483. Correct.

So cos(theta) ≈ -0.3504 / (0.725 * 0.483) ≈ -0.3504 / 0.349 ≈ -1.004.

This is approximately -1, which would mean the angle is 180°, which is impossible.

This suggests that vectors SQ and SP are pointing in nearly opposite directions, which would mean that points Q, S, P are colinear? But S is on QP, so they are colinear.

Wait, if S is on QP, then vectors SQ and SP are in opposite directions, hence the angle between them is 180°, which is correct.

But this contradicts the earlier assumption that angle QSP is 35°.

Wait, what's going on here?

Ah! I think I made a mistake in identifying the angles.

If ST is parallel to QR, and PQ is a transversal, then the corresponding angles would be angle between PQ and ST at S, and angle between PQ and QR at Q.

But angle between PQ and ST at S is not angle QSP, but angle between PQ and ST at S.

Since ST is parallel to QR, and PQ is a transversal, the corresponding angle at S would be the angle between PQ and ST, which is angle between PQ and ST at S.

But since S is on PQ, the line PQ passes through S, so the angle between PQ and ST at S is the angle between PQ (which is the line itself) and ST.

Since ST is parallel to QR, and PQ is a transversal, the corresponding angle would be the angle between PQ and QR at Q, which is angle PQR = 35°.

But since ST is parallel to QR, the angle between PQ and ST at S is equal to angle PQR = 35°.

But how is this angle measured?

At point S, the line ST is going in some direction, and PQ is passing through S. The angle between PQ and ST at S is 35°.

But since S is on PQ, the line PQ is straight, so the angle between PQ and ST at S is the angle between the line PQ (which is straight) and the line ST.

This angle is 35°, so if we consider the direction of ST, it makes 35° with PQ.

But since ST is parallel to QR, and QR is horizontal in the diagram, then the angle between PQ and ST is 35°, which would mean that ST is inclined at 35° from PQ.

But since PQ is going from Q(0,0) to P(1.1, 0.5), its slope is (0.5 - 0)/(1.1 - 0) = 0.5 / 1.1 ≈ 0.4545, so angle of PQ with x-axis is arctan(0.4545) ≈ 24.5°.

If ST is making 35° with PQ, then the angle of ST with x-axis is 24.5° + 35° = 59.5°, or 24.5° - 35° = -10.5°.

But in the diagram, ST is horizontal (since S and T have the same y-coordinate), so it's parallel to x-axis, which is 0°.

But if ST is horizontal, then the angle between PQ and ST is the angle between PQ (≈24.5°) and horizontal (0°), which is 24.5°, not 35°.

This contradicts the earlier assumption that it should be 35°.

Therefore, my earlier reasoning about corresponding angles was incorrect.

This suggests that my initial approach was wrong.

Let me try to find the correct relationship.

Since ST is parallel to QR, and QR is horizontal, then ST is horizontal.

Therefore, the line ST is horizontal.

In the diagram, PQ is a line from Q(0,0) to P(1.1, 0.5), which has a slope of 0.5 / 1.1 ≈ 0.4545, as before.

At point S, which is on PQ, the line ST is horizontal. Therefore, the angle between PQ and ST at S is the angle between PQ and the horizontal line ST.

This angle can be calculated as the angle of PQ with the x-axis, which is arctan(0.5 / 1.1) ≈ 24.5°, as before.

But the problem states that angle Q is 35°, which is the angle at Q between RQ and QS.

Since RQ is along the x-axis from Q(0,0) to R(1,0), and QS is from Q(0,0) to S(0.66, 0.3), the angle at Q is between RQ (along x-axis) and QS.

Let me calculate this angle.

Vector QR is (1, 0) - (0, 0) = (1, 0)

Vector QS is (0.66, 0.3) - (0, 0) = (0.66, 0.3)

The angle at Q is between QR and QS, which can be calculated using the dot product:

cos(theta) = (QR • QS) / (|QR| |QS|)

QR • QS = (1)(0.66) + (0)(0.3) = 0.66

|QR| = 1

|QS| = sqrt(0.66² + 0.3²) ≈ sqrt(0.4356 + 0.09) ≈ sqrt(0.5256) ≈ 0.725

So cos(theta) ≈ 0.66 / 0.725 ≈ 0.909

Therefore, theta ≈ arccos(0.909) ≈ 24.5°, which matches the angle of PQ with x-axis.

But the problem states that angle Q is 35°, but according to this calculation, it's approximately 24.5°.

This is a contradiction.

Wait, this suggests that the Asymptote code's coordinates do not match the problem's given angles.

The problem says angle Q is 35°, but in the Asymptote code's coordinates, it's approximately 24.5°.

This means that the diagram is just an example, and the actual problem has different proportions.

Therefore, I cannot rely on the coordinates to calculate the angle, as they are not to scale.

Therefore, I need to solve it using pure geometry, without relying on specific coordinates.

Let me start over, using the given angles and the parallel lines.

Given: ST || QR, angle at P is 40°, angle at Q is 35°. Find angle STR.

Let me draw triangle PQR, with ST parallel to QR, S on PQ, T on PR.

Given angle at P (between SP and PR) is 40°, angle at Q (between RQ and QS) is 35°.

Since ST || QR, let's use the properties of parallel lines and transversals.

Let me consider the transversal PQ intersecting ST and QR. Since ST || QR, the corresponding angles are equal.

The angle between PQ and ST at S is equal to the angle between PQ and QR at Q.

Let me denote this angle as x.

So, angle between PQ and ST at S is x, and angle between PQ and QR at Q is x.

But angle at Q is given as 35°, which is the angle between RQ and QS.

Wait, if the angle between PQ and QR at Q is x, and angle at Q is 35°, which is between RQ and QS, then x is part of that 35°?

Wait, this is getting too vague. Let me try to label the diagram with variables.

Let me denote:

At point Q, angle between RQ and QS is 35°.

Since ST || QR, and PQ is a transversal, the corresponding angle at S (between PQ and ST) is equal to the angle at Q (between RQ and QS).

Wait, if ST || QR, then the corresponding angle at S (between PQ and ST) is equal to the angle at Q (between PQ and QR).

Wait, but angle at Q between PQ and QR is part of the given angle at Q.

Wait, let me clarify.

At point Q, there are two lines: QR and QS. The angle between them is 35°.

If we consider PQ as a transversal, and ST || QR, then the corresponding angle at S (between PQ and ST) is equal to the angle at Q between PQ and QR.

Let me denote the angle between PQ and QR at Q as y. Then, the angle between PQ and ST at S is also y.

But the angle at Q between RQ and QS is 35°, which is the angle between QR and QS.

If I can relate y to this 35°, I can find y.

Let me consider triangle QRS? No, S is on PQ.

Wait, at point Q, the lines are QR, QS, and PQ.

The angle between QR and QS is 35°, and the angle between PQ and QR is y.

Therefore, the angle between PQ and QS is 35° - y? Or y - 35°? Depends on the configuration.

This is getting too confusing without a diagram.

Let me try to use the fact that ST || QR and apply the basic proportionality theorem.

Since ST || QR, and S is on PQ, T is on PR, then QS/QP = RT/PR.

Let me denote QS/QP = RT/PR = k.

Let me denote k as the ratio.

Let me also consider the angles.

At point P, angle between SP and PR is 40°.

At point Q, angle between RQ and QS is 35°.

Let me consider triangle PQS and triangle PRT.

Wait, not sure.

Alternatively, let me use the Law of Sines in triangle PQS and triangle PRT.

But I need more information.

Alternatively, let me consider the angles in triangle PQR.

Let me denote angle at P as 40°, but wait, no.

Wait, the given angle at P is between SP and PR, which is 40°.

Let me denote angle QPR as α, which is the angle at P between PQ and PR.

Then, the given angle at P (between SP and PR) is 40°, and since S is on PQ, the angle between SP and PQ is α - 40°?

Wait, if angle between SP and PR is 40°, and angle between PQ and PR is α, then the angle between SP and PQ is α - 40°.

But I need to relate this to other parts.

This is getting too abstract. Maybe I should use the fact that ST || QR and apply the Law of Sines in triangles involving ST and QR.

Let me consider triangle PQR and the line ST parallel to QR.

Since ST || QR, the triangles PST and PQR are similar? No, because S is on PQ and T is on PR, but ST is parallel to QR, so by the basic proportionality theorem, the triangles PST and PQR are similar?

Wait, if a line divides two sides of a triangle proportionally, then it is parallel to the third side. Conversely, if a line is parallel to the third side, it divides the other two sides proportionally.

So, since ST || QR, then triangles PST and PQR are similar? No, because ST is not necessarily connecting to the third side.

Wait, if ST is parallel to QR, then triangles PST and PQR are similar? No, because S is on PQ and T is on PR, so the line ST is cutting PQ and PR, and is parallel to QR, so triangles PST and PQR are similar.

Wait, yes! Because if a line parallel to one side of a triangle intersects the other two sides, then it creates a smaller similar triangle.

Wait, in this case, ST is parallel to QR, and S is on PQ, T is on PR, so triangle PST is similar to triangle PQR.

Wait, but ST is parallel to QR, so triangle PST ~ triangle PQR by AA similarity.

Because angle at P is common, and angle PST = angle PQR (since ST || QR).

Yes!

Therefore, triangle PST ~ triangle PQR.

Therefore, corresponding angles are equal.

So, angle PST = angle PQR = 35°, and angle PTS = angle PRQ.

Wait, let me confirm.

In triangle PST and PQR:

- Angle at P is common.

- Angle PST = angle PQR (since ST || QR, corresponding angles).

Therefore, triangles PST ~ PQR by AA similarity.

Therefore, the ratio of sides is equal.

Let me denote the ratio of similarity as k = PS / PQ = PT / PR = ST / QR.

Now, let me use this to find some relations.

Let me denote angle at P as 40°, which is angle between SP and PR.

Wait, in triangle PQR, angle at P is angle QPR, which is the angle between PQ and PR.

In triangle PST, angle at P is the same, angle QPR.

But in triangle PST, angle at P is also part of the triangle.

Wait, but we know angle at P in triangle PST is the same as in triangle PQR.

But we also know that in triangle PST, angle PST = 35°, and angle PTS = angle PRQ.

Let me denote angle PRQ as β.

In triangle PQR, angles are:

- At Q: 35°

- At P: let's call it γ

- At R: β

So, γ + 35° + β = 180° => γ + β = 145°

In triangle PST, angles are:

- At P: γ

- At S: 35°

- At T: β

Therefore, γ + 35° + β = 180°, which is the same as above.

But this doesn't give new information.

However, we know that in triangle PST, angle at P is γ, and in the problem, the angle at P (between SP and PR) is 40°.

Wait, the angle at P in triangle PST is the same as angle QPR in triangle PQR, which is γ.

But the problem states that the angle at P (between SP and PR) is 40°, which is the same as angle QPR, which is γ.

Therefore, γ = 40°.

Therefore, in triangle PQR, γ = 40°, angle at Q is 35°, so angle at R is β = 180° - 40° - 35° = 105°.

Wait, this is different from earlier, but now it makes sense.

So, in triangle PQR, angles are:

- At P: 40°

- At Q: 35°

- At R: 105°

Now, since triangles PST ~ PQR, the ratio of sides is k = PS / PQ = PT / PR = ST / QR.

Let me denote this ratio as k.

Now, we need to find angle STR.

Let me consider triangle STR.

Point T is on PR, and we need angle at T between S and R.

Let me consider triangle STR. To find angle STR, I can use the Law of Sines or Law of Cosines if I know some sides or other angles.

Alternatively, since we know some angles in triangle PQR and the similarity, maybe we can find relations.

Let me consider triangle STR.

At point T, we have angle STR, which is what we need.

Let me consider triangle STR and triangle TPR or something.

Alternatively, since ST || QR, and we know angle at R is 105°, maybe angle STR is related.

Wait, let me think about the angles at point T.

In triangle PQR, angle at R is 105°, which is angle PRQ.

Since ST || QR, the angle between PR and ST at T is equal to angle PRQ = 105°.

Wait, this is the corresponding angle.

So, at point T, the angle between PR and ST is 105°, which is angle PTR.

But angle PTR is part of triangle PTR.

Now, in triangle STR, we have angle at T: angle STR, and angle at R: angle SRT, and angle at S: angle TSR.

But how to relate these?

Wait, if I can find angle SRT, then I can find angle STR.

Angle SRT is the angle at R between SR and RT.

But RT is part of PR, and SR is a line from R to S.

In triangle PQR, angle at R is 105°, which is between PR and QR.

Since ST || QR, and S is on PQ, the line SR is a transversal.

Maybe we can find angle SRT using some relations.

Alternatively, since triangles PST ~ PQR, the ratio of sides is k = PS / PQ.

Let me denote PQ = a, PR = b, QR = c.

Then, PS = k * PQ = k * a, PT = k * PR = k * b, and ST = k * QR = k * c.

But without knowing the actual lengths, it's hard to proceed.

However, we can use the Law of Sines in triangle PQR.

In triangle PQR:

- angle at P: 40°

- angle at Q: 35°

- angle at R: 105°

By the Law of Sines:

PQ / sin(105°) = PR / sin(35°) = QR / sin(40°)

Let me denote PQ = a, PR = b, QR = c.

So:

a / sin(105°) = b / sin(35°) = c / sin(40°)

Let me compute the ratios.

Let me choose a value for one side to make calculations easier. Let me set PQ = a = 1.

Then, PR = b = sin(35°) / sin(105°) * a ≈ (0.5736) / (0.9659) ≈ 0.5937

QR = c = sin(40°) / sin(105°) * a ≈ (0.6428) / (0.9659) ≈ 0.665

Now, since triangles PST ~ PQR with ratio k, then PS = k * PQ = k * 1 = k, PT = k * PR = k * 0.5937, and ST = k * QR = k * 0.665.

But also, since S is on PQ, PS + SQ = PQ = 1, so SQ = 1 - k.

Similarly, PT + TR = PR = 0.5937, so TR = 0.5937 - k * 0.5937 = 0.5937(1 - k).

Now, let me consider triangle STR. To find angle STR, I need to know some sides or other angles.

Let me try to use the Law of Sines in triangle STR.

But I need to know some sides.

Alternatively, let me consider triangle STR and triangle TPR.

Wait, at point T, we have angle PTR = 105° (from earlier), and angle STR is what we need.

If I can find the other angles in triangle STR, I can find angle STR.

Let me consider triangle STR.

At point S, angle is angle TSR.

At point R, angle is angle SRT.

At point T, angle is angle STR.

If I can find angle TSR or angle SRT, I can find angle STR.

Let me try to find angle SRT.

Angle SRT is the angle at R between SR and RT.

In triangle PQR, angle at R is 105°, which is between PR and QR.

Since ST || QR, and SR is a line from R to S, which is on PQ.

Let me consider the line SR.

In triangle PQR, we can find the coordinates of S if we know the ratio k.

But since I set PQ = 1, and PS = k, then S divides PQ in the ratio k : (1 - k).

But without knowing k, it's hard to proceed.

Wait, but from the similarity of triangles PST and PQR, the ratio k is the same for all sides.

But how to find k?

Wait, we don't have enough information to find k.

Wait, but maybe we can relate it to the given angles.

Wait, in triangle PST, we know angle at S is 35°, angle at P is 40°, so angle at T is 180 - 35 - 40 = 105°.

Wait, in triangle PST, angles are:

- At P: 40°

- At S: 35°

- At T: 105°

Therefore, angle at T in triangle PST is 105°, which is angle PTS.

But angle PTS is the same as angle PTR, which we previously said is 105°, which matches.

Now, in triangle STR, we need angle STR.

Let me consider triangle STR and triangle PTS.

Wait, angle at T in triangle PTS is 105°, which is angle PTR.

If I can find some relationship between triangles STR and PTS, maybe.

Alternatively, let me consider the straight line at point T.

At point T, the angles around it are:

- angle STR (which we need)

- angle PTR (105°)

- angle STP (which is part of triangle PST)

But since ST is a straight line, the sum of angles around T should be 360°, but I need to consider the specific angles.

Actually, at point T, there are two angles: angle STR (between S and R) and angle PTR (between P and R), and angle STP (between S and P).

These three angles should add up to 180°, since they are on a straight line? No, because T is a point with multiple lines.

Actually, at point T, the lines are ST, TR, and TP.

So the angles at T are:

- Between ST and TR: angle STR (what we need)

- Between TR and TP: angle PTR = 105°

- Between TP and ST: angle STP

These three angles should add up to 360°, but since they are around point T, but if we consider the straight lines, it's more complex.

Actually, if you look at the lines at point T: ST, TR, and TP.

If you consider the angles between ST and TR (angle STR), between TR and TP (angle PTR = 105°), and between TP and ST (angle STP), then these three angles should add up to 360°.

But in reality, since ST and TP are two lines from T, and TR is another line, it's a bit complex.

Alternatively, if we consider the straight line from T to P and T to R, and T to S, it's better to use triangle relationships.

Let me consider triangle STR and triangle STP.

In triangle STP, we know all angles: 40°, 35°, 105°.

In triangle STR, we need angle STR.

If I can find the lengths of sides in these triangles, I can use the Law of Sines or Cosines.

Let me assume some lengths.

Let me set PQ = 1, as before.

From earlier, PR ≈ 0.5937, QR ≈ 0.665.

From the similarity, k = PS / PQ = PS / 1 = PS.

But we need to find k.

Wait, but how?

Wait, in triangle PST, we know the angles, and if we can relate it to the sides, maybe we can find k.

Let me use the Law of Sines in triangle PST.

In triangle PST:

- angle at P: 40°

- angle at S: 35°

- angle at T: 105°

Let me denote PS = k, PT = k * PR = k * 0.5937, and ST = k * QR = k * 0.665.

By the Law of Sines in triangle PST:

PS / sin(angle at T) = PT / sin(angle at S) = ST / sin(angle at P)

So:

k / sin(105°) = (k * 0.5937) / sin(35°) = (k * 0.665) / sin(40°)

Let me check the first two ratios:

k / sin(105°) = (k * 0.5937) / sin(35°)

Cancel k:

1 / sin(105°) = 0.5937 / sin(35°)

Calculate sin(105°) ≈ sin(60° + 45°) = sin60cos45 + cos60sin45 ≈ (0.8660 * 0.7071) + (0.5 * 0.7071) ≈ 0.6124 + 0.3536 ≈ 0.9660

sin(35°) ≈ 0.5736

So 0.5937 / 0.5736 ≈ 1.035

1 / 0.9660 ≈ 1.035

Yes, this checks out.

Similarly, check the first and third ratios:

k / sin(105°) = (k * 0.665) / sin(40°)

Cancel k:

1 / sin(105°) = 0.665 / sin(40°)

Calculate sin(40°) ≈ 0.6428

0.665 / 0.6428 ≈ 1.035

1 / 0.9660 ≈ 1.035

Also checks out.

Therefore, the ratio is consistent.

Now, to find angle STR, let's consider triangle STR.

Let me use the Law of Sines in triangle STR.

But I need to know some sides or angles.

Let me consider triangle STR and triangle PTR.

Wait, in triangle PTR, we know angle at T is 105°, and we can find other angles if we know sides.

Alternatively, let me consider triangle STR and use the Law of Sines.

Let me denote:

In triangle STR:

- angle at T: angle STR = x (what we need)

- angle at R: angle SRT = y

- angle at S: angle TSR = z

We know that x + y + z = 180°

If I can find y or z, I can find x.

Let me try to find angle at R, y = angle SRT.

Angle SRT is the angle at R between SR and RT.

In triangle PQR, angle at R is 105°, which is between PR and QR.

Since ST || QR, and SR is a line from R to S, which is on PQ, we can find angle SRT.

Let me consider triangle SRT and triangle QRT?

Wait, not sure.

Alternatively, let me use the Law of Sines in triangle SRT and triangle SQR.

Wait, this is getting too complex.

Let me try to use coordinates with the given angles.

Let me set up a coordinate system where Q is at (0, 0), R is at (1, 0), and P is somewhere in the plane.

Given angle at Q is 35°, angle at P is 40°, and ST || QR.

Let me try to find coordinates of P.

Let me assume QR is from (0, 0) to (1, 0).

Let me let Q be at (0, 0), R at (1, 0).

Let me let P be at some point (x, y).

Given angle at Q is 35°, which is the angle between RQ (from Q to R, along x-axis) and QS (from Q to S).

But S is on PQ.

Also, angle at P is 40°, which is the angle between SP and PR.

This is complex, but let me try.

Let me assume that S divides PQ in some ratio, say k:1-k, and T divides PR in the same ratio, since ST || QR.

But without knowing k, it's difficult.

Alternatively, let me use the Law of Sines in triangle PQR.

In triangle PQR, angles are 40°, 35°, 105°.

Let me set QR = 1.

Then, by the Law of Sines:

PQ / sin(105°) = PR / sin(35°) = QR / sin(40°) = 1 / sin(40°)

So:

PQ = sin(105°) / sin(40°) ≈ 0.9659 / 0.6428 ≈ 1.503

PR = sin(35°) / sin(40°) ≈ 0.5736 / 0.6428 ≈ 0.892

Now, since ST || QR, and S is on PQ, T is on PR, with PS / PQ = PT / PR = k.

Let me choose k = 0.6, as in the Asymptote code, but this is just an example.

But since the actual value of k is not given, I need to find it.

Wait, but how?

Wait, in the problem, there is no additional information, so the angle STR should be determined solely by the given angles and the parallel lines.

This suggests that the angle STR is constant, regardless of the position of S and T, as long as ST || QR and the given angles at P and Q.

This is possible, and it might be a fixed value.

Let me try to find it using the Law of Sines in triangle STR.

Let me consider the following:

In triangle PQR, with angles 40°, 35°, 105°, and ST || QR.

Let me consider the line ST, which is parallel to QR, and let me find the angles in triangle STR.

At point T, angle PTR = 105° (as established earlier).

In triangle STR, angle at T is angle STR, and angle at R is angle SRT.

Let me consider the line SR.

In triangle PQR, let me find the angle at S.

Wait, in triangle PQR, we can find the lengths and then use the Law of Sines to find other angles.

But this is getting too involved.

Wait, let me consider the following:

Since ST || QR, the angle between ST and PR is equal to the angle between QR and PR, which is 105°.

This is angle PTR = 105°.

Now, in triangle STR, we have angle at T = 105°, and we need to find angle STR.

Wait, no, angle at T in triangle STR is not 105°, it's angle STR.

Wait, I think I'm getting confused again.

Let me try to use the fact that in triangle PST, we have angles 40°, 35°, 105°, and in triangle STR, we need to find angle STR.

If I can find the relationship between these triangles.

Let me consider that in triangle PST, angle at T is 105°, and in triangle STR, angle at T is what we need.

If I can find the relationship between these two angles.

At point T, the angle between ST and PR is 105° (angle PTR), and the angle between ST and TR is angle STR.

If I can find the angle between PR and TR, which is zero, since T is on PR.

Wait, no.

At point T, the line TR is part of PR, and ST is a line from T to S.

So, the angle between ST and PR at T is 105°, and the angle between ST and TR is the same as the angle between ST and PR, which is 105°.

Wait, but TR is part of PR, so the angle between ST and TR is the same as the angle between ST and PR, which is 105°.

But this would mean that angle STR is 105°, but this can't be, because in the coordinate example, it was approximately 78.7°.

This suggests a mistake in reasoning.

Wait, if at point T, the angle between ST and PR is 105°, and TR is part of PR, then the angle between ST and TR is 105°, which would mean angle STR = 105°.

But this contradicts the coordinate calculation.

However, in the coordinate example, with the given angles, the calculation gave approximately 78.7°, which is different from 105°.

This suggests that my assumption is wrong.

Let me re-examine.

If ST || QR, and PR is a transversal, then the corresponding angle at T (between ST and PR) is equal to the corresponding angle at R (between QR and PR).

But the angle at R between QR and PR is angle PRQ, which is 105°.

Therefore, the angle between ST and PR at T is 105°, which is angle PTR.

But angle PTR is at T between P and R, while angle STR is at T between S and R.

These are different angles.

At point T, the lines are ST, TR, and TP.

The angle between ST and TR is angle STR, and the angle between TR and TP is angle PTR = 105°.

The angle between ST and TP is angle STP.

These three angles at T add up to 180°, because they are on a straight line? No, because TP and TR are on the same line PR, so the angles at T are:

- Between ST and TR: angle STR

- Between TR and TP: angle PTR = 105°

- Between TP and ST: angle STP

But since TP and TR are on the same line, the angle between TP and TR is 180°, but angle PTR is part of it.

Actually, at point T, the line PR passes through T, so the angles on one side of PR are angle STR and angle STP, and on the other side is angle PTR.

Wait, this is getting too confusing.

Let me use the fact that in triangle PST, angle at T is 105°, and in triangle STR, angle at T is what we need.

If I can find the relationship between these two angles.

At point T, the line ST is common to both triangles PST and STR.

The angle at T in triangle PST is 105°, which is between ST and PT.

The angle at T in triangle STR is between ST and RT.

Since PT and RT are on the same line PR, the angle between ST and PT (105°) and the angle between ST and RT (angle STR) are supplementary.

Wait, if PT and RT are on the same line, then the angle between ST and PT (105°) and the angle between ST and RT (angle STR) add up to 180°.

Yes! Because PT and RT are in opposite directions along PR.

Therefore, angle between ST and PT (105°) + angle between ST and RT (angle STR) = 180°.

Therefore, angle STR = 180° - 105° = 75°.

Ah! This makes sense!

So, angle STR is 75°.

This matches with the earlier calculation in the coordinate example, where it was approximately 78.7°, but that was due to the specific coordinates not matching the given angles.

But with the correct angles, it should be 75°.

Let me verify this logic.

At point T, the line PR passes through T, so the angles on either side of PR are supplementary.

The angle between ST and PT is 105° (from triangle PST), and the angle between ST and RT is angle STR.

Since PT and RT are in opposite directions, the sum of these two angles is 180°.

Therefore, angle STR = 180° - 105° = 75°.

Yes, this makes sense.

Therefore, the measure of angle STR is 75 degrees.

**Final Answer**
\boxed{75}

To determine the measure of angle $ \angle STR $, we are given the following:

- $ \overline{ST} \parallel \overline{QR} $
- $ \angle P = 40^\circ $
- $ \angle Q = 35^\circ $

We are to find $ \angle STR $, the angle at point $ T $ between segments $ \overline{ST} $ and $ \overline{TR} $.

---

### Step 1: Understand the Geometry

We are given that $ \overline{ST} \parallel \overline{QR} $, and that points $ S $ and $ T $ lie on segments $ \overline{PQ} $ and $ \overline{PR} $, respectively. This setup implies that triangle $ \triangle PST $ is similar to triangle $ \triangle PQR $ by the **Basic Proportionality Theorem** (also known as Thales' Theorem), since a line parallel to one side of a triangle intersects the other two sides proportionally.

---

### Step 2: Use Triangle Angle Sum to Find $ \angle R $

In triangle $ \triangle PQR $, we are given:

- $ \angle Q = 35^\circ $
- $ \angle P = 40^\circ $

Using the triangle angle sum property:

$$
\angle R = 180^\circ - \angle P - \angle Q = 180^\circ - 40^\circ - 35^\circ = 105^\circ
$$

So, $ \angle PRQ = 105^\circ $.

---

### Step 3: Use Parallel Lines to Find $ \angle PTR $

Since $ \overline{ST} \parallel \overline{QR} $, and $ \overline{PR} $ is a transversal, the corresponding angles are equal. Therefore:

$$
\angle PTR = \angle PRQ = 105^\circ
$$

This is the angle at point $ T $ between segments $ \overline{PT} $ and $ \overline{TR} $.

---

### Step 4: Use Straight Line Angle Sum at Point $ T $

At point $ T $, the lines $ \overline{ST} $, $ \overline{TR} $, and $ \overline{TP} $ meet. Since $ \overline{PT} $ and $ \overline{TR} $ are collinear (they lie on the same line $ \overline{PR} $), the angles formed at point $ T $ by $ \overline{ST} $ with $ \overline{PT} $ and $ \overline{TR} $ are supplementary.

That is:

$$
\angle STR + \angle PTR = 180^\circ
$$

Substitute the known value of $ \angle PTR $:

$$
\angle STR = 180^\circ - 105^\circ = 75^\circ
$$

---

### Final Answer

$$
\boxed{75}
$$

Solution: From $\triangle PQR$, we have $\angle R = 180^\circ - \angle Q - \angle P = 105^\circ$.  Since $\overline{ST}\parallel \overline{QR}$, we have $\angle STR = 180^\circ - \angle R = \boxed{75^\circ}$.