Explanation:
Okay, so I need to find the measure of angle α in this hexagon inscribed in a circle. Let me start by recalling some properties of cyclic polygons. Since the hexagon is inscribed in a circle, it's a cyclic hexagon. I remember that in cyclic polygons, the opposite angles have some relationship, but I'm not sure exactly what. Wait, for cyclic quadrilaterals, opposite angles are supplementary. But this is a hexagon, so maybe there's a similar property?

Alternatively, maybe I can use the fact that the measure of an inscribed angle is half the measure of its intercepted arc. That seems more useful here. Let me try to visualize the diagram. The Asymptote code defines points A through F around a circle centered at O. The labels mention angles at points F, B, and D. Specifically, angle at F is labeled 105 degrees, angle at B is 110 degrees, and angle α is at point D. The hexagon is drawn in the order A-B-C-D-E-F-A.

First, let me note that in a cyclic polygon, the sum of all the interior angles can be calculated. For an n-sided polygon, the sum is (n-2)*180 degrees. For a hexagon, that would be (6-2)*180 = 720 degrees. So the sum of all interior angles is 720 degrees. However, since it's cyclic, maybe there's another way to relate the angles?

But maybe I need to use the arcs. Each interior angle of a cyclic polygon is related to the arcs. For a cyclic quadrilateral, each angle is half the sum of the measures of the arcs intercepted by the two opposite angles. Wait, maybe for a cyclic polygon with more sides, each interior angle is half the sum of the measures of the arcs intercepted by the two adjacent sides?

Wait, let me think again. For a cyclic polygon, each interior angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, no. Let me recall that in a cyclic polygon, each interior angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, actually, for a triangle, the inscribed angle theorem says that an angle is half the measure of its intercepted arc. For quadrilaterals, opposite angles intercept arcs that add up to 360 degrees, so they are supplementary. For a pentagon or hexagon, maybe similar relationships?

Alternatively, maybe I can think of each interior angle as half the sum of the intercepted arcs. Wait, let me think of a single interior angle. Let's take angle at point B. In the diagram, angle at B is labeled 110 degrees. Let me try to figure out which arcs it intercepts. In a cyclic polygon, each interior angle is formed by two chords, and the measure of the angle is half the sum of the measures of the intercepted arcs. Wait, no, actually, for an inscribed angle, it's half the measure of the intercepted arc. But for an interior angle of a polygon, it's formed by two adjacent sides, which are chords of the circle. So, the angle at vertex B is formed by chords AB and BC. The intercepted arcs would be the arc from A to C not containing B, right? Wait, no. Wait, in a cyclic polygon, the measure of an interior angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, maybe not. Let me check.

Alternatively, maybe the measure of an interior angle is half the difference of the intercepted arcs? Wait, I need to recall the exact formula. Let me think of a cyclic quadrilateral. In a cyclic quadrilateral, the opposite angles are supplementary. Each angle is half the sum of the intercepted arcs. Wait, if you have a cyclic quadrilateral ABCD, then angle A is half the measure of arc BCD, and angle C is half the measure of arc BAD. Since arc BCD + arc BAD = 360 degrees, then angle A + angle C = 180 degrees. So, in general, for a cyclic quadrilateral, each angle is half the measure of the arc opposite to it. Wait, but in a cyclic quadrilateral, the opposite angles intercept arcs that add up to 360 degrees. So, angle A intercepts arc BCD, which is 360 - arc ABC. Wait, maybe I need to think more carefully.

Alternatively, maybe for any polygon inscribed in a circle, the measure of an interior angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, for example, in a triangle, each angle is half the measure of its intercepted arc. For a quadrilateral, each angle is half the measure of the arc opposite to it? Wait, no. Let me think again.

Let me take a specific example. Let's take angle at point B in the hexagon. The angle at B is formed by chords AB and BC. The intercepted arcs would be arc AC (the arc that's opposite to angle B). Wait, but in a cyclic polygon, the measure of an interior angle is equal to half the measure of the arc that is opposite to it? Wait, no. Wait, in a cyclic quadrilateral, angle A is half the measure of arc BCD, which is the arc opposite to angle A. Similarly, angle B is half the measure of arc CDA. So, in general, for a cyclic polygon, each interior angle is equal to half the measure of the arc that is opposite to it. Wait, but in a hexagon, how does that work?

Alternatively, maybe the measure of an interior angle is half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, but each side is a chord, so each angle is formed by two chords. The intercepted arcs would be the arcs that are opposite to the angle. Wait, maybe I need to use the formula for the measure of an inscribed angle. If I can figure out which arcs are intercepted by the angles, then I can relate them.

Alternatively, maybe I can use the fact that the sum of all the central angles is 360 degrees. Since the hexagon is inscribed in a circle, each side corresponds to a central angle. If I can find the measures of these central angles, then maybe I can relate them to the interior angles.

But perhaps there's a better way. Let me try to think step by step.

First, let me note that in a cyclic polygon, the sum of the measures of the arcs between each pair of adjacent vertices is 360 degrees. So, if I denote the arcs between each pair of consecutive vertices as follows:

Let me label the arcs between the points. Let me denote arc AB as a, arc BC as b, arc CD as c, arc DE as d, arc EF as e, and arc FA as f. Then, since the total circumference is 360 degrees, we have:

a + b + c + d + e + f = 360 degrees.

Now, each interior angle of the hexagon can be related to these arcs. For example, the interior angle at vertex B is formed by chords AB and BC. The measure of angle B is equal to half the sum of the measures of the arcs intercepted by the two angles. Wait, no. Wait, in a cyclic polygon, the measure of an interior angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, but each interior angle is formed by two adjacent sides, which intercept arcs. Wait, actually, for an inscribed angle, the angle is half the measure of its intercepted arc. However, in a polygon, the interior angle is not an inscribed angle unless it's a triangle. Wait, maybe I need to think of the interior angle as an inscribed angle?

Wait, no. Let me think again. Let me take angle at point B. The angle at point B is formed by the chords AB and BC. If I extend those chords, they form an angle at point B. However, in the circle, the measure of angle ABC (the interior angle at B) is equal to half the measure of the arc that is opposite to it, which would be arc AC. Wait, but arc AC is made up of arcs AB + BC? Wait, no. If you have points A, B, C on the circle, then arc AC can be the arc that goes through B or the arc that goes the other way around the circle. Since the polygon is convex (as it's inscribed in a circle), the interior angle at B would intercept the arc AC that does not contain B. Wait, but in that case, the measure of angle ABC would be half the measure of arc AC. Wait, but arc AC is equal to arc AB + arc BC? No, arc AC is the arc from A to C not passing through B. Wait, if the points are in order A, B, C, D, E, F around the circle, then arc AC would be arc AB + arc BC. But if we take the arc AC that does not contain B, then it would be the other arc, which would be arc A to C going the other way around the circle, which would be arc AF + arc FE + arc ED + arc DC? Wait, no, that seems too long. Wait, maybe I need to clarify.

Let me think again. If the hexagon is labeled in order A, B, C, D, E, F around the circle, then the arcs between consecutive points are AB, BC, CD, DE, EF, FA. So, the arc from A to C that does not contain B would be arc A to C going through F, E, D? Wait, but that would be a major arc. However, the measure of angle ABC is an inscribed angle. Wait, but inscribed angles are formed by two chords with the vertex on the circumference. So, angle ABC is formed by chords AB and BC, so the intercepted arc is arc AC that does not contain B. Therefore, the measure of angle ABC is half the measure of arc AC. But arc AC is equal to arc AB + arc BC? Wait, no. If arc AB is a, arc BC is b, then arc AC (the arc that goes through B) is a + b. But the arc AC that does not contain B would be the rest of the circle, which is 360 - (a + b). Therefore, the measure of angle ABC is half of (360 - (a + b))? Wait, that seems too big. Wait, no. Wait, if angle ABC is an inscribed angle intercepting arc AC that does not contain B, then the measure of angle ABC is half the measure of arc AC. But arc AC that does not contain B is equal to 360 - (arc AB + arc BC) = 360 - (a + b). Therefore, angle ABC = 1/2*(360 - a - b). But angle ABC is given as 110 degrees. Therefore:

110 = 1/2*(360 - a - b)

Multiply both sides by 2:

220 = 360 - a - b

Therefore:

a + b = 360 - 220 = 140 degrees.

Similarly, let's consider angle at point F, which is labeled 105 degrees. Let me apply the same logic. Angle at F is formed by chords EF and FA. The intercepted arc is arc EA that does not contain F. Let me check. The arc EA that does not contain F would be arc E to A going through B, C, D? Wait, the points are A, B, C, D, E, F. So, arc from E to A not containing F would be arc E to A passing through B, C, D? Wait, but that would be arc E to B to C to D to A? Wait, but arc EA not containing F would be the arc that goes from E to A without passing through F. Since the points are in order A, B, C, D, E, F around the circle, then the arc from E to A that does not contain F would be the arc going through B, C, D. Wait, but that arc would be arc E to D to C to B to A? Wait, but arc EA not containing F would be the arc that is opposite to F. Let me think again.

Alternatively, maybe the intercepted arc for angle at F is arc ED? Wait, no. Let me think of angle at F. The angle is formed by chords EF and FA. So, the intercepted arc is the arc opposite to angle F, which is arc EA. Wait, but arc EA can be split into arc EF + arc FA? No, arc EA is from E to A. If we go through F, that would be arc EF + arc FA. But if we go the other way, that would be arc ED + DC + CB + BA. Wait, but since the polygon is convex, the intercepted arc for angle F should be the arc that is opposite to angle F, which is the arc that doesn't contain F. Therefore, arc EA not containing F. Since the points are in order A, B, C, D, E, F, then arc EA not containing F would be arc E to A passing through B, C, D. Therefore, the measure of angle F is half the measure of arc EA. Let me denote arc EA as arc ED + DC + CB + BA. Wait, but arc EA can be calculated as arc ED + DC + CB + BA? Wait, arc from E to A through B, C, D would be arc ED + DC + CB + BA? Wait, arc ED is from E to D, then DC is D to C, CB is C to B, BA is B to A. So yes, that adds up to arc EA. However, arc EA can also be expressed as arc EF + arc FA, which is e + f. But since the total circumference is 360 degrees, arc EA (the one not containing F) is equal to 360 - (arc EF + arc FA) = 360 - (e + f). Therefore, angle at F is half of that:

105 = 1/2*(360 - e - f)

Multiply both sides by 2:

210 = 360 - e - f

Therefore:

e + f = 360 - 210 = 150 degrees.

Okay, so now I have two equations:

1. a + b = 140 degrees (from angle at B)
2. e + f = 150 degrees (from angle at F)

Now, I need more equations. Let me consider angle α, which is at point D. The problem asks for the measure of α. Let me try to find an equation involving α. Let me apply the same logic. Angle at D is formed by chords CD and DE. The intercepted arc is arc CE that does not contain D. Let me figure out what arc CE is. The arc from C to E not containing D. Since the points are in order A, B, C, D, E, F, the arc from C to E not containing D would be arc C to E passing through F, A, B? Wait, no. If we go from C to E not containing D, then since D is between C and E, the arc that does not contain D would be the other arc from C to E, which goes through F, A, B. Wait, but that would be arc C to F to A to B to E? Wait, that seems complicated. Let me think again.

Alternatively, arc CE not containing D would be the arc that goes from C to E passing through F, A, B? Wait, but that would be a major arc. Let me calculate its measure. The measure of arc CE not containing D is equal to 360 degrees minus the measure of arc CD + arc DE. Wait, arc CD is c, arc DE is d. Therefore, arc CE not containing D is 360 - (c + d). Therefore, angle at D is half of that:

α = 1/2*(360 - c - d)

Therefore, α = 180 - (c + d)/2.

So, if I can find c + d, then I can find α. Alternatively, maybe I can find other equations involving these arcs.

Let me also consider other angles. However, the problem only gives me two angles: angle at B is 110 degrees, angle at F is 105 degrees, and angle α at D. Maybe there are other angles that I can relate? Wait, the Asymptote code labels angle at F as 105 degrees, angle at B as 110 degrees, and angle α at D. Maybe there are other angles that are not labeled? But since the problem only gives me two angles, maybe I need to find relations between the arcs.

We have the total of all arcs:

a + b + c + d + e + f = 360 degrees.

From earlier, we have:

a + b = 140 degrees,

e + f = 150 degrees.

Therefore, substituting into the total:

140 + c + d + 150 = 360

So, c + d = 360 - 140 - 150 = 70 degrees.

Therefore, c + d = 70 degrees.

Then, going back to the expression for α:

α = 180 - (c + d)/2 = 180 - 70/2 = 180 - 35 = 145 degrees.

Wait, so is α equal to 145 degrees? Let me check my steps again to make sure.

First, angle at B: intercepted arc is arc AC not containing B, which is 360 - (a + b). Therefore, angle B = 1/2*(360 - a - b) = 110 degrees. Therefore, 360 - a - b = 220, so a + b = 140. Correct.

Angle at F: intercepted arc is arc EA not containing F, which is 360 - (e + f). Therefore, angle F = 1/2*(360 - e - f) = 105 degrees. Therefore, 360 - e - f = 210, so e + f = 150. Correct.

Total arcs: a + b + c + d + e + f = 360. Therefore, 140 + c + d + 150 = 360 => c + d = 70. Correct.

Then, angle at D: intercepted arc is arc CE not containing D, which is 360 - (c + d). Therefore, angle D = 1/2*(360 - c - d) = 1/2*(360 - 70) = 1/2*(290) = 145 degrees. Therefore, α = 145 degrees. That seems straightforward.

But let me check if there's another way to get this, or if I made an assumption that might be wrong.

Wait, let me verify the intercepted arc for angle D. The angle at D is formed by chords CD and DE. The intercepted arc should be the arc opposite to angle D, which is arc CE. But is that arc CE the one not containing D? Yes, because if you have angle at D, the intercepted arc is the arc that is opposite, which doesn't include D. Since the points are in order C, D, E, the arc CE that doesn't contain D would be the arc going the other way around the circle, passing through F, A, B. Therefore, its measure is 360 - (arc CD + arc DE) = 360 - (c + d). Therefore, angle D is half of that. So, yes, that seems correct.

Therefore, with c + d = 70 degrees, angle D is 1/2*(360 - 70) = 1/2*(290) = 145 degrees. Therefore, α is 145 degrees. Is that the answer? Wait, but let me check if there are other angles that I can use to verify.

Alternatively, maybe I can find other angles. For example, if I can find the measure of angle at A, or angle at C, or angle at E, but since they are not given, maybe I can't. However, let me check if the total sum of all interior angles is 720 degrees. Let me see.

We have angles at B = 110°, at F = 105°, and at D = 145°. Let me assume that the other angles are something else. Let me try to compute the sum.

But maybe I can find expressions for the other angles in terms of arcs. Let me try.

For example, angle at A: formed by chords FA and AB. The intercepted arc is arc FB not containing A. Wait, similar to before. Let me think. The intercepted arc for angle at A would be arc opposite to A, which is arc from F to B not containing A. Since the points are in order A, B, C, D, E, F, the arc from F to B not containing A would be arc F to E to D to C to B. Which is arc FE + ED + DC + CB. Which is e + d + c + b. Therefore, measure of angle at A is 1/2*(e + d + c + b). But since a + b = 140, and e + f = 150, and c + d = 70, then e + d + c + b = (e + f) + (c + d) - f + b? Wait, maybe not. Let me think again.

Wait, angle at A intercepts arc FB not containing A. The measure of arc FB not containing A is equal to 360 - arc AB - arc AF? Wait, arc AB is a, arc AF is f. Wait, arc from F to B not containing A would be arc F to B passing through E, D, C. Which is arc FE + ED + DC + CB. Which is e + d + c + b. Alternatively, since the total circumference is 360, arc FB not containing A is equal to 360 - arc AB - arc AF? Wait, arc AB is a, arc AF is f. But arc AB + arc AF is a + f. However, arc FB not containing A is 360 - arc AB - arc AF? Wait, no. If you have arc AB and arc AF, they are adjacent? Wait, arc AB is from A to B, arc AF is from A to F. So, arc FB not containing A would be the rest of the circle, which is 360 - arc AB - arc AF? Wait, no. If you have points A, B, C, D, E, F on the circle, then arc from F to B not containing A would be arc F to E to D to C to B, which is arc FE + ED + DC + CB. Which is e + d + c + b. Alternatively, since arc AB is a, arc BC is b, arc CD is c, arc DE is d, arc EF is e, arc FA is f. Then, arc FB not containing A is arc FE + ED + DC + CB = e + d + c + b. Therefore, measure of angle at A is 1/2*(e + d + c + b). But since e + f = 150, and c + d = 70, and a + b = 140. Let me see:

e + d + c + b = (e + f) + (c + d) - f + b? Wait, no. Let me express e + d + c + b as (e + f) + (c + d) - f + b? Wait, maybe not. Let me think:

We have e + d + c + b = (e + f) + (c + d) - f + b? Wait, no. Let me think of e + d + c + b = (e + d + c + b). Since we know that a + b = 140, and e + f = 150, and c + d = 70. Let me see:

If I add up all the arcs:

a + b + c + d + e + f = 360.

We have:

a + b = 140,

c + d = 70,

e + f = 150.

So, adding them up: 140 + 70 + 150 = 360. Correct.

Therefore, e + d + c + b = (e + f) + (c + d) - f + b? Wait, no. Let me think:

e + d + c + b = (e + f) + (c + d) - f + b? Wait, that seems convoluted. Let me think of e + d + c + b = (e + d + c + b). Since we know that a + b = 140, so b = 140 - a. Similarly, e + f = 150, so e = 150 - f. But maybe that's not helpful. Alternatively, since e + d + c + b = (e + f) + (c + d) - f + b? Wait, no. Let me think:

If I take e + d + c + b, that's equal to (e + f) + (c + d) - f + b - something? Wait, maybe not. Let me think of e + d + c + b = (e + d + c + b). Since we know that e + f = 150, so e = 150 - f. Similarly, c + d = 70. And b is part of a + b = 140. But unless we have more information, maybe we can't find e + d + c + b. Wait, but maybe we can express it in terms of known quantities. Let me see:

We have:

a + b = 140,

c + d = 70,

e + f = 150.

Therefore, e + d + c + b = (e + f) + (c + d) - f + b? Wait, no. Let me think:

If I take e + d + c + b, that's equal to (e + f) + (c + d) - f + b - something? Wait, maybe not. Let me think of e + d + c + b = (e + d + c + b). Since e + f = 150, then e = 150 - f. Similarly, since c + d = 70, then c + d = 70. And b is part of a + b = 140. But unless we have more info, maybe we can't find e + d + c + b. Wait, but maybe we can use the total sum. Since a + b + c + d + e + f = 360, then e + d + c + b = 360 - a - f. But since a + b = 140, then a = 140 - b. Therefore, e + d + c + b = 360 - (140 - b) - f = 360 - 140 + b - f = 220 + b - f. But since e + f = 150, then f = 150 - e. Therefore, substituting:

e + d + c + b = 220 + b - (150 - e) = 220 + b - 150 + e = 70 + b + e. But since e + f = 150, and f = 150 - e, but I don't know if that helps. This seems too convoluted. Maybe I need another approach.

Alternatively, since angle at A is 1/2*(e + d + c + b). But since we don't know angle at A, maybe we can't use that. However, if I consider that the sum of all interior angles is 720 degrees, and we know three angles: angle B = 110°, angle F = 105°, angle D = 145°, then the sum of the remaining three angles (A, C, E) is 720 - 110 - 105 - 145 = 720 - 360 = 360 degrees. So angles A + C + E = 360 degrees. But without more information, I can't find individual angles. However, maybe I can find expressions for angles A, C, E in terms of arcs.

Let me try. For angle at A: as above, angle A = 1/2*(arc FB not containing A) = 1/2*(e + d + c + b). Similarly, angle at C: formed by chords BC and CD. The intercepted arc is arc BD not containing C. Let me calculate that. Arc BD not containing C would be arc B to D passing through E, F, A. Which is arc BE + EF + FA + AB? Wait, no. From B to D not containing C would be arc B to E to F to A to D? Wait, points are A, B, C, D, E, F. So arc from B to D not containing C would be arc B to E to F to A to D? Wait, that seems too long. Alternatively, arc BD not containing C is the arc that goes from B to D passing through E, F, A. Which is arc BE + EF + FA + AD? Wait, no. Let me think again. The arc BD not containing C would be the arc that goes from B to D without passing through C. Since the points are in order A, B, C, D, E, F, the arc from B to D not containing C would be the arc going through E, F, A. Therefore, arc BE + EF + FA + AD? Wait, no. From B to D, passing through E, F, A. So, arc from B to E is arc BE, which is arc BC + CD + DE? Wait, no. Wait, arc from B to E not containing C would be arc B to E passing through F, A, D? Wait, this is getting too confusing. Maybe there's a better way.

Alternatively, since angle at C is formed by chords BC and CD. The intercepted arc is arc BD not containing C. The measure of angle C is half the measure of arc BD. Since arc BD can be calculated as arc BC + CD? No, that would be arc BD passing through C. The arc BD not containing C is the other arc, which is 360 - (arc BC + arc CD) = 360 - (b + c). Therefore, angle at C is 1/2*(360 - b - c). Similarly, angle at E is formed by chords DE and EF. The intercepted arc is arc DF not containing E. Which would be 360 - (arc DE + arc EF) = 360 - (d + e). Therefore, angle at E is 1/2*(360 - d - e).

So, angles:

Angle A = 1/2*(e + d + c + b)

Angle C = 1/2*(360 - b - c)

Angle E = 1/2*(360 - d - e)

Therefore, sum of angles A + C + E:

= 1/2*(e + d + c + b) + 1/2*(360 - b - c) + 1/2*(360 - d - e)

Let me compute this:

First term: 1/2*(e + d + c + b)

Second term: 1/2*(360 - b - c)

Third term: 1/2*(360 - d - e)

Adding them together:

= [ (e + d + c + b) + (360 - b - c) + (360 - d - e) ] / 2

Simplify numerator:

e + d + c + b + 360 - b - c + 360 - d - e

Let me combine like terms:

e - e = 0

d - d = 0

c - c = 0

b - b = 0

Then, 360 + 360 = 720

Therefore, numerator is 720, so total sum is 720 / 2 = 360 degrees. Which matches our previous result that angles A + C + E = 360 degrees. Therefore, this doesn't give us new information. So, we can't find individual angles unless we have more equations.

But since we already found α = 145 degrees through the previous method, and the steps seem consistent, maybe that is the correct answer. However, let me check if there's another way to verify.

Alternatively, maybe I can use the fact that the sum of all arcs is 360 degrees, and we have:

a + b = 140,

c + d = 70,

e + f = 150.

Therefore, if I can find relations between other angles, but since we don't have more angles given, maybe not. However, let me check if angle α is indeed 145 degrees.

Alternatively, maybe I can think of the hexagon as composed of triangles or something else. But since it's cyclic, maybe not necessary.

Alternatively, let me check if the answer makes sense. If angle at D is 145 degrees, which is quite large, but given that the other angles are 110 and 105, which are also relatively large, maybe it's possible. Let me check if the total sum of angles would be okay.

We have angles at B = 110°, F = 105°, D = 145°, and angles A, C, E sum to 360°. So total sum is 110 + 105 + 145 + 360 = 720°, which is correct. So that seems okay.

Alternatively, maybe I can check with specific values. Let me assume some values for the arcs. For example, suppose that c + d = 70°, so let me pick arbitrary values for c and d. Let me say c = 30°, d = 40°, just for example. Then, angle at D would be 1/2*(360 - 30 - 40) = 1/2*(290) = 145°, which matches. Then, with c = 30°, d = 40°, then a + b = 140°, e + f = 150°, and total arcs sum to 360°. So, that seems okay. Therefore, regardless of the specific values of c and d, as long as their sum is 70°, angle D is 145°, so that seems consistent.

Therefore, I think my answer is correct. Therefore, α is 145 degrees.

**Final Answer**
\boxed{145}

To determine the measure of angle $\alpha$ in the given cyclic hexagon, we utilize the geometric properties of inscribed angles in a circle.

---

### Step 1: Understand the Geometry

We are given a **cyclic hexagon** inscribed in a circle. This means all six vertices lie on the circumference of the circle. The key property we use is that the **measure of an inscribed angle** is **half the measure of the arc it intercepts**.

Let’s denote the arcs between consecutive vertices as follows:

- Arc $AB = a$
- Arc $BC = b$
- Arc $CD = c$
- Arc $DE = d$
- Arc $EF = e$
- Arc $FA = f$

Since the total circumference of the circle is $360^\circ$, we have:

$$
a + b + c + d + e + f = 360^\circ
$$

---

### Step 2: Use Given Angles to Form Equations

We are given:

- $\angle B = 110^\circ$
- $\angle F = 105^\circ$
- $\angle D = \alpha$ (to be found)

#### For $\angle B$:

This angle is formed by chords $AB$ and $BC$, and it intercepts the arc $AC$ that **does not contain** point $B$. The measure of this arc is:

$$
\text{Arc } AC = 360^\circ - (a + b)
$$

So:

$$
\angle B = \frac{1}{2}(360^\circ - a - b) = 110^\circ
$$

Multiply both sides by 2:

$$
360^\circ - a - b = 220^\circ \Rightarrow a + b = 140^\circ
$$

#### For $\angle F$:

This angle is formed by chords $EF$ and $FA$, and it intercepts the arc $EA$ that **does not contain** point $F$. The measure of this arc is:

$$
\text{Arc } EA = 360^\circ - (e + f)
$$

So:

$$
\angle F = \frac{1}{2}(360^\circ - e - f) = 105^\circ
$$

Multiply both sides by 2:

$$
360^\circ - e - f = 210^\circ \Rightarrow e + f = 150^\circ
$$

---

### Step 3: Use Total Arc Sum to Find $c + d$

We now use the total arc sum:

$$
a + b + c + d + e + f = 360^\circ
$$

Substitute the known values:

$$
140^\circ + c + d + 150^\circ = 360^\circ \Rightarrow c + d = 70^\circ
$$

---

### Step 4: Use Arcs to Find $\alpha = \angle D$

Angle $\alpha$ is formed by chords $CD$ and $DE$, and it intercepts the arc $CE$ that **does not contain** point $D$. The measure of this arc is:

$$
\text{Arc } CE = 360^\circ - (c + d)
$$

So:

$$
\alpha = \frac{1}{2}(360^\circ - c - d) = \frac{1}{2}(360^\circ - 70^\circ) = \frac{1}{2}(290^\circ) = 145^\circ
$$

---

### Final Answer

$$
\boxed{145}
$$

[Assistant]
Explanation:
Okay, so I need to find the measure of angle α in this hexagon inscribed in a circle. Let me start by recalling some properties of cyclic polygons. Since the hexagon is inscribed in a circle, it's a cyclic hexagon. I remember that in cyclic polygons, the opposite angles have some relationship, but I'm not sure exactly what. Wait, for cyclic quadrilaterals, opposite angles are supplementary. But this is a hexagon, so maybe there's a similar property?

Alternatively, maybe I can use the fact that the measure of an inscribed angle is half the measure of its intercepted arc. That seems more useful here. Let me try to visualize the diagram. The Asymptote code defines points A through F around a circle centered at O. The labels mention angles at points F, B, and D. Specifically, angle at F is labeled 105 degrees, angle at B is 110 degrees, and angle α is at point D. The hexagon is drawn in the order A-B-C-D-E-F-A.

First, let me note that in a cyclic polygon, the sum of all the interior angles can be calculated. For an n-sided polygon, the sum is (n-2)*180 degrees. For a hexagon, that would be (6-2)*180 = 720 degrees. So the sum of all interior angles is 720 degrees. However, since it's cyclic, maybe there's another way to relate the angles?

But maybe I need to use the arcs. Each interior angle of a cyclic polygon is related to the arcs. For a cyclic quadrilateral, each angle is half the sum of the measures of the arcs intercepted by the two opposite angles. Wait, maybe for a cyclic polygon with more sides, each interior angle is half the sum of the measures of the arcs intercepted by the two adjacent sides?

Wait, let me think again. For a cyclic polygon, each interior angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, no. Let me recall that in a cyclic polygon, each interior angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, actually, for a triangle, the inscribed angle theorem says that an angle is half the measure of its intercepted arc. For quadrilaterals, opposite angles intercept arcs that add up to 360 degrees, so they are supplementary. For a pentagon or hexagon, maybe similar relationships?

Alternatively, maybe I can think of each interior angle as half the sum of the intercepted arcs. Wait, let me think of a single interior angle. Let's take angle at point B. In the diagram, angle at B is labeled 110 degrees. Let me try to figure out which arcs it intercepts. In a cyclic polygon, each interior angle is formed by two chords, and the measure of the angle is half the sum of the measures of the intercepted arcs. Wait, no, actually, for an inscribed angle, it's half the measure of the intercepted arc. But for an interior angle of a polygon, it's formed by two adjacent sides, which are chords of the circle. So, the angle at vertex B is formed by chords AB and BC. The intercepted arcs would be the arc from A to C not containing B, right? Wait, no. Wait, in a cyclic polygon, the measure of an interior angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, maybe not. Let me check.

Alternatively, maybe the measure of an interior angle is half the difference of the intercepted arcs? Wait, I need to recall the exact formula. Let me think of a cyclic quadrilateral. In a cyclic quadrilateral, the opposite angles are supplementary. Each angle is half the sum of the intercepted arcs. Wait, if you have a cyclic quadrilateral ABCD, then angle A is half the measure of arc BCD, and angle C is half the measure of arc BAD. Since arc BCD + arc BAD = 360 degrees, then angle A + angle C = 180 degrees. So, in general, for a cyclic quadrilateral, each angle is half the measure of the arc opposite to it. Wait, but in a cyclic quadrilateral, the opposite angles intercept arcs that add up to 360 degrees. So, angle A intercepts arc BCD, which is 360 - arc ABC. Wait, maybe I need to think more carefully.

Alternatively, maybe for any polygon inscribed in a circle, the measure of an interior angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, for example, in a triangle, each angle is half the measure of its intercepted arc. For a quadrilateral, each angle is half the measure of the arc opposite to it? Wait, no. Let me think again.

Let me take a specific example. Let's take angle at point B in the hexagon. The angle at B is formed by chords AB and BC. The intercepted arcs would be arc AC (the arc that's opposite to angle B). Wait, but in a cyclic polygon, the measure of an interior angle is equal to half the measure of the arc that is opposite to it? Wait, no. Wait, in a cyclic quadrilateral, angle A is half the measure of arc BCD, which is the arc opposite to angle A. Similarly, angle B is half the measure of arc CDA. So, in general, for a cyclic polygon, each interior angle is equal to half the measure of the arc that is opposite to it. Wait, but in a hexagon, how does that work?

Alternatively, maybe the measure of an interior angle is half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, but each side is a chord, so each angle is formed by two chords. The intercepted arcs would be the arcs that are opposite to the angle. Wait, maybe I need to use the formula for the measure of an inscribed angle. If I can figure out which arcs are intercepted by the angles, then I can relate them.

Alternatively, maybe I can use the fact that the sum of all the central angles is 360 degrees. Since the hexagon is inscribed in a circle, each side corresponds to a central angle. If I can find the measures of these central angles, then maybe I can relate them to the interior angles.

But perhaps there's a better way. Let me try to think step by step.

First, let me note that in a cyclic polygon, the sum of the measures of the arcs between each pair of adjacent vertices is 360 degrees. So, if I denote the arcs between each pair of consecutive vertices as follows:

Let me label the arcs between the points. Let me denote arc AB as a, arc BC as b, arc CD as c, arc DE as d, arc EF as e, and arc FA as f. Then, since the total circumference is 360 degrees, we have:

a + b + c + d + e + f = 360 degrees.

Now, each interior angle of the hexagon can be related to these arcs. For example, the interior angle at vertex B is formed by chords AB and BC. The measure of angle B is equal to half the sum of the measures of the arcs intercepted by the two angles. Wait, no. Wait, in a cyclic polygon, the measure of an interior angle is equal to half the sum of the measures of the arcs intercepted by the two adjacent sides. Wait, but each interior angle is formed by two adjacent sides, which intercept arcs. Wait, actually, for an inscribed angle, the angle is half the measure of its intercepted arc. However, in a polygon, the interior angle is not an inscribed angle unless it's a triangle. Wait, maybe I need to think of the interior angle as an inscribed angle?

Wait, no. Let me think again. Let me take angle at point B. The angle at point B is formed by the chords AB and BC. If I extend those chords, they form an angle at point B. However, in the circle, the measure of angle ABC (the interior angle at B) is equal to half the measure of the arc that is opposite to it, which would be arc AC. Wait, but arc AC is made up of arcs AB + BC? Wait, no. If you have points A, B, C on the circle, then arc AC can be the arc that goes through B or the arc that goes the other way around the circle. Since the polygon is convex (as it's inscribed in a circle), the interior angle at B would intercept the arc AC that does not contain B. Wait, but in that case, the measure of angle ABC would be half the measure of arc AC. Wait, but arc AC is equal to arc AB + arc BC? No, arc AC is the arc from A to C not passing through B. Wait, if the points are in order A, B, C, D, E, F around the circle, then arc AC would be arc AB + arc BC. But if we take the arc AC that does not contain B, then it would be the other arc, which would be arc A to C going the other way around the circle, which would be arc AF + arc FE + arc ED + arc DC? Wait, no, that seems too long. Wait, maybe I need to clarify.

Let me think again. If the hexagon is labeled in order A, B, C, D, E, F around the circle, then the arcs between consecutive points are AB, BC, CD, DE, EF, FA. So, the arc from A to C that does not contain B would be arc A to C going through F, E, D? Wait, but that would be a major arc. However, the measure of angle ABC is an inscribed angle. Wait, but inscribed angles are formed by two chords with the vertex on the circumference. So, angle ABC is formed by chords AB and BC, so the intercepted arc is arc AC that does not contain B. Therefore, the measure of angle ABC is half the measure of arc AC. But arc AC is equal to arc AB + arc BC? Wait, no. If arc AB is a, arc BC is b, then arc AC (the arc that goes through B) is a + b. But the arc AC that does not contain B would be the rest of the circle, which is 360 - (a + b). Therefore, the measure of angle ABC is half of (360 - (a + b))? Wait, that seems too big. Wait, no. Wait, if angle ABC is an inscribed angle intercepting arc AC that does not contain B, then the measure of angle ABC is half the measure of arc AC. But arc AC that does not contain B is equal to 360 - (arc AB + arc BC) = 360 - (a + b). Therefore, angle ABC = 1/2*(360 - a - b). But angle ABC is given as 110 degrees. Therefore:

110 = 1/2*(360 - a - b)

Multiply both sides by 2:

220 = 360 - a - b

Therefore:

a + b = 360 - 220 = 140 degrees.

Similarly, let's consider angle at point F, which is labeled 105 degrees. Let me apply the same logic. Angle at F is formed by chords EF and FA. The intercepted arc is arc EA that does not contain F. Let me check. The arc EA that does not contain F would be arc E to A going through B, C, D? Wait, the points are A, B, C, D, E, F. So, arc from E to A not containing F would be arc E to A passing through B, C, D? Wait, but that would be arc E to B to C to D to A? Wait, but arc EA not containing F would be the arc that goes from E to A without passing through F. Since the points are in order A, B, C, D, E, F around the circle, then the arc from E to A that does not contain F would be the arc going through B, C, D. Wait, but that arc would be arc E to D to C to B to A? Wait, but arc EA not containing F would be the arc that is opposite to F. Let me think again.

Alternatively, maybe the intercepted arc for angle at F is arc ED? Wait, no. Let me think of angle at F. The angle is formed by chords EF and FA. So, the intercepted arc is the arc opposite to angle F, which is arc EA. Wait, but arc EA can be split into arc EF + arc FA? No, arc EA is from E to A. If we go through F, that would be arc EF + arc FA. But if we go the other way, that would be arc ED + DC + CB + BA. Wait, but since the polygon is convex, the intercepted arc for angle F should be the arc that is opposite to angle F, which is the arc that doesn't contain F. Therefore, arc EA not containing F. Since the points are in order A, B, C, D, E, F, then arc EA not containing F would be arc E to A passing through B, C, D. Therefore, the measure of angle F is half the measure of arc EA. Let me denote arc EA as arc ED + DC + CB + BA. Wait, but arc EA can be calculated as arc ED + DC + CB + BA? Wait, arc from E to A through B, C, D would be arc ED + DC + CB + BA? Wait, arc ED is from E to D, then DC is D to C, CB is C to B, BA is B to A. So yes, that adds up to arc EA. However, arc EA can also be expressed as arc EF + arc FA, which is e + f. But since the total circumference is 360 degrees, arc EA (the one not containing F) is equal to 360 - (arc EF + arc FA) = 360 - (e + f). Therefore, angle at F is half of that:

105 = 1/2*(360 - e - f)

Multiply both sides by 2:

210 = 360 - e - f

Therefore:

e + f = 360 - 210 = 150 degrees.

Okay, so now I have two equations:

1. a + b = 140 degrees (from angle at B)
2. e + f = 150 degrees (from angle at F)

Now, I need more equations. Let me consider angle α, which is at point D. The problem asks for the measure of α. Let me try to find an equation involving α. Let me apply the same logic. Angle at D is formed by chords CD and DE. The intercepted arc is arc CE that does not contain D. Let me figure out what arc CE is. The arc from C to E not containing D. Since the points are in order A, B, C, D, E, F, the arc from C to E not containing D would be arc C to E passing through F, A, B? Wait, no. If we go from C to E not containing D, then since D is between C and E, the arc that does not contain D would be the other arc from C to E, which goes through F, A, B. Wait, but that would be arc C to F to A to B to E? Wait, that seems complicated. Let me think again.

Alternatively, arc CE not containing D would be the arc that goes from C to E passing through F, A, B? Wait, but that would be a major arc. Let me calculate its measure. The measure of arc CE not containing D is equal to 360 degrees minus the measure of arc CD + arc DE. Wait, arc CD is c, arc DE is d. Therefore, arc CE not containing D is 360 - (c + d). Therefore, angle at D is half of that:

α = 1/2*(360 - c - d)

Therefore, α = 180 - (c + d)/2.

So, if I can find c + d, then I can find α. Alternatively, maybe I can find other equations involving these arcs.

Let me also consider other angles. However, the problem only gives me two angles: angle at B is 110 degrees, angle at F is 105 degrees, and angle α at D. Maybe there are other angles that I can relate? Wait, the Asymptote code labels angle at F as 105 degrees, angle at B as 110 degrees, and angle α at D. Maybe there are other angles that are not labeled? But since the problem only gives me two angles, maybe I need to find relations between the arcs.

We have the total of all arcs:

a + b + c + d + e + f = 360 degrees.

From earlier, we have:

a + b = 140 degrees,

e + f = 150 degrees.

Therefore, substituting into the total:

140 + c + d + 150 = 360

So, c + d = 360 - 140 - 150 = 70 degrees.

Therefore, c + d = 70 degrees.

Then, going back to the expression for α:

α = 180 - (c + d)/2 = 180 - 70/2 = 180 - 35 = 145 degrees.

Wait, so is α equal to 145 degrees? Let me check my steps again to make sure.

First, angle at B: intercepted arc is arc AC not containing B, which is 360 - (a + b). Therefore, angle B = 1/2*(360 - a - b) = 110 degrees. Therefore, 360 - a - b = 220, so a + b = 140. Correct.

Angle at F: intercepted arc is arc EA not containing F, which is 360 - (e + f). Therefore, angle F = 1/2*(360 - e - f) = 105 degrees. Therefore, 360 - e - f = 210, so e + f = 150. Correct.

Total arcs: a + b + c + d + e + f = 360. Therefore, 140 + c + d + 150 = 360 => c + d = 70. Correct.

Then, angle at D: intercepted arc is arc CE not containing D, which is 360 - (c + d). Therefore, angle D = 1/2*(360 - c - d) = 1/2*(360 - 70) = 1/2*(290) = 145 degrees. Therefore, α = 145 degrees. That seems straightforward.

But let me check if there's another way to get this, or if I made an assumption that might be wrong.

Wait, let me verify the intercepted arc for angle D. The angle at D is formed by chords CD and DE. The intercepted arc should be the arc opposite to angle D, which is arc CE. But is that arc CE the one not containing D? Yes, because if you have angle at D, the intercepted arc is the arc that is opposite, which doesn't include D. Since the points are in order C, D, E, the arc CE that doesn't contain D would be the arc going the other way around the circle, passing through F, A, B. Therefore, its measure is 360 - (arc CD + arc DE) = 360 - (c + d). Therefore, angle D is half of that. So, yes, that seems correct.

Therefore, with c + d = 70 degrees, angle D is 1/2*(360 - 70) = 1/2*(290) = 145 degrees. Therefore, α is 145 degrees. Is that the answer? Wait, but let me check if there are other angles that I can use to verify.

Alternatively, maybe I can find other angles. For example, if I can find the measure of angle at A, or angle at C, or angle at E, but since they are not given, maybe I can't. However, let me check if the total sum of all interior angles is 720 degrees. Let me see.

We have angles at B = 110°, at F = 105°, and at D = 145°. Let me assume that the other angles are something else. Let me try to compute the sum.

But maybe I can find expressions for the other angles in terms of arcs. Let me try.

For example, angle at A: formed by chords FA and AB. The intercepted arc is arc FB not containing A. Wait, similar to before. Let me think. The intercepted arc for angle at A would be arc opposite to A, which is arc from F to B not containing A. Since the points are in order A, B, C, D, E, F, the arc from F to B not containing A would be arc F to E to D to C to B. Which is arc FE + ED + DC + CB. Which is e + d + c + b. Therefore, measure of angle at A is 1/2*(e + d + c + b). But since a + b = 140, and e + f = 150, and c + d = 70, then e + d + c + b = (e + f) + (c + d) - f + b? Wait, maybe not. Let me think again.

Wait, angle at A intercepts arc FB not containing A. The measure of arc FB not containing A is equal to 360 - arc AB - arc AF? Wait, arc AB is a, arc AF is f. Wait, arc from F to B not containing A would be arc F to B passing through E, D, C. Which is arc FE + ED + DC + CB. Which is e + d + c + b. Alternatively, since the total circumference is 360, arc FB not containing A is equal to 360 - arc AB - arc AF? Wait, arc AB is a, arc AF is f. But arc AB + arc AF is a + f. However, arc FB not containing A is 360 - arc AB - arc AF? Wait, no. If you have arc AB and arc AF, they are adjacent? Wait, arc AB is from A to B, arc AF is from A to F. So, arc FB not containing A would be the rest of the circle, which is 360 - arc AB - arc AF? Wait, no. If you have points A, B, C, D, E, F on the circle, then arc from F to B not containing A would be arc F to E to D to C to B, which is arc FE + ED + DC + CB. Which is e + d + c + b. Alternatively, since arc AB is a, arc BC is b, arc CD is c, arc DE is d, arc EF is e, arc FA is f. Then, arc FB not containing A is arc FE + ED + DC + CB = e + d + c + b. Therefore, measure of angle at A is 1/2*(e + d + c + b). But since e + f = 150, and c + d = 70, and a + b = 140. Let me see:

e + d + c + b = (e + f) + (c + d) - f + b? Wait, no. Let me express e + d + c + b as (e + f) + (c + d) - f + b? Wait, maybe not. Let me think:

We have e + d + c + b = (e + f) + (c + d) - f + b? Wait, no. Let me think of e + d + c + b = (e + d + c + b). Since we know that a + b = 140, and e + f = 150, and c + d = 70. Let me see:

If I add up all the arcs:

a + b + c + d + e + f = 360.

We have:

a + b = 140,

c + d = 70,

e + f = 150.

So, adding them up: 140 + 70 + 150 = 360. Correct.

Therefore, e + d + c + b = (e + f) + (c + d) - f + b? Wait, no. Let me think:

e + d + c + b = (e + f) + (c + d) - f + b? Wait, that seems convoluted. Let me think of e + d + c + b = (e + d + c + b). Since we know that a + b = 140, so b = 140 - a. Similarly, e + f = 150, so e = 150 - f. But maybe that's not helpful. Alternatively, since e + d + c + b = (e + f) + (c + d) - f + b? Wait, no. Let me think:

If I take e + d + c + b, that's equal to (e + f) + (c + d) - f + b - something? Wait, maybe not. Let me think of e + d + c + b = (e + d + c + b). Since we know that e + f = 150, so e = 150 - f. Similarly, c + d = 70. And b is part of a + b = 140. But unless we have more information, maybe we can't find e + d + c + b. Wait, but maybe we can express it in terms of known quantities. Let me see:

We have:

a + b = 140,

c + d = 70,

e + f = 150.

Therefore, e + d + c + b = (e + f) + (c + d) - f + b? Wait, no. Let me think:

If I take e + d + c + b, that's equal to (e + f) + (c + d) - f + b - something? Wait, maybe not. Let me think of e + d + c + b = (e + d + c + b). Since e + f = 150, then e = 150 - f. Similarly, since c + d = 70, then c + d = 70. And b is part of a + b = 140. But unless we have more info, maybe we can't find e + d + c + b. Wait, but maybe we can use the total sum. Since a + b + c + d + e + f = 360, then e + d + c + b = 360 - a - f. But since a + b = 140, then a = 140 - b. Therefore, e + d + c + b = 360 - (140 - b) - f = 360 - 140 + b - f = 220 + b - f. But since e + f = 150, then f = 150 - e. Therefore, substituting:

e + d + c + b = 220 + b - (150 - e) = 220 + b - 150 + e = 70 + b + e. But since e + f = 150, and f = 150 - e, but I don't know if that helps. This seems too convoluted. Maybe I need another approach.

Alternatively, since angle at A is 1/2*(e + d + c + b). But since we don't know angle at A, maybe we can't use that. However, if I consider that the sum of all interior angles is 720 degrees, and we know three angles: angle B = 110°, angle F = 105°, angle D = 145°, then the sum of the remaining three angles (A, C, E) is 720 - 110 - 105 - 145 = 720 - 360 = 360 degrees. So angles A + C + E = 360 degrees. But without more information, I can't find individual angles. However, maybe I can find expressions for angles A, C, E in terms of arcs.

Let me try. For angle at A: as above, angle A = 1/2*(arc FB not containing A) = 1/2*(e + d + c + b). Similarly, angle at C: formed by chords BC and CD. The intercepted arc is arc BD not containing C. Let me calculate that. Arc BD not containing C would be arc B to D passing through E, F, A. Which is arc BE + EF + FA + AB? Wait, no. From B to D not containing C would be arc B to E to F to A to D? Wait, points are A, B, C, D, E, F. So arc from B to D not containing C would be arc B to E to F to A to D? Wait, that seems too long. Alternatively, arc BD not containing C is the arc that goes from B to D passing through E, F, A. Which is arc BE + EF + FA + AD? Wait, no. Let me think again. The arc BD not containing C would be the arc that goes from B to D without passing through C. Since the points are in order A, B, C, D, E, F, the arc from B to D not containing C would be the arc going through E, F, A. Therefore, arc BE + EF + FA + AD? Wait, no. From B to D, passing through E, F, A. So, arc from B to E is arc BE, which is arc BC + CD + DE? Wait, no. Wait, arc from B to E not containing C would be arc B to E passing through F, A, D? Wait, this is getting too confusing. Maybe there's a better way.

Alternatively, since angle at C is formed by chords BC and CD. The intercepted arc is arc BD not containing C. The measure of angle C is half the measure of arc BD. Since arc BD can be calculated as arc BC + CD? No, that would be arc BD passing through C. The arc BD not containing C is the other arc, which is 360 - (arc BC + arc CD) = 360 - (b + c). Therefore, angle at C is 1/2*(360 - b - c). Similarly, angle at E is formed by chords DE and EF. The intercepted arc is arc DF not containing E. Which would be 360 - (arc DE + arc EF) = 360 - (d + e). Therefore, angle at E is 1/2*(360 - d - e).

So, angles:

Angle A = 1/2*(e + d + c + b)

Angle C = 1/2*(360 - b - c)

Angle E = 1/2*(360 - d - e)

Therefore, sum of angles A + C + E:

= 1/2*(e + d + c + b) + 1/2*(360 - b - c) + 1/2*(360 - d - e)

Let me compute this:

First term: 1/2*(e + d + c + b)

Second term: 1/2*(360 - b - c)

Third term: 1/2*(360 - d - e)

Adding them together:

= [ (e + d + c + b) + (360 - b - c) + (360 - d - e) ] / 2

Simplify numerator:

e + d + c + b + 360 - b - c + 360 - d - e

Let me combine like terms:

e - e = 0

d - d = 0

c - c = 0

b - b = 0

Then, 360 + 360 = 720

Therefore, numerator is 720, so total sum is 720 / 2 = 360 degrees. Which matches our previous result that angles A + C + E = 360 degrees. Therefore, this doesn't give us new information. So, we can't find individual angles unless we have more equations.

But since we already found α = 145 degrees through the previous method, and the steps seem consistent, maybe that is the correct answer. However, let me check if there's another way to verify.

Alternatively, maybe I can use the fact that the sum of all arcs is 360 degrees, and we have:

a + b = 140,

c + d = 70,

e + f = 150.

Therefore, if I can find relations between other angles, but since we don't have more angles given, maybe not. However, let me check if angle α is indeed 145 degrees.

Alternatively, maybe I can think of the hexagon as composed of triangles or something else. But since it's cyclic, maybe not necessary.

Alternatively, let me check if the answer makes sense. If angle at D is 145 degrees, which is quite large, but given that the other angles are 110 and 105, which are also relatively large, maybe it's possible. Let me check if the total sum of angles would be okay.

We have angles at B = 110°, F = 105°, D = 145°, and angles A, C, E sum to 360°. So total sum is 110 + 105 + 145 + 360 = 720°, which is correct. So that seems okay.

Alternatively, maybe I can check with specific values. Let me assume some values for the arcs. For example, suppose that c + d = 70°, so let me pick arbitrary values for c and d. Let me say c = 30°, d = 40°, just for example. Then, angle at D would be 1/2*(360 - 30 - 40) = 1/2*(290) = 145°, which matches. Then, with c = 30°, d = 40°, then a + b = 140°, e + f = 150°, and total arcs sum to 360°. So, that seems okay. Therefore, regardless of the specific values of c and d, as long as their sum is 70°, angle D is 145°, so that seems consistent.

Therefore, I think my answer is correct. Therefore, α is 145 degrees.

**Final Answer**
\boxed{145}

To determine the measure of angle $\alpha$ in the given cyclic hexagon, we utilize the geometric properties of inscribed angles in a circle.

---

### Step 1: Understand the Geometry

We are given a **cyclic hexagon** inscribed in a circle. This means all six vertices lie on the circumference of the circle. The key property we use is that the **measure of an inscribed angle** is **half the measure of the arc it intercepts**.

Let’s denote the arcs between consecutive vertices as follows:

- Arc $AB = a$
- Arc $BC = b$
- Arc $CD = c$
- Arc $DE = d$
- Arc $EF = e$
- Arc $FA = f$

Since the total circumference of the circle is $360^\circ$, we have:

$$
a + b + c + d + e + f = 360^\circ
$$

---

### Step 2: Use Given Angles to Form Equations

We are given:

- $\angle B = 110^\circ$
- $\angle F = 105^\circ$
- $\angle D = \alpha$ (to be found)

#### For $\angle B$:

This angle is formed by chords $AB$ and $BC$, and it intercepts the arc $AC$ that **does not contain** point $B$. The measure of this arc is:

$$
\text{Arc } AC = 360^\circ - (a + b)
$$

So:

$$
\angle B = \frac{1}{2}(360^\circ - a - b) = 110^\circ
$$

Multiply both sides by 2:

$$
360^\circ - a - b = 220^\circ \Rightarrow a + b = 140^\circ
$$

#### For $\angle F$:

This angle is formed by chords $EF$ and $FA$, and it intercepts the arc $EA$ that **does not contain** point $F$. The measure of this arc is:

$$
\text{Arc } EA = 360^\circ - (e + f)
$$

So:

$$
\angle F = \frac{1}{2}(360^\circ - e - f) = 105^\circ
$$

Multiply both sides by 2:

$$
360^\circ - e - f = 210^\circ \Rightarrow e + f = 150^\circ
$$

---

### Step 3: Use Total Arc Sum to Find $c + d$

We now use the total arc sum:

$$
a + b + c + d + e + f = 360^\circ
$$

Substitute the known values:

$$
140^\circ + c + d + 150^\circ = 360^\circ \Rightarrow c + d = 70^\circ
$$

---

### Step 4: Use Arcs to Find $\alpha = \angle D$

Angle $\alpha$ is formed by chords $CD$ and $DE$, and it intercepts the arc $CE$ that **does not contain** point $D$. The measure of this arc is:

$$
\text{Arc } CE = 360^\circ - (c + d)
$$

So:

$$
\alpha = \frac{1}{2}(360^\circ - c - d) = \frac{1}{2}(360^\circ - 70^\circ) = \frac{1}{2}(290^\circ) = 145^\circ
$$

---

### Final Answer

$$
\boxed{145}
$$

Solution:
Labeling our vertices will help a great deal, as will drawing a few radii: [asy]
pair pA, pB, pC, pD, pE, pF, pO;
pO = (0, 0);
pA = pO + dir(-10);
pB = pO + dir(60);
pC = pO + dir(130);
pD = pO + dir(170);
pE = pO + dir(-160);
pF = pO + dir(-80);
draw(pA--pB--pC--pD--pE--pF--pA);
draw(pA--pO--pC--pO--pE--pO, red);
draw(circle(pO, 1));
label("$O$", pO, NE);
label("$A$", pA, E);
label("$B$", pB, NE);
label("$C$", pC, NW);
label("$D$", pD, W);
label("$E$", pE, SW);
label("$F$", pF, S);
label("$105^\circ$", pF, N * 2);
label("$110^\circ$", pB, SW * 1.5);
label("$\alpha$", pD, E);
[/asy] First of all, we see that $\angle ABC = 110^\circ$ must be half of the major arc ${AEC},$ thus arc ${AEC} = 2 \cdot \angle ABC.$ Then, the minor arc ${AC}$ must be $360^\circ - 2 \cdot \angle ABC = 360^\circ - 2 \cdot 110^\circ = 140^\circ.$

Likewise, the minor arc ${EA}$ must be $360^\circ - 2 \cdot \angle EFA = 360^\circ - 2 \cdot 105^\circ = 150^\circ,$ and the minor arc ${CE}$ is $360^\circ - 2 \alpha.$ Now, arc ${AC},$ ${CE},$ and ${EA}$ must add up to $360^\circ,$ which means that \begin{align*}
360^\circ &= (360^\circ - 2 \alpha) + 140^\circ + 150^\circ\\
360^\circ &= 650^\circ - 2\alpha\\
2\alpha &= 290^\circ\\
\alpha &= \boxed{145^\circ}.
\end{align*}