Explanation:
Okay, so I need to compute the product of the number 5 plus the square root of 3 and its radical conjugate. Hmm, first, I should remember what a radical conjugate is. If I have a number that's in the form a plus b times the square root of c, then its radical conjugate would be a minus b times the square root of c, right? So, in this case, the number is 5 plus the square root of 3. Therefore, its radical conjugate should be 5 minus the square root of 3. Let me just confirm that... Yeah, I think that's correct. The radical conjugate changes the sign between the terms. So, if there's a plus, it becomes a minus, and vice versa. So, that seems right.

Alright, so the problem is asking for the product of these two numbers: (5 + √3) and (5 - √3). So, I need to multiply them together. How do I do that? Well, I remember that when you multiply two binomials, you can use the FOIL method: First, Outer, Inner, Last. Let me try that.

First, multiply the first terms: 5 times 5. That's 25. Then the Outer terms: 5 times -√3. That would be -5√3. Then the Inner terms: √3 times 5. That would be 5√3. And finally, the Last terms: √3 times -√3. That would be - (√3)^2. Let me write that down step by step.

So, expanding (5 + √3)(5 - √3):

First: 5 * 5 = 25

Outer: 5 * (-√3) = -5√3

Inner: √3 * 5 = 5√3

Last: √3 * (-√3) = - (√3 * √3) = - ( (√3)^2 ) = -3

Now, add all those terms together: 25 - 5√3 + 5√3 - 3.

Wait, let's combine like terms. The -5√3 and +5√3 should cancel each other out, right? Because they are additive inverses. So, -5√3 + 5√3 equals zero. Then, we have 25 - 3. That's 22. So, the product is 22? That seems straightforward, but let me check if I did everything correctly.

Alternatively, I remember there's a formula for multiplying a binomial by its conjugate. It's called the difference of squares formula. The formula is (a + b)(a - b) = a² - b². Let me apply that here. If I let a be 5 and b be √3, then (5 + √3)(5 - √3) equals 5² - (√3)². Calculating that, 5 squared is 25, and (√3) squared is 3. So, 25 - 3 is 22. Yep, same answer. So that confirms it. So, the product is 22.

Wait, just to be thorough, let me go back to my first method. So, expanding with FOIL:

First: 25

Outer: -5√3

Inner: +5√3

Last: -3

Adding them up: 25 - 5√3 + 5√3 - 3. So, combining the middle terms: (-5√3 + 5√3) is 0, so we have 25 - 3 = 22. Yep, exactly. So both methods give me the same answer, which is reassuring.

But let me also think about why the radical conjugate is useful. I think when you multiply a number by its conjugate, you eliminate the radical in the product. In this case, the result is 22, which is a rational number. So that's why it's called a conjugate; it helps to rationalize denominators or simplify expressions. For example, if you had something like 1/(5 + √3), multiplying numerator and denominator by the conjugate 5 - √3 would give you a rational denominator. But in this case, the question is just asking for the product, so it's straightforward.

But maybe I should double-check my arithmetic. Let me compute (5 + √3)(5 - √3) again step by step. Let me expand it as 5*(5 - √3) + √3*(5 - √3). Let's compute each part separately.

First part: 5*(5 - √3) = 5*5 - 5*√3 = 25 - 5√3

Second part: √3*(5 - √3) = √3*5 - √3*√3 = 5√3 - (√3)^2 = 5√3 - 3

Now, adding those two parts together: (25 - 5√3) + (5√3 - 3). Combine like terms:

25 - 3 = 22

-5√3 + 5√3 = 0

So total is 22. Same result. Okay, that seems solid.

Alternatively, maybe using decimal approximations to verify? Let me try that. Let me approximate √3. √3 is approximately 1.732. So 5 + √3 ≈ 5 + 1.732 = 6.732. The conjugate is 5 - √3 ≈ 5 - 1.732 = 3.268. Multiplying 6.732 by 3.268. Let me compute that.

First, approximate 6.732 * 3.268. Let me compute this step by step. Maybe break it down:

Let me note that 6.732 * 3.268 ≈ ?

Well, 6.732 * 3 = 20.196

6.732 * 0.268 ≈ Let me compute that. 6.732 * 0.2 = 1.3464; 6.732 * 0.06 = 0.40392; 6.732 * 0.008 = 0.053856. Adding those together: 1.3464 + 0.40392 = 1.75032 + 0.053856 ≈ 1.804176.

So total approximate value is 20.196 + 1.804176 ≈ 22.000176. That's approximately 22. Which matches our exact calculation. So that gives me more confidence that the answer is indeed 22. So even though the decimal approximation is slightly over 22 due to rounding errors in the approximated values of √3 and the multiplications, it's very close to 22, which is consistent with our exact answer.

Therefore, I think I can be confident that the product of 5 + √3 and its radical conjugate 5 - √3 is 22.

But just to recap, the key steps were recognizing the radical conjugate, applying the difference of squares formula, or expanding via FOIL, both leading to the same answer. Then verifying with decimal approximation. All methods converge to 22, so the answer must be correct.

I think that's thorough. I can't see any mistakes in my reasoning. So yeah, the answer is 22.

**Final Answer**
The product of the number $5+\sqrt{3}$ and its radical conjugate is \boxed{22}.

To solve the problem, we are asked to compute the product of the number $5 + \sqrt{3}$ and its **radical conjugate**.

---

### Step 1: Identify the Radical Conjugate

The **radical conjugate** of an expression in the form $ a + \sqrt{b} $ is $ a - \sqrt{b} $. Therefore, the radical conjugate of $ 5 + \sqrt{3} $ is:
$$
5 - \sqrt{3}
$$

---

### Step 2: Multiply the Expression by Its Conjugate

We now compute the product:
$$
(5 + \sqrt{3})(5 - \sqrt{3})
$$

This is a classic case for applying the **difference of squares** identity:
$$
(a + b)(a - b) = a^2 - b^2
$$

Applying the identity here:
- $ a = 5 $
- $ b = \sqrt{3} $

So:
$$
(5 + \sqrt{3})(5 - \sqrt{3}) = 5^2 - (\sqrt{3})^2 = 25 - 3 = 22
$$

---

### Step 3: Confirm the Result with Expansion (Optional)

Alternatively, we can expand directly using the distributive property (FOIL method):

$$
(5 + \sqrt{3})(5 - \sqrt{3}) = 5 \cdot 5 + 5 \cdot (-\sqrt{3}) + \sqrt{3} \cdot 5 + \sqrt{3} \cdot (-\sqrt{3})
$$

Breaking it down:
- $ 5 \cdot 5 = 25 $
- $ 5 \cdot (-\sqrt{3}) = -5\sqrt{3} $
- $ \sqrt{3} \cdot 5 = 5\sqrt{3} $
- $ \sqrt{3} \cdot (-\sqrt{3}) = -(\sqrt{3})^2 = -3 $

Combine:
$$
25 - 5\sqrt{3} + 5\sqrt{3} - 3 = 25 - 3 = 22
$$

This confirms the result.

---

### Final Answer

$$
\boxed{22}
$$

[Assistant]
Explanation:
Okay, so I need to compute the product of the number 5 plus the square root of 3 and its radical conjugate. Hmm, first, I should remember what a radical conjugate is. If I have a number that's in the form a plus b times the square root of c, then its radical conjugate would be a minus b times the square root of c, right? So, in this case, the number is 5 plus the square root of 3. Therefore, its radical conjugate should be 5 minus the square root of 3. Let me just confirm that... Yeah, I think that's correct. The radical conjugate changes the sign between the terms. So, if there's a plus, it becomes a minus, and vice versa. So, that seems right.

Alright, so the problem is asking for the product of these two numbers: (5 + √3) and (5 - √3). So, I need to multiply them together. How do I do that? Well, I remember that when you multiply two binomials, you can use the FOIL method: First, Outer, Inner, Last. Let me try that.

First, multiply the first terms: 5 times 5. That's 25. Then the Outer terms: 5 times -√3. That would be -5√3. Then the Inner terms: √3 times 5. That would be 5√3. And finally, the Last terms: √3 times -√3. That would be - (√3)^2. Let me write that down step by step.

So, expanding (5 + √3)(5 - √3):

First: 5 * 5 = 25

Outer: 5 * (-√3) = -5√3

Inner: √3 * 5 = 5√3

Last: √3 * (-√3) = - (√3 * √3) = - ( (√3)^2 ) = -3

Now, add all those terms together: 25 - 5√3 + 5√3 - 3.

Wait, let's combine like terms. The -5√3 and +5√3 should cancel each other out, right? Because they are additive inverses. So, -5√3 + 5√3 equals zero. Then, we have 25 - 3. That's 22. So, the product is 22? That seems straightforward, but let me check if I did everything correctly.

Alternatively, I remember there's a formula for multiplying a binomial by its conjugate. It's called the difference of squares formula. The formula is (a + b)(a - b) = a² - b². Let me apply that here. If I let a be 5 and b be √3, then (5 + √3)(5 - √3) equals 5² - (√3)². Calculating that, 5 squared is 25, and (√3) squared is 3. So, 25 - 3 is 22. Yep, same answer. So that confirms it. So, the product is 22.

Wait, just to be thorough, let me go back to my first method. So, expanding with FOIL:

First: 25

Outer: -5√3

Inner: +5√3

Last: -3

Adding them up: 25 - 5√3 + 5√3 - 3. So, combining the middle terms: (-5√3 + 5√3) is 0, so we have 25 - 3 = 22. Yep, exactly. So both methods give me the same answer, which is reassuring.

But let me also think about why the radical conjugate is useful. I think when you multiply a number by its conjugate, you eliminate the radical in the product. In this case, the result is 22, which is a rational number. So that's why it's called a conjugate; it helps to rationalize denominators or simplify expressions. For example, if you had something like 1/(5 + √3), multiplying numerator and denominator by the conjugate 5 - √3 would give you a rational denominator. But in this case, the question is just asking for the product, so it's straightforward.

But maybe I should double-check my arithmetic. Let me compute (5 + √3)(5 - √3) again step by step. Let me expand it as 5*(5 - √3) + √3*(5 - √3). Let's compute each part separately.

First part: 5*(5 - √3) = 5*5 - 5*√3 = 25 - 5√3

Second part: √3*(5 - √3) = √3*5 - √3*√3 = 5√3 - (√3)^2 = 5√3 - 3

Now, adding those two parts together: (25 - 5√3) + (5√3 - 3). Combine like terms:

25 - 3 = 22

-5√3 + 5√3 = 0

So total is 22. Same result. Okay, that seems solid.

Alternatively, maybe using decimal approximations to verify? Let me try that. Let me approximate √3. √3 is approximately 1.732. So 5 + √3 ≈ 5 + 1.732 = 6.732. The conjugate is 5 - √3 ≈ 5 - 1.732 = 3.268. Multiplying 6.732 by 3.268. Let me compute that.

First, approximate 6.732 * 3.268. Let me compute this step by step. Maybe break it down:

Let me note that 6.732 * 3.268 ≈ ?

Well, 6.732 * 3 = 20.196

6.732 * 0.268 ≈ Let me compute that. 6.732 * 0.2 = 1.3464; 6.732 * 0.06 = 0.40392; 6.732 * 0.008 = 0.053856. Adding those together: 1.3464 + 0.40392 = 1.75032 + 0.053856 ≈ 1.804176.

So total approximate value is 20.196 + 1.804176 ≈ 22.000176. That's approximately 22. Which matches our exact calculation. So that gives me more confidence that the answer is indeed 22. So even though the decimal approximation is slightly over 22 due to rounding errors in the approximated values of √3 and the multiplications, it's very close to 22, which is consistent with our exact answer.

Therefore, I think I can be confident that the product of 5 + √3 and its radical conjugate 5 - √3 is 22.

But just to recap, the key steps were recognizing the radical conjugate, applying the difference of squares formula, or expanding via FOIL, both leading to the same answer. Then verifying with decimal approximation. All methods converge to 22, so the answer must be correct.

I think that's thorough. I can't see any mistakes in my reasoning. So yeah, the answer is 22.

**Final Answer**
The product of the number $5+\sqrt{3}$ and its radical conjugate is \boxed{22}.

To solve the problem, we are asked to compute the product of the number $5 + \sqrt{3}$ and its **radical conjugate**.

---

### Step 1: Identify the Radical Conjugate

The **radical conjugate** of an expression in the form $ a + \sqrt{b} $ is $ a - \sqrt{b} $. Therefore, the radical conjugate of $ 5 + \sqrt{3} $ is:
$$
5 - \sqrt{3}
$$

---

### Step 2: Multiply the Expression by Its Conjugate

We now compute the product:
$$
(5 + \sqrt{3})(5 - \sqrt{3})
$$

This is a classic case for applying the **difference of squares** identity:
$$
(a + b)(a - b) = a^2 - b^2
$$

Applying the identity here:
- $ a = 5 $
- $ b = \sqrt{3} $

So:
$$
(5 + \sqrt{3})(5 - \sqrt{3}) = 5^2 - (\sqrt{3})^2 = 25 - 3 = 22
$$

---

### Step 3: Confirm the Result with Expansion (Optional)

Alternatively, we can expand directly using the distributive property (FOIL method):

$$
(5 + \sqrt{3})(5 - \sqrt{3}) = 5 \cdot 5 + 5 \cdot (-\sqrt{3}) + \sqrt{3} \cdot 5 + \sqrt{3} \cdot (-\sqrt{3})
$$

Breaking it down:
- $ 5 \cdot 5 = 25 $
- $ 5 \cdot (-\sqrt{3}) = -5\sqrt{3} $
- $ \sqrt{3} \cdot 5 = 5\sqrt{3} $
- $ \sqrt{3} \cdot (-\sqrt{3}) = -(\sqrt{3})^2 = -3 $

Combine:
$$
25 - 5\sqrt{3} + 5\sqrt{3} - 3 = 25 - 3 = 22
$$

This confirms the result.

---

### Final Answer

$$
\boxed{22}
$$

Solution: The radical conjugate of this number is $5-\sqrt{3},$ so the product of the two numbers is \[(5+\sqrt3)(5-\sqrt3) = 5^2 - (\sqrt3)^2 = 25 - 3 = \boxed{22}.\]


Explanation:
Okay, let me try to solve this equation: 10^x minus 10 equals 9990. Hmm, so the equation is 10^x - 10 = 9990. I need to find the value of x. Let me think about how to approach this.

First, maybe I should isolate the term with the exponent, which is 10^x. To do that, I can add 10 to both sides of the equation. Let me write that down:

10^x - 10 + 10 = 9990 + 10

Simplifying both sides, the left side becomes 10^x because -10 and +10 cancel each other out. On the right side, 9990 plus 10 is... let me calculate that. 9990 + 10 is 10,000. So now the equation is:

10^x = 10,000

Alright, so now I need to solve for x in the equation 10^x = 10,000. I remember that 10 raised to some power equals 10,000. Let me recall the powers of 10.

10^1 is 10, 10^2 is 100, 10^3 is 1,000, 10^4 is 10,000. Oh! So 10^4 equals 10,000. That means x must be 4. Wait, let me double-check that. If x is 4, then 10^4 is 10*10*10*10, which is 10,000. Yep, that's correct.

But let me make sure I didn't skip any steps or make any mistakes earlier. Starting from the original equation:

10^x - 10 = 9990

Adding 10 to both sides gives 10^x = 9990 + 10. Let me confirm 9990 + 10 again. 9990 + 10 is indeed 10,000. So that step is correct. Then recognizing that 10^4 is 10,000, so x is 4.

Is there another way to approach this? Maybe using logarithms? Let me try that method to verify.

Starting again from 10^x = 10,000. Taking the logarithm base 10 of both sides:

log₁₀(10^x) = log₁₀(10,000)

Simplifying the left side, log base 10 of 10^x is just x, because logarithm and exponentiation are inverse functions. So:

x = log₁₀(10,000)

But log base 10 of 10,000 is the exponent you need to raise 10 to get 10,000. Which again is 4. Therefore, x = 4.

So both methods lead me to the same answer, which is reassuring. Let me check once again by substituting x = 4 back into the original equation to verify.

Original equation: 10^x - 10 = 9990

Substitute x with 4:

10^4 - 10 = 10,000 - 10 = 9990

Which matches the right side of the equation. Perfect, that checks out.

Wait, just to be thorough, is there a possibility that there's another solution? For example, sometimes with exponents, especially if dealing with different bases or negative exponents, there might be multiple solutions. But in this case, since the base is 10, which is positive and not equal to 1, the function 10^x is strictly increasing. That means it's one-to-one, so there can only be one solution. Therefore, x = 4 is the unique solution.

Alternatively, if I think about the graph of the function f(x) = 10^x - 10, it's an exponential curve shifted down by 10 units. Since 10^x is always positive and increases rapidly, the equation f(x) = 9990 will have exactly one solution where the curve crosses the line y = 9990. Which again is at x = 4.

I think that covers all the bases. I don't see any mistakes in my reasoning, and all the checks confirm that x is indeed 4. So I'm confident that the answer is correct.

**Final Answer**
The value of $ x $ is \boxed{4}.

To solve the equation:

$$
10^x - 10 = 9990
$$

we aim to isolate the exponential term $10^x$ and then solve for $x$.

---

### Step 1: Isolate the Exponential Expression

Add 10 to both sides:

$$
10^x = 9990 + 10
$$

$$
10^x = 10000
$$

---

### Step 2: Recognize Powers of 10

Observe that:

$$
10^4 = 10 \times 10 \times 10 \times 10 = 10000
$$

Thus, we can deduce:

$$
x = 4
$$

---

### Step 3: Verification

To confirm, substitute $x = 4$ back into the original equation:

$$
10^4 - 10 = 10000 - 10 = 9990
$$

Which matches the right-hand side of the equation, confirming the solution is correct.

---

### Step 4: Alternative Approach – Logarithms

To solve more formally, take the base-10 logarithm of both sides of the equation $10^x = 10000$:

$$
\log_{10}(10^x) = \log_{10}(10000)
$$

$$
x = \log_{10}(10000) = 4
$$

This again gives $x = 4$, supporting the earlier result.

---

### Final Answer

$$
\boxed{4}
$$

[Assistant]
Explanation:
Okay, let me try to solve this equation: 10^x minus 10 equals 9990. Hmm, so the equation is 10^x - 10 = 9990. I need to find the value of x. Let me think about how to approach this.

First, maybe I should isolate the term with the exponent, which is 10^x. To do that, I can add 10 to both sides of the equation. Let me write that down:

10^x - 10 + 10 = 9990 + 10

Simplifying both sides, the left side becomes 10^x because -10 and +10 cancel each other out. On the right side, 9990 plus 10 is... let me calculate that. 9990 + 10 is 10,000. So now the equation is:

10^x = 10,000

Alright, so now I need to solve for x in the equation 10^x = 10,000. I remember that 10 raised to some power equals 10,000. Let me recall the powers of 10.

10^1 is 10, 10^2 is 100, 10^3 is 1,000, 10^4 is 10,000. Oh! So 10^4 equals 10,000. That means x must be 4. Wait, let me double-check that. If x is 4, then 10^4 is 10*10*10*10, which is 10,000. Yep, that's correct.

But let me make sure I didn't skip any steps or make any mistakes earlier. Starting from the original equation:

10^x - 10 = 9990

Adding 10 to both sides gives 10^x = 9990 + 10. Let me confirm 9990 + 10 again. 9990 + 10 is indeed 10,000. So that step is correct. Then recognizing that 10^4 is 10,000, so x is 4.

Is there another way to approach this? Maybe using logarithms? Let me try that method to verify.

Starting again from 10^x = 10,000. Taking the logarithm base 10 of both sides:

log₁₀(10^x) = log₁₀(10,000)

Simplifying the left side, log base 10 of 10^x is just x, because logarithm and exponentiation are inverse functions. So:

x = log₁₀(10,000)

But log base 10 of 10,000 is the exponent you need to raise 10 to get 10,000. Which again is 4. Therefore, x = 4.

So both methods lead me to the same answer, which is reassuring. Let me check once again by substituting x = 4 back into the original equation to verify.

Original equation: 10^x - 10 = 9990

Substitute x with 4:

10^4 - 10 = 10,000 - 10 = 9990

Which matches the right side of the equation. Perfect, that checks out.

Wait, just to be thorough, is there a possibility that there's another solution? For example, sometimes with exponents, especially if dealing with different bases or negative exponents, there might be multiple solutions. But in this case, since the base is 10, which is positive and not equal to 1, the function 10^x is strictly increasing. That means it's one-to-one, so there can only be one solution. Therefore, x = 4 is the unique solution.

Alternatively, if I think about the graph of the function f(x) = 10^x - 10, it's an exponential curve shifted down by 10 units. Since 10^x is always positive and increases rapidly, the equation f(x) = 9990 will have exactly one solution where the curve crosses the line y = 9990. Which again is at x = 4.

I think that covers all the bases. I don't see any mistakes in my reasoning, and all the checks confirm that x is indeed 4. So I'm confident that the answer is correct.

**Final Answer**
The value of $ x $ is \boxed{4}.

To solve the equation:

$$
10^x - 10 = 9990
$$

we aim to isolate the exponential term $10^x$ and then solve for $x$.

---

### Step 1: Isolate the Exponential Expression

Add 10 to both sides:

$$
10^x = 9990 + 10
$$

$$
10^x = 10000
$$

---

### Step 2: Recognize Powers of 10

Observe that:

$$
10^4 = 10 \times 10 \times 10 \times 10 = 10000
$$

Thus, we can deduce:

$$
x = 4
$$

---

### Step 3: Verification

To confirm, substitute $x = 4$ back into the original equation:

$$
10^4 - 10 = 10000 - 10 = 9990
$$

Which matches the right-hand side of the equation, confirming the solution is correct.

---

### Step 4: Alternative Approach – Logarithms

To solve more formally, take the base-10 logarithm of both sides of the equation $10^x = 10000$:

$$
\log_{10}(10^x) = \log_{10}(10000)
$$

$$
x = \log_{10}(10000) = 4
$$

This again gives $x = 4$, supporting the earlier result.

---

### Final Answer

$$
\boxed{4}
$$

Solution: Since $10^x - 10 = 9990,$ we have $$10^x = 9990+10=10000.$$If $10^x = 10000,$ then $x=\boxed{4},$ since $10000$ ends in four zeroes.


Explanation:
Okay, so I need to find the sum of the measures of angle J and angle H in this circle problem. Let me first try to visualize the diagram based on the Asymptote code provided. There's a circle with center J, and points H, O, and N. HO and HN are tangent to the circle at points O and N respectively. The diagram shows J connected to N and O, and then H connected to N and O, forming quadrilateral JNHO? Wait, the Asymptote code draws J--N--H--O--cycle, so that's a four-sided figure with vertices J, N, H, O. So, J is the center of the circle, and HO and HN are tangent to the circle at O and N.

First, let me recall that tangent lines to a circle are perpendicular to the radius at the point of tangency. That means that JO is perpendicular to HO, and JN is perpendicular to HN. So, angles HJO and HJN are right angles (90 degrees each).

So maybe I can use that information. Let me try to sketch this mentally. The circle has center J, point H outside the circle, from which two tangent lines are drawn to points O and N on the circle. Then connecting J to N and J to O, so JO and JN are radii.

Now, quadrilateral JNHO has two right angles at O and N. Wait, but quadrilateral JNHO? Let me check. If I connect J to N, N to H, H to O, and O back to J? So, the sides are JN, NH, HO, and OJ. Yes. So angles at O and N are right angles. Therefore, quadrilateral JNHO has two right angles.

But what is the measure of angle J and angle H in this quadrilateral? Wait, angle J is the angle at point J, which is the center of the circle, between points N and O. So angle NJO. Wait, but if JO and JN are radii, then angle J is the central angle between points N and O. Similarly, angle H is the angle at point H between points N and O, which would be angle NH O.

So the problem is asking for m∠J + m∠H, where ∠J is the central angle between N and O, and ∠H is the angle at H between N and O.

So perhaps I need to find those two angles and add them together.

Let me recall that in circle geometry, the angle between a tangent and a chord is equal to the measure of the inscribed angle on the opposite side of the chord. Wait, but here, angle at H is formed by two tangents, right? Because HO and HN are both tangents from point H to the circle. So angle at H is the angle between two tangents, which is called the "angle between two tangents" and there's a formula for that.

Yes, the measure of the angle between two tangents from an external point is equal to half the difference of the measures of the intercepted arcs. Specifically, if two tangents are drawn from an external point H to a circle, then the angle at H is equal to half the difference of the measures of the intercepted arcs.

In this case, the two intercepted arcs would be the major arc NO and the minor arc NO. Since the tangents are HO and HN, they intercept arcs NO. Depending on the position of N and O, but since they are two points on the circle, there are two arcs between them: minor arc NO and major arc NO.

So the measure of angle H would be (measure of major arc NO - measure of minor arc NO)/2.

But since the entire circle is 360 degrees, major arc NO + minor arc NO = 360 degrees. So if we let the measure of minor arc NO be x degrees, then major arc NO is 360 - x degrees. Therefore, angle H would be [(360 - x) - x]/2 = (360 - 2x)/2 = 180 - x.

But also, the central angle ∠J (which is angle NJO) is equal to the measure of the arc NO. Since angle at center is equal to the measure of its intercepted arc. Therefore, if ∠J is the central angle for arc NO, then measure of ∠J is x degrees. Therefore, the measure of angle H is 180 - x, so the sum of m∠J + m∠H is x + (180 - x) = 180 degrees.

Wait, that can't be. Wait, if that's true, then regardless of where points N and O are on the circle, the sum would be 180 degrees? Is that possible?

Wait, maybe I made a mistake here. Let me check again.

First, angle H is formed by two tangents from point H to the circle. The formula for the angle between two tangents is indeed half the difference of the intercepted arcs. So angle at H is 1/2*(measure of major arc NO - measure of minor arc NO).

But since the central angle for arc NO is the measure of minor arc NO, which is x, and major arc NO is 360 - x. Therefore, angle H = ( (360 - x) - x ) / 2 = (360 - 2x)/2 = 180 - x.

Then, angle J, the central angle, is x. Therefore, adding them together, angle J + angle H is x + 180 - x = 180 degrees.

Therefore, the sum is 180 degrees.

Wait, but that seems too straightforward. Let me check with a specific example. Suppose that arc NO is 120 degrees. Then angle J would be 120 degrees. Then the major arc NO is 240 degrees. Then angle H would be (240 - 120)/2 = 60 degrees. Then sum is 120 + 60 = 180.

Alternatively, if arc NO is 90 degrees. Then angle J is 90 degrees, major arc NO is 270 degrees. Then angle H is (270 - 90)/2 = 90 degrees. Sum is 180.

Another example: if arc NO is 180 degrees, then angle J is 180 degrees, major arc NO is 180 degrees, so angle H is (180 - 180)/2 = 0 degrees. Wait, but that can't be. If arc NO is 180 degrees, then points N and O are endpoints of a diameter. Then tangents from H would be... Wait, if N and O are endpoints of a diameter, then the tangents from H would be symmetrical with respect to the diameter NO. However, if angle H is 0 degrees, that would imply that H is somewhere on the line NO? Wait, but if H is on the line NO, then the tangent lines would be... Wait, if H is on the circle, but H is an external point. If H is on the line NO, then depending on where it is. Wait, but if arc NO is 180 degrees, so the diameter, and if H is outside, then the two tangent would be symmetric.

Wait, maybe in that case, angle H is 0? That seems odd. Wait, but let's think. If points N and O are endpoints of a diameter, and H is a point such that HO and HN are both tangents. Wait, if H is outside the circle, then there are two tangents from H to the circle. If N and O are endpoints of a diameter, then the two tangents from H would be symmetric across the line JO (or JN, since J is the center). However, if H is such that the two tangent points are N and O, then H lies along the line perpendicular to the diameter NO at its midpoint? Wait, no. Wait, if N and O are endpoints of diameter, then the line JO is the same as the line JO. If you have tangents from H to points N and O, then H must lie on the line that is the radical axis or something? Wait, maybe not. Let me think.

Wait, if H is such that HN and HO are tangents to the circle at N and O, then H lies on the external tangent lines. However, if N and O are endpoints of diameter, then the tangents at N and O are perpendicular to the radius. So the tangent at N is perpendicular to JN, and tangent at O is perpendicular to JO. Since JN and JO are in opposite directions (as N and O are endpoints of diameter), then the two tangents would be two lines perpendicular to JN and JO respectively. If JN is one direction, JO is opposite. Therefore, the tangent at N is perpendicular to JN, so direction is perpendicular to JN, and tangent at O is perpendicular to JO, which is same direction as tangent at N? Wait, no. If JO is opposite to JN, then the tangent at O is perpendicular to JO, which is same as perpendicular to -JN. Therefore, same as tangent at N? Wait, no. If you have a line perpendicular to JN, and another line perpendicular to JO (which is -JN), then they are both perpendicular to the same line, meaning they are parallel? Therefore, if the two tangents are parallel? But if H is outside the circle, then two tangents from H to the circle would meet at H. But if the two tangent lines are parallel, then there's no such point H. Unless the lines are the same, but they can't be the same if they are tangents at different points. Therefore, in this case, if N and O are endpoints of diameter, then the tangents at N and O are parallel, so there is no external point H from which both tangents can be drawn to points N and O. Therefore, arc NO cannot be 180 degrees. Therefore, in reality, arc NO is less than 180 degrees? Wait, but the problem didn't specify where N and O are. So maybe in the diagram, arc NO is less than 180 degrees, so that the tangents HO and HN exist.

Therefore, perhaps my calculation is okay. So in general, angle J + angle H = 180 degrees? Therefore, the answer is 180 degrees?

But let me think again. Maybe I can approach this using triangle properties instead.

Since JO and JN are radii, and HO and HN are tangents, then JO perpendicular to HO and JN perpendicular to HN. Therefore, triangles JHO and JHN are right triangles with right angles at O and N respectively.

So, in triangle JHO, right-angled at O, and triangle JHN, right-angled at N.

If we can find some relations between these triangles.

Let me denote ∠J as angle OJN? Wait, angle at J is the angle between JO and JN. Wait, JO and JN are two radii, so angle between them is the central angle. Let me denote that as ∠OJN = θ degrees. Then, angle J is θ.

Then, in triangle JHO, right-angled at O, we have angle at H as ∠OHN, which is angle H. Wait, but angle at H is common to both triangles? No, angle at H is the angle between HO and HN. Wait, in triangle JHO, there's angle at H, and in triangle JHN, there's angle at H. But since both triangles share angle at H, which is the angle we are interested in.

Wait, maybe I can use the fact that the two right triangles share the hypotenuse JH.

Wait, in triangle JHO and JHN, they both have hypotenuse HJ, and legs JO and HO, and JN and HN, respectively.

Since JO and JN are radii, they are equal in length. Therefore, triangles JHO and JHN are congruent? Because they have hypotenuse HJ in common, and legs JO and JN equal. Therefore, they are congruent by hypotenuse-leg theorem. Therefore, their corresponding angles are equal.

Therefore, angles at H in both triangles are equal. Wait, but that's the same angle at H. So that doesn't give me new information.

Alternatively, maybe I can consider the quadrilateral JNHO. It has two right angles at O and N. So the sum of all interior angles of a quadrilateral is 360 degrees. Therefore, angle J + angle H + angle O + angle N = 360 degrees.

Since angles O and N are each 90 degrees, then angle J + angle H = 360 - 90 - 90 = 180 degrees.

Oh! That's a much simpler way to see it. Since quadrilateral JNHO has two right angles, the sum of the other two angles must be 180 degrees. Therefore, angle J + angle H = 180 degrees.

Therefore, the answer is 180 degrees.

Wait, so regardless of the specific configuration, as long as HO and HN are tangents at O and N, and J is the center, then the quadrilateral JNHO has angles at O and N of 90 degrees, hence angle J + angle H = 180 degrees.

Therefore, the answer is 180 degrees.

But let me check again. If I consider the quadrilateral JNHO, it's not necessarily a convex quadrilateral? Wait, in the Asymptote code, they draw J--N--H--O--cycle. So points J, N, H, O connected in that order. So assuming that the diagram is drawn in such a way that the quadrilateral is convex. Therefore, the sum of internal angles is 360 degrees. Therefore, angle J + angle H + 90 + 90 = 360, so angle J + angle H = 180.

Yes, that seems solid. Therefore, regardless of where the points are, as long as HO and HN are tangents, then angle J + angle H is 180 degrees.

But let me think again if there's any possibility that the quadrilateral is self-intersecting or something? If points are arranged differently. But given the Asymptote code, they define J at (0,0), H at (6,0), then construct a circle with center J and radius 3. Then they find midpoint between J and H, which is at (3,0), then draw a circle with radius 3 centered at (3,0). The intersection points of the two circles are points O and N.

So, the first circle is centered at (0,0) with radius 3, the second circle is centered at (3,0) with radius 3. The intersection points of two circles with centers 3 units apart and both radius 3. So, they intersect at two points. The line connecting the centers is the x-axis from (0,0) to (3,0). The intersection points would be above and below the x-axis.

So points O and N would be located symmetrically above and below the x-axis. Then, drawing J--N--H--O--cycle would result in a quadrilateral where N is one intersection point above, O is the other below? Or both above? Wait, the Asymptote code says:

pair M = midpoint(J--H); which is (3,0). Then path secCirc = Circle(M,3). So the two circles intersect at points tangentPoints[0] and tangentPoints[1]. Since the circles are symmetric about the x-axis, the intersection points are symmetric with respect to the x-axis.

Therefore, if O is one intersection point, say, above the x-axis, then N is the other one below? Or vice versa? Wait, the code says O = tangentPoints[0]; N = tangentPoints[1]; but depending on the order. But regardless, the points are symmetric. Therefore, connecting J--N--H--O--cycle would make a quadrilateral with two sides from J to N and J to O, and sides from N to H and H to O.

But since H is at (6,0), which is to the right of the circle. So H is outside the circle. Then, connecting N (which is above the x-axis) to H to O (which is below the x-axis).

But in any case, quadrilateral JNHO would have two right angles at O and N. So regardless of the exact positions, the sum of angles at J and H is 180 degrees.

Therefore, the answer is 180 degrees.

But just to make sure, let me try to compute coordinates.

Let me try to assign coordinates as per Asymptote code. Let me set J at (0,0), H at (6,0). The circle centered at J with radius 3. The other circle is centered at midpoint of JH, which is at (3,0), and radius 3.

Equation of first circle: x² + y² = 9.

Equation of second circle: (x - 3)² + y² = 9.

Subtracting the two equations: (x - 3)² + y² - (x² + y²) = 9 - 9 => (x² -6x +9 + y²) - x² - y² = 0 => -6x +9 = 0 => x = 9/6 = 3/2.

So points of intersection lie on x = 3/2. Plugging back into first equation: (3/2)² + y² = 9 => 9/4 + y² = 9 => y² = 9 - 9/4 = 27/4 => y = ± (3√3)/2.

Therefore, points O and N are at (3/2, (3√3)/2) and (3/2, -(3√3)/2). Let me assume O is (3/2, (3√3)/2) and N is (3/2, -(3√3)/2).

So coordinates:

J = (0, 0)

H = (6, 0)

N = (3/2, - (3√3)/2)

O = (3/2, (3√3)/2)

Now, let me compute angle J and angle H.

First, angle at J: which is angle between points N, J, O. Since J is the center, and N and O are points on the circle. So angle NJO is the central angle between N and O. The coordinates of N and O are symmetric with respect to the x-axis. So the angle between them can be calculated.

The coordinates of N are (1.5, - (3√3)/2) and O are (1.5, (3√3)/2). So vector from J to N is (1.5, - (3√3)/2), and vector from J to O is (1.5, (3√3)/2).

The angle between these two vectors can be found using the dot product.

Dot product of vectors JO and JN:

JO • JN = (1.5)(1.5) + [ ( - (3√3)/2 ) ( (3√3)/2 ) ]

Wait, no. Wait, JO is from J to O: (1.5, (3√3)/2), and JN is from J to N: (1.5, - (3√3)/2). Wait, but actually, angle at J is between JN and JO. Wait, no. Wait, angle J is the angle between JN and JO? Wait, in the quadrilateral JNHO, angle at J is between JN and JO? Wait, the quadrilateral is J connected to N connected to H connected to O connected to J. Therefore, angle at J is between JN and JO? Wait, no. Wait, when you have the quadrilateral J-N-H-O-J. So the sides at J are JN and JO. Therefore, angle at J is between JN and JO. Therefore, yes, the central angle between points N and O.

Therefore, the vectors are from J to N and from J to O, which are (1.5, - (3√3)/2 ) and (1.5, (3√3)/2 ). Therefore, the angle between them can be found by the dot product formula.

First, let me compute the dot product:

JO • JN = (1.5)(1.5) + [ ( (3√3)/2 ) * ( - (3√3)/2 ) ]

Wait, no: JO is vector to O: (1.5, (3√3)/2), and JN is vector to N: (1.5, - (3√3)/2). Therefore, the dot product is:

(1.5)(1.5) + [ ( (3√3)/2 ) * ( - (3√3)/2 ) ]

Calculating:

1.5 * 1.5 = 2.25

( (3√3)/2 ) * ( - (3√3)/2 ) = - (9 * 3)/4 = -27/4 = -6.75

Therefore, total dot product is 2.25 - 6.75 = -4.5

Now, the magnitude of JO and JN vectors are both radii, which is 3 units. Wait, let me confirm:

The length of JO is sqrt( (1.5)^2 + ( (3√3)/2 )^2 )

Calculating:

(1.5)^2 = 2.25

( (3√3)/2 )^2 = 9*3 / 4 = 27/4 = 6.75

Total is 2.25 + 6.75 = 9, sqrt(9) = 3. Correct.

Therefore, the dot product is |JO| |JN| cos(theta), where theta is the angle between them. Therefore:

-4.5 = 3 * 3 * cos(theta) => -4.5 = 9 cos(theta) => cos(theta) = -4.5 / 9 = -0.5

Therefore, theta = arccos(-0.5) which is 120 degrees.

So measure of angle J is 120 degrees.

Now let's compute measure of angle H.

Angle at H is the angle between HO and HN. Points H is at (6,0), O is at (1.5, (3√3)/2 ), N is at (1.5, - (3√3)/2 ).

So vectors from H to O and from H to N.

Vector HO: O - H = (1.5 - 6, (3√3)/2 - 0 ) = (-4.5, (3√3)/2 )

Vector HN: N - H = (1.5 - 6, - (3√3)/2 - 0 ) = (-4.5, - (3√3)/2 )

So angle at H is the angle between vectors HO and HN.

Let me compute the angle between these two vectors using the dot product formula.

First, compute the dot product:

HO • HN = (-4.5)(-4.5) + [ (3√3)/2 * (-3√3)/2 ]

Calculating:

(-4.5)^2 = 20.25

( (3√3)/2 * (-3√3)/2 ) = - (9 * 3)/4 = -27/4 = -6.75

Therefore, dot product = 20.25 - 6.75 = 13.5

Now compute the magnitudes of HO and HN. Since HO and HN are both tangent segments from H to the circle, they should have the same length.

Let me compute the length of HO:

sqrt( (-4.5)^2 + ( (3√3)/2 )^2 )

= sqrt( 20.25 + 6.75 )

= sqrt(27)

= 3√3

Similarly for HN.

Therefore, |HO| = |HN| = 3√3

Thus, the dot product is |HO||HN| cos(theta), so:

13.5 = (3√3)(3√3) cos(theta) = 9 * 3 cos(theta) = 27 cos(theta)

Therefore, cos(theta) = 13.5 / 27 = 0.5

Therefore, theta = arccos(0.5) = 60 degrees

So measure of angle H is 60 degrees.

Therefore, angle J + angle H = 120 + 60 = 180 degrees.

Therefore, this specific case gives 180 degrees.

So that confirms my previous conclusion. Therefore, regardless of the position of points N and O, as long as HO and HN are tangents, angle J + angle H is 180 degrees.

Alternatively, since we saw that in quadrilateral JNHO, two angles are 90 degrees each, so the other two angles must add to 180. Therefore, answer is 180 degrees.

Therefore, the answer is \boxed{180} degrees.

**Final Answer**
\boxed{180}

To determine the sum of the measures of $ \angle J $ and $ \angle H $ in the given geometric configuration, we start by analyzing the structure of the quadrilateral $ JNHO $, which is formed by:

- $ J $, the center of the circle,
- $ H $, an external point from which two tangents $ HO $ and $ HN $ are drawn to the circle,
- $ O $ and $ N $, the points of tangency on the circle.

---

### Step 1: Understanding the Angle at $ J $

Since $ HO $ and $ HN $ are tangents to the circle at $ O $ and $ N $, the line segments $ JO $ and $ JN $ are radii of the circle and are perpendicular to the tangents at the points of tangency. This implies:

$$
\angle JO H = 90^\circ \quad \text{and} \quad \angle JN H = 90^\circ
$$

Therefore, in quadrilateral $ JNHO $, two of its interior angles at $ O $ and $ N $ are right angles.

---

### Step 2: Applying the Quadrilateral Angle Sum

The sum of the interior angles of any quadrilateral is always $ 360^\circ $. Given that two of the angles are right angles (each $ 90^\circ $), the remaining two angles — $ \angle J $ and $ \angle H $ — must satisfy:

$$
\angle J + \angle H + 90^\circ + 90^\circ = 360^\circ
$$

Simplifying:

$$
\angle J + \angle H = 360^\circ - 180^\circ = 180^\circ
$$

---

### Step 3: Verifying with Coordinate Geometry (Optional)

Alternatively, using coordinate geometry, by assigning coordinates based on the Asymptote diagram and calculating the vectors $ \vec{JO} $, $ \vec{JN} $, and the resulting angle between them (at point $ J $), we can also compute $ \angle J = 120^\circ $. Similarly, computing $ \angle H $ as the angle between vectors from $ H $ to points $ O $ and $ N $, we find it to be $ 60^\circ $. Adding these gives:

$$
\angle J + \angle H = 120^\circ + 60^\circ = 180^\circ
$$

Which confirms the earlier result.

---

### Final Answer

$$
\boxed{180}
$$

[Assistant]
Explanation:
Okay, so I need to find the sum of the measures of angle J and angle H in this circle problem. Let me first try to visualize the diagram based on the Asymptote code provided. There's a circle with center J, and points H, O, and N. HO and HN are tangent to the circle at points O and N respectively. The diagram shows J connected to N and O, and then H connected to N and O, forming quadrilateral JNHO? Wait, the Asymptote code draws J--N--H--O--cycle, so that's a four-sided figure with vertices J, N, H, O. So, J is the center of the circle, and HO and HN are tangent to the circle at O and N.

First, let me recall that tangent lines to a circle are perpendicular to the radius at the point of tangency. That means that JO is perpendicular to HO, and JN is perpendicular to HN. So, angles HJO and HJN are right angles (90 degrees each).

So maybe I can use that information. Let me try to sketch this mentally. The circle has center J, point H outside the circle, from which two tangent lines are drawn to points O and N on the circle. Then connecting J to N and J to O, so JO and JN are radii.

Now, quadrilateral JNHO has two right angles at O and N. Wait, but quadrilateral JNHO? Let me check. If I connect J to N, N to H, H to O, and O back to J? So, the sides are JN, NH, HO, and OJ. Yes. So angles at O and N are right angles. Therefore, quadrilateral JNHO has two right angles.

But what is the measure of angle J and angle H in this quadrilateral? Wait, angle J is the angle at point J, which is the center of the circle, between points N and O. So angle NJO. Wait, but if JO and JN are radii, then angle J is the central angle between points N and O. Similarly, angle H is the angle at point H between points N and O, which would be angle NH O.

So the problem is asking for m∠J + m∠H, where ∠J is the central angle between N and O, and ∠H is the angle at H between N and O.

So perhaps I need to find those two angles and add them together.

Let me recall that in circle geometry, the angle between a tangent and a chord is equal to the measure of the inscribed angle on the opposite side of the chord. Wait, but here, angle at H is formed by two tangents, right? Because HO and HN are both tangents from point H to the circle. So angle at H is the angle between two tangents, which is called the "angle between two tangents" and there's a formula for that.

Yes, the measure of the angle between two tangents from an external point is equal to half the difference of the measures of the intercepted arcs. Specifically, if two tangents are drawn from an external point H to a circle, then the angle at H is equal to half the difference of the measures of the intercepted arcs.

In this case, the two intercepted arcs would be the major arc NO and the minor arc NO. Since the tangents are HO and HN, they intercept arcs NO. Depending on the position of N and O, but since they are two points on the circle, there are two arcs between them: minor arc NO and major arc NO.

So the measure of angle H would be (measure of major arc NO - measure of minor arc NO)/2.

But since the entire circle is 360 degrees, major arc NO + minor arc NO = 360 degrees. So if we let the measure of minor arc NO be x degrees, then major arc NO is 360 - x degrees. Therefore, angle H would be [(360 - x) - x]/2 = (360 - 2x)/2 = 180 - x.

But also, the central angle ∠J (which is angle NJO) is equal to the measure of the arc NO. Since angle at center is equal to the measure of its intercepted arc. Therefore, if ∠J is the central angle for arc NO, then measure of ∠J is x degrees. Therefore, the measure of angle H is 180 - x, so the sum of m∠J + m∠H is x + (180 - x) = 180 degrees.

Wait, that can't be. Wait, if that's true, then regardless of where points N and O are on the circle, the sum would be 180 degrees? Is that possible?

Wait, maybe I made a mistake here. Let me check again.

First, angle H is formed by two tangents from point H to the circle. The formula for the angle between two tangents is indeed half the difference of the intercepted arcs. So angle at H is 1/2*(measure of major arc NO - measure of minor arc NO).

But since the central angle for arc NO is the measure of minor arc NO, which is x, and major arc NO is 360 - x. Therefore, angle H = ( (360 - x) - x ) / 2 = (360 - 2x)/2 = 180 - x.

Then, angle J, the central angle, is x. Therefore, adding them together, angle J + angle H is x + 180 - x = 180 degrees.

Therefore, the sum is 180 degrees.

Wait, but that seems too straightforward. Let me check with a specific example. Suppose that arc NO is 120 degrees. Then angle J would be 120 degrees. Then the major arc NO is 240 degrees. Then angle H would be (240 - 120)/2 = 60 degrees. Then sum is 120 + 60 = 180.

Alternatively, if arc NO is 90 degrees. Then angle J is 90 degrees, major arc NO is 270 degrees. Then angle H is (270 - 90)/2 = 90 degrees. Sum is 180.

Another example: if arc NO is 180 degrees, then angle J is 180 degrees, major arc NO is 180 degrees, so angle H is (180 - 180)/2 = 0 degrees. Wait, but that can't be. If arc NO is 180 degrees, then points N and O are endpoints of a diameter. Then tangents from H would be... Wait, if N and O are endpoints of a diameter, then the tangents from H would be symmetrical with respect to the diameter NO. However, if angle H is 0 degrees, that would imply that H is somewhere on the line NO? Wait, but if H is on the line NO, then the tangent lines would be... Wait, if H is on the circle, but H is an external point. If H is on the line NO, then depending on where it is. Wait, but if arc NO is 180 degrees, so the diameter, and if H is outside, then the two tangent would be symmetric.

Wait, maybe in that case, angle H is 0? That seems odd. Wait, but let's think. If points N and O are endpoints of a diameter, and H is a point such that HO and HN are both tangents. Wait, if H is outside the circle, then there are two tangents from H to the circle. If N and O are endpoints of a diameter, then the two tangents from H would be symmetric across the line JO (or JN, since J is the center). However, if H is such that the two tangent points are N and O, then H lies along the line perpendicular to the diameter NO at its midpoint? Wait, no. Wait, if N and O are endpoints of diameter, then the line JO is the same as the line JO. If you have tangents from H to points N and O, then H must lie on the line that is the radical axis or something? Wait, maybe not. Let me think.

Wait, if H is such that HN and HO are tangents to the circle at N and O, then H lies on the external tangent lines. However, if N and O are endpoints of diameter, then the tangents at N and O are perpendicular to the radius. So the tangent at N is perpendicular to JN, and tangent at O is perpendicular to JO. Since JN and JO are in opposite directions (as N and O are endpoints of diameter), then the two tangents would be two lines perpendicular to JN and JO respectively. If JN is one direction, JO is opposite. Therefore, the tangent at N is perpendicular to JN, so direction is perpendicular to JN, and tangent at O is perpendicular to JO, which is same direction as tangent at N? Wait, no. If JO is opposite to JN, then the tangent at O is perpendicular to JO, which is same as perpendicular to -JN. Therefore, same as tangent at N? Wait, no. If you have a line perpendicular to JN, and another line perpendicular to JO (which is -JN), then they are both perpendicular to the same line, meaning they are parallel? Therefore, if the two tangents are parallel? But if H is outside the circle, then two tangents from H to the circle would meet at H. But if the two tangent lines are parallel, then there's no such point H. Unless the lines are the same, but they can't be the same if they are tangents at different points. Therefore, in this case, if N and O are endpoints of diameter, then the tangents at N and O are parallel, so there is no external point H from which both tangents can be drawn to points N and O. Therefore, arc NO cannot be 180 degrees. Therefore, in reality, arc NO is less than 180 degrees? Wait, but the problem didn't specify where N and O are. So maybe in the diagram, arc NO is less than 180 degrees, so that the tangents HO and HN exist.

Therefore, perhaps my calculation is okay. So in general, angle J + angle H = 180 degrees? Therefore, the answer is 180 degrees?

But let me think again. Maybe I can approach this using triangle properties instead.

Since JO and JN are radii, and HO and HN are tangents, then JO perpendicular to HO and JN perpendicular to HN. Therefore, triangles JHO and JHN are right triangles with right angles at O and N respectively.

So, in triangle JHO, right-angled at O, and triangle JHN, right-angled at N.

If we can find some relations between these triangles.

Let me denote ∠J as angle OJN? Wait, angle at J is the angle between JO and JN. Wait, JO and JN are two radii, so angle between them is the central angle. Let me denote that as ∠OJN = θ degrees. Then, angle J is θ.

Then, in triangle JHO, right-angled at O, we have angle at H as ∠OHN, which is angle H. Wait, but angle at H is common to both triangles? No, angle at H is the angle between HO and HN. Wait, in triangle JHO, there's angle at H, and in triangle JHN, there's angle at H. But since both triangles share angle at H, which is the angle we are interested in.

Wait, maybe I can use the fact that the two right triangles share the hypotenuse JH.

Wait, in triangle JHO and JHN, they both have hypotenuse HJ, and legs JO and HO, and JN and HN, respectively.

Since JO and JN are radii, they are equal in length. Therefore, triangles JHO and JHN are congruent? Because they have hypotenuse HJ in common, and legs JO and JN equal. Therefore, they are congruent by hypotenuse-leg theorem. Therefore, their corresponding angles are equal.

Therefore, angles at H in both triangles are equal. Wait, but that's the same angle at H. So that doesn't give me new information.

Alternatively, maybe I can consider the quadrilateral JNHO. It has two right angles at O and N. So the sum of all interior angles of a quadrilateral is 360 degrees. Therefore, angle J + angle H + angle O + angle N = 360 degrees.

Since angles O and N are each 90 degrees, then angle J + angle H = 360 - 90 - 90 = 180 degrees.

Oh! That's a much simpler way to see it. Since quadrilateral JNHO has two right angles, the sum of the other two angles must be 180 degrees. Therefore, angle J + angle H = 180 degrees.

Therefore, the answer is 180 degrees.

Wait, so regardless of the specific configuration, as long as HO and HN are tangents at O and N, and J is the center, then the quadrilateral JNHO has angles at O and N of 90 degrees, hence angle J + angle H = 180 degrees.

Therefore, the answer is 180 degrees.

But let me check again. If I consider the quadrilateral JNHO, it's not necessarily a convex quadrilateral? Wait, in the Asymptote code, they draw J--N--H--O--cycle. So points J, N, H, O connected in that order. So assuming that the diagram is drawn in such a way that the quadrilateral is convex. Therefore, the sum of internal angles is 360 degrees. Therefore, angle J + angle H + 90 + 90 = 360, so angle J + angle H = 180.

Yes, that seems solid. Therefore, regardless of where the points are, as long as HO and HN are tangents, then angle J + angle H is 180 degrees.

But let me think again if there's any possibility that the quadrilateral is self-intersecting or something? If points are arranged differently. But given the Asymptote code, they define J at (0,0), H at (6,0), then construct a circle with center J and radius 3. Then they find midpoint between J and H, which is at (3,0), then draw a circle with radius 3 centered at (3,0). The intersection points of the two circles are points O and N.

So, the first circle is centered at (0,0) with radius 3, the second circle is centered at (3,0) with radius 3. The intersection points of two circles with centers 3 units apart and both radius 3. So, they intersect at two points. The line connecting the centers is the x-axis from (0,0) to (3,0). The intersection points would be above and below the x-axis.

So points O and N would be located symmetrically above and below the x-axis. Then, drawing J--N--H--O--cycle would result in a quadrilateral where N is one intersection point above, O is the other below? Or both above? Wait, the Asymptote code says:

pair M = midpoint(J--H); which is (3,0). Then path secCirc = Circle(M,3). So the two circles intersect at points tangentPoints[0] and tangentPoints[1]. Since the circles are symmetric about the x-axis, the intersection points are symmetric with respect to the x-axis.

Therefore, if O is one intersection point, say, above the x-axis, then N is the other one below? Or vice versa? Wait, the code says O = tangentPoints[0]; N = tangentPoints[1]; but depending on the order. But regardless, the points are symmetric. Therefore, connecting J--N--H--O--cycle would make a quadrilateral with two sides from J to N and J to O, and sides from N to H and H to O.

But since H is at (6,0), which is to the right of the circle. So H is outside the circle. Then, connecting N (which is above the x-axis) to H to O (which is below the x-axis).

But in any case, quadrilateral JNHO would have two right angles at O and N. So regardless of the exact positions, the sum of angles at J and H is 180 degrees.

Therefore, the answer is 180 degrees.

But just to make sure, let me try to compute coordinates.

Let me try to assign coordinates as per Asymptote code. Let me set J at (0,0), H at (6,0). The circle centered at J with radius 3. The other circle is centered at midpoint of JH, which is at (3,0), and radius 3.

Equation of first circle: x² + y² = 9.

Equation of second circle: (x - 3)² + y² = 9.

Subtracting the two equations: (x - 3)² + y² - (x² + y²) = 9 - 9 => (x² -6x +9 + y²) - x² - y² = 0 => -6x +9 = 0 => x = 9/6 = 3/2.

So points of intersection lie on x = 3/2. Plugging back into first equation: (3/2)² + y² = 9 => 9/4 + y² = 9 => y² = 9 - 9/4 = 27/4 => y = ± (3√3)/2.

Therefore, points O and N are at (3/2, (3√3)/2) and (3/2, -(3√3)/2). Let me assume O is (3/2, (3√3)/2) and N is (3/2, -(3√3)/2).

So coordinates:

J = (0, 0)

H = (6, 0)

N = (3/2, - (3√3)/2)

O = (3/2, (3√3)/2)

Now, let me compute angle J and angle H.

First, angle at J: which is angle between points N, J, O. Since J is the center, and N and O are points on the circle. So angle NJO is the central angle between N and O. The coordinates of N and O are symmetric with respect to the x-axis. So the angle between them can be calculated.

The coordinates of N are (1.5, - (3√3)/2) and O are (1.5, (3√3)/2). So vector from J to N is (1.5, - (3√3)/2), and vector from J to O is (1.5, (3√3)/2).

The angle between these two vectors can be found using the dot product.

Dot product of vectors JO and JN:

JO • JN = (1.5)(1.5) + [ ( - (3√3)/2 ) ( (3√3)/2 ) ]

Wait, no. Wait, JO is from J to O: (1.5, (3√3)/2), and JN is from J to N: (1.5, - (3√3)/2). Wait, but actually, angle at J is between JN and JO. Wait, no. Wait, angle J is the angle between JN and JO? Wait, in the quadrilateral JNHO, angle at J is between JN and JO? Wait, the quadrilateral is J connected to N connected to H connected to O connected to J. Therefore, angle at J is between JN and JO? Wait, no. Wait, when you have the quadrilateral J-N-H-O-J. So the sides at J are JN and JO. Therefore, angle at J is between JN and JO. Therefore, yes, the central angle between points N and O.

Therefore, the vectors are from J to N and from J to O, which are (1.5, - (3√3)/2 ) and (1.5, (3√3)/2 ). Therefore, the angle between them can be found by the dot product formula.

First, let me compute the dot product:

JO • JN = (1.5)(1.5) + [ ( (3√3)/2 ) * ( - (3√3)/2 ) ]

Wait, no: JO is vector to O: (1.5, (3√3)/2), and JN is vector to N: (1.5, - (3√3)/2). Therefore, the dot product is:

(1.5)(1.5) + [ ( (3√3)/2 ) * ( - (3√3)/2 ) ]

Calculating:

1.5 * 1.5 = 2.25

( (3√3)/2 ) * ( - (3√3)/2 ) = - (9 * 3)/4 = -27/4 = -6.75

Therefore, total dot product is 2.25 - 6.75 = -4.5

Now, the magnitude of JO and JN vectors are both radii, which is 3 units. Wait, let me confirm:

The length of JO is sqrt( (1.5)^2 + ( (3√3)/2 )^2 )

Calculating:

(1.5)^2 = 2.25

( (3√3)/2 )^2 = 9*3 / 4 = 27/4 = 6.75

Total is 2.25 + 6.75 = 9, sqrt(9) = 3. Correct.

Therefore, the dot product is |JO| |JN| cos(theta), where theta is the angle between them. Therefore:

-4.5 = 3 * 3 * cos(theta) => -4.5 = 9 cos(theta) => cos(theta) = -4.5 / 9 = -0.5

Therefore, theta = arccos(-0.5) which is 120 degrees.

So measure of angle J is 120 degrees.

Now let's compute measure of angle H.

Angle at H is the angle between HO and HN. Points H is at (6,0), O is at (1.5, (3√3)/2 ), N is at (1.5, - (3√3)/2 ).

So vectors from H to O and from H to N.

Vector HO: O - H = (1.5 - 6, (3√3)/2 - 0 ) = (-4.5, (3√3)/2 )

Vector HN: N - H = (1.5 - 6, - (3√3)/2 - 0 ) = (-4.5, - (3√3)/2 )

So angle at H is the angle between vectors HO and HN.

Let me compute the angle between these two vectors using the dot product formula.

First, compute the dot product:

HO • HN = (-4.5)(-4.5) + [ (3√3)/2 * (-3√3)/2 ]

Calculating:

(-4.5)^2 = 20.25

( (3√3)/2 * (-3√3)/2 ) = - (9 * 3)/4 = -27/4 = -6.75

Therefore, dot product = 20.25 - 6.75 = 13.5

Now compute the magnitudes of HO and HN. Since HO and HN are both tangent segments from H to the circle, they should have the same length.

Let me compute the length of HO:

sqrt( (-4.5)^2 + ( (3√3)/2 )^2 )

= sqrt( 20.25 + 6.75 )

= sqrt(27)

= 3√3

Similarly for HN.

Therefore, |HO| = |HN| = 3√3

Thus, the dot product is |HO||HN| cos(theta), so:

13.5 = (3√3)(3√3) cos(theta) = 9 * 3 cos(theta) = 27 cos(theta)

Therefore, cos(theta) = 13.5 / 27 = 0.5

Therefore, theta = arccos(0.5) = 60 degrees

So measure of angle H is 60 degrees.

Therefore, angle J + angle H = 120 + 60 = 180 degrees.

Therefore, this specific case gives 180 degrees.

So that confirms my previous conclusion. Therefore, regardless of the position of points N and O, as long as HO and HN are tangents, angle J + angle H is 180 degrees.

Alternatively, since we saw that in quadrilateral JNHO, two angles are 90 degrees each, so the other two angles must add to 180. Therefore, answer is 180 degrees.

Therefore, the answer is \boxed{180} degrees.

**Final Answer**
\boxed{180}

To determine the sum of the measures of $ \angle J $ and $ \angle H $ in the given geometric configuration, we start by analyzing the structure of the quadrilateral $ JNHO $, which is formed by:

- $ J $, the center of the circle,
- $ H $, an external point from which two tangents $ HO $ and $ HN $ are drawn to the circle,
- $ O $ and $ N $, the points of tangency on the circle.

---

### Step 1: Understanding the Angle at $ J $

Since $ HO $ and $ HN $ are tangents to the circle at $ O $ and $ N $, the line segments $ JO $ and $ JN $ are radii of the circle and are perpendicular to the tangents at the points of tangency. This implies:

$$
\angle JO H = 90^\circ \quad \text{and} \quad \angle JN H = 90^\circ
$$

Therefore, in quadrilateral $ JNHO $, two of its interior angles at $ O $ and $ N $ are right angles.

---

### Step 2: Applying the Quadrilateral Angle Sum

The sum of the interior angles of any quadrilateral is always $ 360^\circ $. Given that two of the angles are right angles (each $ 90^\circ $), the remaining two angles — $ \angle J $ and $ \angle H $ — must satisfy:

$$
\angle J + \angle H + 90^\circ + 90^\circ = 360^\circ
$$

Simplifying:

$$
\angle J + \angle H = 360^\circ - 180^\circ = 180^\circ
$$

---

### Step 3: Verifying with Coordinate Geometry (Optional)

Alternatively, using coordinate geometry, by assigning coordinates based on the Asymptote diagram and calculating the vectors $ \vec{JO} $, $ \vec{JN} $, and the resulting angle between them (at point $ J $), we can also compute $ \angle J = 120^\circ $. Similarly, computing $ \angle H $ as the angle between vectors from $ H $ to points $ O $ and $ N $, we find it to be $ 60^\circ $. Adding these gives:

$$
\angle J + \angle H = 120^\circ + 60^\circ = 180^\circ
$$

Which confirms the earlier result.

---

### Final Answer

$$
\boxed{180}
$$

Solution: Since $\overline{OH}$ and $\overline{NH}$ are tangent to radii of the circle at $O$ and $N$, we have $\angle O =\angle N = 90^\circ$.  The sum of the measures of the interior angles of quadrilateral $JOHN$ is $360^\circ$, so $\angle J + \angle H = 360^\circ - \angle O - \angle N = \boxed{180^\circ}$.


Explanation:
Okay, so I need to find the remainder when the sum of numbers from 1 to 10 is divided by 9. Let me think about how to approach this.

First, maybe I should calculate the sum of numbers from 1 to 10. I remember there's a formula for the sum of an arithmetic series. The formula is n(n + 1)/2, right? Where n is the last number in the sequence. In this case, n is 10. Let me apply that formula.

So, plugging in n = 10: 10*(10 + 1)/2. Let me calculate that. 10 + 1 is 11, so 10*11 is 110. Then divide by 2, which gives 55. So the sum of numbers from 1 to 10 is 55.

Now, the question asks for the remainder when this sum is divided by 9. So I need to compute 55 divided by 9 and find the remainder. Let me do that division.

First, how many times does 9 go into 55? Let me see: 9*6 is 54, and 9*7 is 63. Since 63 is too big, it goes in 6 times. So 9*6 = 54. Subtract that from 55, and the remainder is 55 - 54 = 1.

Wait, so the remainder is 1? Let me check that again to be sure.

Alternatively, maybe I can use another method to verify. Maybe adding the numbers step by step and then taking the modulus at each step? Let me try that.

Starting with 1. Then 1 + 2 = 3. Then 3 + 3 = 6. Then 6 + 4 = 10. Then 10 + 5 = 15. Then 15 + 6 = 21. Then 21 + 7 = 28. Then 28 + 8 = 36. Then 36 + 9 = 45. Then 45 + 10 = 55. So that's the same sum as before. Now, if I take each step modulo 9, maybe that would help? Let me try:

Start with 1. 1 mod 9 is 1. Then 1 + 2 = 3 mod 9 is 3. 3 + 3 = 6 mod 9 is 6. 6 + 4 = 10 mod 9 is 1 (since 10 - 9 = 1). Then 1 + 5 = 6 mod 9 is 6. 6 + 6 = 12 mod 9 is 3 (since 12 - 9 = 3). 3 + 7 = 10 mod 9 is 1. 1 + 8 = 9 mod 9 is 0. 0 + 9 = 9 mod 9 is 0. 0 + 10 = 10 mod 9 is 1.

So according to this step-by-step modulus calculation, the remainder is 1. That matches the previous result. So that seems to confirm that the remainder is 1.

Alternatively, maybe there's another way to compute this using properties of modular arithmetic. Let me think.

Since we need the sum from 1 to 10 mod 9, maybe I can compute each term mod 9 first and then add them up. Let's try that.

Numbers from 1 to 10: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

Compute each modulo 9:

1 mod 9 = 1

2 mod 9 = 2

3 mod 9 = 3

4 mod 9 = 4

5 mod 9 = 5

6 mod 9 = 6

7 mod 9 = 7

8 mod 9 = 8

9 mod 9 = 0

10 mod 9 = 1 (since 10 - 9 = 1)

Now add all these up: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 0 + 1.

Let me compute that:

1 + 2 = 3

3 + 3 = 6

6 + 4 = 10

10 + 5 = 15

15 + 6 = 21

21 + 7 = 28

28 + 8 = 36

36 + 0 = 36

36 + 1 = 37

So the total is 37. Now, find 37 mod 9.

How many times does 9 go into 37? 9*4 = 36, so 37 - 36 = 1. So 37 mod 9 is 1. Again, same result.

Therefore, regardless of the method I use, the remainder is 1.

Wait, but just to make sure I didn't make a mistake in my calculations, let me check once again.

First method: sum is 55, 55 divided by 9 is 6*9=54, remainder 1. Correct.

Second method: step-by-step addition with modulus, ended up with 1.

Third method: adding each term's modulus and then taking modulus again, got 37, which mod 9 is 1.

So all three methods give the same answer, which is 1.

Alternatively, is there another way? Maybe using the formula for the sum and then taking modulus?

So the formula gives sum = 10*11/2 = 55. Then 55 mod 9.

Alternatively, compute 55 mod 9.

But 9*6=54, so 55-54=1.

Alternatively, think of 55 as 5*11. But maybe not helpful.

Alternatively, use the property that (a + b) mod m = [(a mod m) + (b mod m)] mod m.

But since we already did that, maybe there's no need.

Alternatively, maybe think of 1 to 10 as numbers 1 to 9 plus 10.

Sum from 1 to 9 is 45, as 9*10/2=45. Then add 10, so 45 + 10 = 55. Then 45 mod 9 is 0, since 45 is divisible by 9. Then 0 + 10 mod 9 = 10 mod 9 = 1.

So that's another way.

Yes, that works. So if I split the sum into 1 to 9 and then 10. Since the sum from 1 to 9 is 45, which is divisible by 9, so 45 mod 9 is 0, then adding 10. 10 mod 9 is 1, so total remainder is 1.

So that's another confirmation.

Therefore, regardless of how I approach it, the remainder is 1.

Therefore, I'm confident the answer is 1.

But just to make sure, let me check with actual division.

Divide 55 by 9. 9*6=54. 55-54=1. So quotient is 6, remainder 1.

Yes.

Alternatively, I can check with another modulus. For example, if I have 55 divided by 9, remainder 1. So 55 = 9*6 + 1. Which is correct.

Alternatively, if I think about 9*6 is 54, which is 54, and then 55 is one more. So yes, the remainder is 1.

Alternatively, maybe I can use negative numbers? Let me think. For example, 55 divided by 9.

But that might complicate.

Alternatively, think of numbers mod 9. For example, the numbers 1 through 10 mod 9.

Which is 1,2,3,4,5,6,7,8,0,1. Adding them up gives 1+2+3+4+5+6+7+8+0+1. Which is (1+2+3+4+5+6+7+8) + 0 +1.

Sum from 1 to 8 is 36. Then 36 + 0 + 1 is 37. Then 37 mod 9.

37 divided by 9 is 4*9=36, remainder 1.

Yes.

Alternatively, maybe I can think of 37 as 36 +1. 36 is 4*9, so remainder 1.

Yes.

So all methods lead to remainder 1.

Therefore, the answer is 1.

I think that's thorough enough. I considered multiple methods, checked each step, verified with different approaches. All lead to the same answer. So I feel confident that the remainder is indeed 1 when 1 + 2 + ... + 10 is divided by 9.

**Final Answer**
The remainder is \boxed{1}.

To solve the problem of finding the remainder when the sum $ 1 + 2 + 3 + \dots + 10 $ is divided by 9, we can proceed in several steps using both direct computation and modular arithmetic techniques.

---

### Step 1: Compute the Total Sum

We use the formula for the sum of the first $ n $ natural numbers:

$$
\text{Sum} = \frac{n(n+1)}{2}
$$

For $ n = 10 $:

$$
\text{Sum} = \frac{10 \cdot 11}{2} = 55
$$

So, the total sum of the numbers from 1 to 10 is **55**.

---

### Step 2: Find the Remainder When Divided by 9

We now calculate $ 55 \mod 9 $:

$$
55 \div 9 = 6 \text{ remainder } 1
$$

So,

$$
55 \equiv 1 \mod 9
$$

Therefore, the **remainder** when dividing the sum of 1 to 10 by 9 is **1**.

---

### Step 3: Verifying Using Modular Arithmetic Properties

Alternatively, we can compute the sum by taking each term modulo 9, then summing those results modulo 9 to ensure correctness.

The values from 1 to 10 modulo 9 are:

$$
1, 2, 3, 4, 5, 6, 7, 8, 0, 1
$$

Now, sum them:

$$
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 0 + 1 = 37
$$

Then take $ 37 \mod 9 $:

$$
37 \div 9 = 4 \text{ remainder } 1
$$

So:

$$
37 \equiv 1 \mod 9
$$

This confirms our result.

---

### Step 4: Alternative Strategy — Splitting the Sum

We can further verify by splitting the sum:

$$
(1 + 2 + \dots + 9) = 45
\quad \text{and} \quad
45 \mod 9 = 0
$$

Then add 10:

$$
45 + 10 = 55 \quad \text{and} \quad 55 \mod 9 = 1
$$

Again, this confirms the remainder is **1**.

---

### Final Conclusion

All methods consistently lead to the same result. Therefore, the remainder when $ 1 + 2 + 3 + \dots + 10 $ is divided by 9 is:

$$
\boxed{1}
$$

[Assistant]
Explanation:
Okay, so I need to find the remainder when the sum of numbers from 1 to 10 is divided by 9. Let me think about how to approach this.

First, maybe I should calculate the sum of numbers from 1 to 10. I remember there's a formula for the sum of an arithmetic series. The formula is n(n + 1)/2, right? Where n is the last number in the sequence. In this case, n is 10. Let me apply that formula.

So, plugging in n = 10: 10*(10 + 1)/2. Let me calculate that. 10 + 1 is 11, so 10*11 is 110. Then divide by 2, which gives 55. So the sum of numbers from 1 to 10 is 55.

Now, the question asks for the remainder when this sum is divided by 9. So I need to compute 55 divided by 9 and find the remainder. Let me do that division.

First, how many times does 9 go into 55? Let me see: 9*6 is 54, and 9*7 is 63. Since 63 is too big, it goes in 6 times. So 9*6 = 54. Subtract that from 55, and the remainder is 55 - 54 = 1.

Wait, so the remainder is 1? Let me check that again to be sure.

Alternatively, maybe I can use another method to verify. Maybe adding the numbers step by step and then taking the modulus at each step? Let me try that.

Starting with 1. Then 1 + 2 = 3. Then 3 + 3 = 6. Then 6 + 4 = 10. Then 10 + 5 = 15. Then 15 + 6 = 21. Then 21 + 7 = 28. Then 28 + 8 = 36. Then 36 + 9 = 45. Then 45 + 10 = 55. So that's the same sum as before. Now, if I take each step modulo 9, maybe that would help? Let me try:

Start with 1. 1 mod 9 is 1. Then 1 + 2 = 3 mod 9 is 3. 3 + 3 = 6 mod 9 is 6. 6 + 4 = 10 mod 9 is 1 (since 10 - 9 = 1). Then 1 + 5 = 6 mod 9 is 6. 6 + 6 = 12 mod 9 is 3 (since 12 - 9 = 3). 3 + 7 = 10 mod 9 is 1. 1 + 8 = 9 mod 9 is 0. 0 + 9 = 9 mod 9 is 0. 0 + 10 = 10 mod 9 is 1.

So according to this step-by-step modulus calculation, the remainder is 1. That matches the previous result. So that seems to confirm that the remainder is 1.

Alternatively, maybe there's another way to compute this using properties of modular arithmetic. Let me think.

Since we need the sum from 1 to 10 mod 9, maybe I can compute each term mod 9 first and then add them up. Let's try that.

Numbers from 1 to 10: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

Compute each modulo 9:

1 mod 9 = 1

2 mod 9 = 2

3 mod 9 = 3

4 mod 9 = 4

5 mod 9 = 5

6 mod 9 = 6

7 mod 9 = 7

8 mod 9 = 8

9 mod 9 = 0

10 mod 9 = 1 (since 10 - 9 = 1)

Now add all these up: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 0 + 1.

Let me compute that:

1 + 2 = 3

3 + 3 = 6

6 + 4 = 10

10 + 5 = 15

15 + 6 = 21

21 + 7 = 28

28 + 8 = 36

36 + 0 = 36

36 + 1 = 37

So the total is 37. Now, find 37 mod 9.

How many times does 9 go into 37? 9*4 = 36, so 37 - 36 = 1. So 37 mod 9 is 1. Again, same result.

Therefore, regardless of the method I use, the remainder is 1.

Wait, but just to make sure I didn't make a mistake in my calculations, let me check once again.

First method: sum is 55, 55 divided by 9 is 6*9=54, remainder 1. Correct.

Second method: step-by-step addition with modulus, ended up with 1.

Third method: adding each term's modulus and then taking modulus again, got 37, which mod 9 is 1.

So all three methods give the same answer, which is 1.

Alternatively, is there another way? Maybe using the formula for the sum and then taking modulus?

So the formula gives sum = 10*11/2 = 55. Then 55 mod 9.

Alternatively, compute 55 mod 9.

But 9*6=54, so 55-54=1.

Alternatively, think of 55 as 5*11. But maybe not helpful.

Alternatively, use the property that (a + b) mod m = [(a mod m) + (b mod m)] mod m.

But since we already did that, maybe there's no need.

Alternatively, maybe think of 1 to 10 as numbers 1 to 9 plus 10.

Sum from 1 to 9 is 45, as 9*10/2=45. Then add 10, so 45 + 10 = 55. Then 45 mod 9 is 0, since 45 is divisible by 9. Then 0 + 10 mod 9 = 10 mod 9 = 1.

So that's another way.

Yes, that works. So if I split the sum into 1 to 9 and then 10. Since the sum from 1 to 9 is 45, which is divisible by 9, so 45 mod 9 is 0, then adding 10. 10 mod 9 is 1, so total remainder is 1.

So that's another confirmation.

Therefore, regardless of how I approach it, the remainder is 1.

Therefore, I'm confident the answer is 1.

But just to make sure, let me check with actual division.

Divide 55 by 9. 9*6=54. 55-54=1. So quotient is 6, remainder 1.

Yes.

Alternatively, I can check with another modulus. For example, if I have 55 divided by 9, remainder 1. So 55 = 9*6 + 1. Which is correct.

Alternatively, if I think about 9*6 is 54, which is 54, and then 55 is one more. So yes, the remainder is 1.

Alternatively, maybe I can use negative numbers? Let me think. For example, 55 divided by 9.

But that might complicate.

Alternatively, think of numbers mod 9. For example, the numbers 1 through 10 mod 9.

Which is 1,2,3,4,5,6,7,8,0,1. Adding them up gives 1+2+3+4+5+6+7+8+0+1. Which is (1+2+3+4+5+6+7+8) + 0 +1.

Sum from 1 to 8 is 36. Then 36 + 0 + 1 is 37. Then 37 mod 9.

37 divided by 9 is 4*9=36, remainder 1.

Yes.

Alternatively, maybe I can think of 37 as 36 +1. 36 is 4*9, so remainder 1.

Yes.

So all methods lead to remainder 1.

Therefore, the answer is 1.

I think that's thorough enough. I considered multiple methods, checked each step, verified with different approaches. All lead to the same answer. So I feel confident that the remainder is indeed 1 when 1 + 2 + ... + 10 is divided by 9.

**Final Answer**
The remainder is \boxed{1}.

To solve the problem of finding the remainder when the sum $ 1 + 2 + 3 + \dots + 10 $ is divided by 9, we can proceed in several steps using both direct computation and modular arithmetic techniques.

---

### Step 1: Compute the Total Sum

We use the formula for the sum of the first $ n $ natural numbers:

$$
\text{Sum} = \frac{n(n+1)}{2}
$$

For $ n = 10 $:

$$
\text{Sum} = \frac{10 \cdot 11}{2} = 55
$$

So, the total sum of the numbers from 1 to 10 is **55**.

---

### Step 2: Find the Remainder When Divided by 9

We now calculate $ 55 \mod 9 $:

$$
55 \div 9 = 6 \text{ remainder } 1
$$

So,

$$
55 \equiv 1 \mod 9
$$

Therefore, the **remainder** when dividing the sum of 1 to 10 by 9 is **1**.

---

### Step 3: Verifying Using Modular Arithmetic Properties

Alternatively, we can compute the sum by taking each term modulo 9, then summing those results modulo 9 to ensure correctness.

The values from 1 to 10 modulo 9 are:

$$
1, 2, 3, 4, 5, 6, 7, 8, 0, 1
$$

Now, sum them:

$$
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 0 + 1 = 37
$$

Then take $ 37 \mod 9 $:

$$
37 \div 9 = 4 \text{ remainder } 1
$$

So:

$$
37 \equiv 1 \mod 9
$$

This confirms our result.

---

### Step 4: Alternative Strategy — Splitting the Sum

We can further verify by splitting the sum:

$$
(1 + 2 + \dots + 9) = 45
\quad \text{and} \quad
45 \mod 9 = 0
$$

Then add 10:

$$
45 + 10 = 55 \quad \text{and} \quad 55 \mod 9 = 1
$$

Again, this confirms the remainder is **1**.

---

### Final Conclusion

All methods consistently lead to the same result. Therefore, the remainder when $ 1 + 2 + 3 + \dots + 10 $ is divided by 9 is:

$$
\boxed{1}
$$

Solution: Looking at our sum, we can see that the numbers $1$ through $8$ can be paired off to form $9,$ so we may eliminate them. That is, $1 + 8 = 2 + 7 = 3 + 6 = 4 + 5 = 9.$ Therefore, the only remaining terms are $9$ and $10,$ and $9$ is obviously also divisible by $9,$ hence we only need to find the remainder of $10$ when divided by $9,$ which is $\boxed{1}.$


Explanation:
Okay, let's see. The problem is about Susan and Pam buying tickets to a show with different discounts, and we need to find out how many more dollars Pam pays than Susan. Hmm, let me break this down step by step.

First, let me note down the given information:

- The full price of one ticket is $20.
- Susan buys 4 tickets with a 25% discount coupon.
- Pam buys 5 tickets with a 30% discount coupon.
- We need to find the difference in the amount they paid, specifically how much more Pam paid compared to Susan.

Alright, so I need to calculate the total amount Susan paid and the total amount Pam paid, then subtract Susan's total from Pam's total to find the difference. Let me start with Susan.

Calculating Susan's total:

Susan buys 4 tickets. The original price per ticket is $20. She has a 25% discount. So, first, I need to find out how much she saves with the discount and then subtract that from the total full price, or alternatively, calculate the discounted price per ticket and multiply by the number of tickets. Either way should work. Maybe calculating the discounted price per ticket is better here.

So, 25% discount on $20. Let me compute 25% of $20. 25% is 0.25 in decimal. So 0.25 multiplied by 20.

Let me compute that: 0.25 * 20. 20 divided by 4 is 5, so 25% of 20 is $5. Therefore, the discount per ticket is $5. That means Susan pays $20 - $5 = $15 per ticket.

Alternatively, I can calculate it as 100% - 25% = 75% of the original price. So 75% of $20 is 0.75 * 20. Let me check that: 0.75 * 20. Well, 0.75 is three quarters, so 20 * 3/4 is 15. Yep, same result. So $15 per ticket.

Therefore, Susan buys 4 tickets at $15 each. So total for Susan is 4 * 15.

Calculating 4 * 15: 4*10=40, 4*5=20, so 40+20=60. Therefore, Susan pays $60 total.

Now, let me do the same for Pam.

Pam buys 5 tickets with a 30% discount. Again, let's find the discounted price per ticket.

Original price is $20. 30% discount. So first, let's find 30% of $20.

30% is 0.30. So 0.30 * 20. 0.3 * 20. Well, 0.1*20=2, so 0.3*20 is 6. Therefore, the discount per ticket is $6. So Pam pays $20 - $6 = $14 per ticket.

Alternatively, using the same method as before, 100% - 30% = 70%. So 70% of $20. 0.70 * 20. Let me compute that: 0.7 * 20. Well, 0.7*10=7, so 0.7*20=14. Yep, same result. So $14 per ticket.

Therefore, Pam buys 5 tickets at $14 each. Total for Pam is 5 * 14.

Calculating 5*14: 5*10=50, 5*4=20, so 50+20=70. Therefore, Pam pays $70 total.

Now, the question is asking how many more dollars Pam pays than Susan. That would be Pam's total minus Susan's total. So that's $70 - $60.

Which is $10. So Pam pays $10 more than Susan.

Wait, but let me double-check my calculations to make sure I didn't make a mistake. It's easy to mix up percentages or multiplication.

First, Susan: 4 tickets at 25% off. $20 each. 25% of 20 is 5, so each ticket is 15. 4*15 is 60. That seems correct.

Pam: 5 tickets at 30% off. 30% of 20 is 6, so each ticket is 14. 5*14 is 70. That also seems correct.

Difference: 70 - 60 = 10. So yes, $10. So the answer is 10?

Wait, but let me think again. Maybe there's a different way to approach it? For instance, calculating total discount amount and then subtracting? Let me try that for verification.

For Susan, total original price for 4 tickets: 4*20 = $80. Then, 25% discount on $80. So 25% of 80 is 0.25 * 80 = $20. Therefore, Susan pays 80 - 20 = $60. Same as before.

For Pam, original total price for 5 tickets: 5*20 = $100. 30% discount on $100. So 0.30 * 100 = $30. Therefore, Pam pays 100 - 30 = $70. Same result as before. Then difference is $70 - $60 = $10. So same answer.

Therefore, that seems consistent.

Alternatively, maybe I can compute the amount each paid using decimals.

For Susan:

Each ticket price after discount: 20*(1 - 0.25) = 20*0.75 = 15. Multiply by 4: 15*4=60. Same.

For Pam:

Each ticket price after discount: 20*(1 - 0.30) = 20*0.70 = 14. Multiply by 5: 14*5=70. Same. So difference is 70 - 60 = 10.

So all methods lead to the same answer. Therefore, I think the answer is $10. Therefore, Pam pays $10 more than Susan.

But just to be thorough, let me check if I interpreted the problem correctly. The question says: "How many more dollars does Pam pay than Susan?" So, it's Pam's total minus Susan's total. Which is 70 - 60 = 10. So yes, that's right. So the answer is 10.

Wait, but let me make sure that the problem isn't trying to trick me in some way. For example, maybe the coupons are applied differently? Like, maybe the coupon is per ticket or something? But the problem says Susan buys 4 tickets using a coupon that gives her a 25% discount. So I think that means that the 25% discount is applied to each ticket, or to the total. But in any case, since the discount is percentage, it's standard to apply it per item or total. However, if it's a per ticket discount, it's same as applying per item. Let me check with another approach.

Suppose the coupon gives 25% discount on each ticket. So Susan buys 4 tickets, each with 25% off. So same as before. So each ticket is 15, total 60. Similarly for Pam. So same result. So the answer is 10.

Alternatively, if the coupon was applied to the total purchase, then same thing. For example, if the total for Susan was 4*20 = 80, then 25% off the total is 80 - 20 = 60, same as before. So regardless, it's same result.

So, yeah, seems like 10 is the answer.

But wait, let me check if I miscalculated 5*14? 5 times 10 is 50, 5 times 4 is 20, so 50+20=70. Correct. 4*15 is 60. Correct. 70-60=10. Correct.

Alternatively, maybe the problem is in the percentage? Let me check:

25% discount on Susan's tickets. 25% of 20 is 5, so each ticket is 15. 5 tickets for Pam: 30% discount. 30% of 20 is 6. So 14 per ticket. Correct. So the difference is 70 - 60. Correct.

Alternatively, maybe the problem is in the number of tickets? Susan buys 4, Pam buys 5. Correct. So:

Total for Susan: 4*15=60

Total for Pam: 5*14=70

Difference: 10. Yes.

So I think I've checked this in multiple ways. All leading to the same answer. So I think the answer is 10.

But let me look again at the problem statement:

"One ticket to a show costs $20 at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buys 5 tickets using a coupon that gives her a 30% discount. How many more dollars does Pam pay than Susan?"

Yes, same as before. So, answer is 10. Therefore, the answer is 10 dollars.

**Final Answer**
\boxed{10}

To determine how many more dollars Pam pays than Susan, we begin by calculating the total cost each person incurs after applying their respective discounts.

---

### **Step 1: Susan’s Total Cost**

- **Ticket price per person (full price):** \$20
- **Number of tickets purchased by Susan:** 4
- **Discount offered to Susan:** 25%

**Discount amount per ticket:**
$$
25\% \text{ of } \$20 = 0.25 \times 20 = \$5
$$

**Discounted price per ticket:**
$$
\$20 - \$5 = \$15
$$

**Total amount Susan pays:**
$$
4 \times \$15 = \$60
$$

---

### **Step 2: Pam’s Total Cost**

- **Ticket price per person (full price):** \$20
- **Number of tickets purchased by Pam:** 5
- **Discount offered to Pam:** 30%

**Discount amount per ticket:**
$$
30\% \text{ of } \$20 = 0.30 \times 20 = \$6
$$

**Discounted price per ticket:**
$$
\$20 - \$6 = \$14
$$

**Total amount Pam pays:**
$$
5 \times \$14 = \$70
$$

---

### **Step 3: Difference in Payment**

We now find how much more Pam pays compared to Susan:
$$
\text{Difference} = \$70 - \$60 = \$10
$$

---

### **Final Answer**

$$
\boxed{10}
$$

[Assistant]
Explanation:
Okay, let's see. The problem is about Susan and Pam buying tickets to a show with different discounts, and we need to find out how many more dollars Pam pays than Susan. Hmm, let me break this down step by step.

First, let me note down the given information:

- The full price of one ticket is $20.
- Susan buys 4 tickets with a 25% discount coupon.
- Pam buys 5 tickets with a 30% discount coupon.
- We need to find the difference in the amount they paid, specifically how much more Pam paid compared to Susan.

Alright, so I need to calculate the total amount Susan paid and the total amount Pam paid, then subtract Susan's total from Pam's total to find the difference. Let me start with Susan.

Calculating Susan's total:

Susan buys 4 tickets. The original price per ticket is $20. She has a 25% discount. So, first, I need to find out how much she saves with the discount and then subtract that from the total full price, or alternatively, calculate the discounted price per ticket and multiply by the number of tickets. Either way should work. Maybe calculating the discounted price per ticket is better here.

So, 25% discount on $20. Let me compute 25% of $20. 25% is 0.25 in decimal. So 0.25 multiplied by 20.

Let me compute that: 0.25 * 20. 20 divided by 4 is 5, so 25% of 20 is $5. Therefore, the discount per ticket is $5. That means Susan pays $20 - $5 = $15 per ticket.

Alternatively, I can calculate it as 100% - 25% = 75% of the original price. So 75% of $20 is 0.75 * 20. Let me check that: 0.75 * 20. Well, 0.75 is three quarters, so 20 * 3/4 is 15. Yep, same result. So $15 per ticket.

Therefore, Susan buys 4 tickets at $15 each. So total for Susan is 4 * 15.

Calculating 4 * 15: 4*10=40, 4*5=20, so 40+20=60. Therefore, Susan pays $60 total.

Now, let me do the same for Pam.

Pam buys 5 tickets with a 30% discount. Again, let's find the discounted price per ticket.

Original price is $20. 30% discount. So first, let's find 30% of $20.

30% is 0.30. So 0.30 * 20. 0.3 * 20. Well, 0.1*20=2, so 0.3*20 is 6. Therefore, the discount per ticket is $6. So Pam pays $20 - $6 = $14 per ticket.

Alternatively, using the same method as before, 100% - 30% = 70%. So 70% of $20. 0.70 * 20. Let me compute that: 0.7 * 20. Well, 0.7*10=7, so 0.7*20=14. Yep, same result. So $14 per ticket.

Therefore, Pam buys 5 tickets at $14 each. Total for Pam is 5 * 14.

Calculating 5*14: 5*10=50, 5*4=20, so 50+20=70. Therefore, Pam pays $70 total.

Now, the question is asking how many more dollars Pam pays than Susan. That would be Pam's total minus Susan's total. So that's $70 - $60.

Which is $10. So Pam pays $10 more than Susan.

Wait, but let me double-check my calculations to make sure I didn't make a mistake. It's easy to mix up percentages or multiplication.

First, Susan: 4 tickets at 25% off. $20 each. 25% of 20 is 5, so each ticket is 15. 4*15 is 60. That seems correct.

Pam: 5 tickets at 30% off. 30% of 20 is 6, so each ticket is 14. 5*14 is 70. That also seems correct.

Difference: 70 - 60 = 10. So yes, $10. So the answer is 10?

Wait, but let me think again. Maybe there's a different way to approach it? For instance, calculating total discount amount and then subtracting? Let me try that for verification.

For Susan, total original price for 4 tickets: 4*20 = $80. Then, 25% discount on $80. So 25% of 80 is 0.25 * 80 = $20. Therefore, Susan pays 80 - 20 = $60. Same as before.

For Pam, original total price for 5 tickets: 5*20 = $100. 30% discount on $100. So 0.30 * 100 = $30. Therefore, Pam pays 100 - 30 = $70. Same result as before. Then difference is $70 - $60 = $10. So same answer.

Therefore, that seems consistent.

Alternatively, maybe I can compute the amount each paid using decimals.

For Susan:

Each ticket price after discount: 20*(1 - 0.25) = 20*0.75 = 15. Multiply by 4: 15*4=60. Same.

For Pam:

Each ticket price after discount: 20*(1 - 0.30) = 20*0.70 = 14. Multiply by 5: 14*5=70. Same. So difference is 70 - 60 = 10.

So all methods lead to the same answer. Therefore, I think the answer is $10. Therefore, Pam pays $10 more than Susan.

But just to be thorough, let me check if I interpreted the problem correctly. The question says: "How many more dollars does Pam pay than Susan?" So, it's Pam's total minus Susan's total. Which is 70 - 60 = 10. So yes, that's right. So the answer is 10.

Wait, but let me make sure that the problem isn't trying to trick me in some way. For example, maybe the coupons are applied differently? Like, maybe the coupon is per ticket or something? But the problem says Susan buys 4 tickets using a coupon that gives her a 25% discount. So I think that means that the 25% discount is applied to each ticket, or to the total. But in any case, since the discount is percentage, it's standard to apply it per item or total. However, if it's a per ticket discount, it's same as applying per item. Let me check with another approach.

Suppose the coupon gives 25% discount on each ticket. So Susan buys 4 tickets, each with 25% off. So same as before. So each ticket is 15, total 60. Similarly for Pam. So same result. So the answer is 10.

Alternatively, if the coupon was applied to the total purchase, then same thing. For example, if the total for Susan was 4*20 = 80, then 25% off the total is 80 - 20 = 60, same as before. So regardless, it's same result.

So, yeah, seems like 10 is the answer.

But wait, let me check if I miscalculated 5*14? 5 times 10 is 50, 5 times 4 is 20, so 50+20=70. Correct. 4*15 is 60. Correct. 70-60=10. Correct.

Alternatively, maybe the problem is in the percentage? Let me check:

25% discount on Susan's tickets. 25% of 20 is 5, so each ticket is 15. 5 tickets for Pam: 30% discount. 30% of 20 is 6. So 14 per ticket. Correct. So the difference is 70 - 60. Correct.

Alternatively, maybe the problem is in the number of tickets? Susan buys 4, Pam buys 5. Correct. So:

Total for Susan: 4*15=60

Total for Pam: 5*14=70

Difference: 10. Yes.

So I think I've checked this in multiple ways. All leading to the same answer. So I think the answer is 10.

But let me look again at the problem statement:

"One ticket to a show costs $20 at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buys 5 tickets using a coupon that gives her a 30% discount. How many more dollars does Pam pay than Susan?"

Yes, same as before. So, answer is 10. Therefore, the answer is 10 dollars.

**Final Answer**
\boxed{10}

To determine how many more dollars Pam pays than Susan, we begin by calculating the total cost each person incurs after applying their respective discounts.

---

### **Step 1: Susan’s Total Cost**

- **Ticket price per person (full price):** \$20
- **Number of tickets purchased by Susan:** 4
- **Discount offered to Susan:** 25%

**Discount amount per ticket:**
$$
25\% \text{ of } \$20 = 0.25 \times 20 = \$5
$$

**Discounted price per ticket:**
$$
\$20 - \$5 = \$15
$$

**Total amount Susan pays:**
$$
4 \times \$15 = \$60
$$

---

### **Step 2: Pam’s Total Cost**

- **Ticket price per person (full price):** \$20
- **Number of tickets purchased by Pam:** 5
- **Discount offered to Pam:** 30%

**Discount amount per ticket:**
$$
30\% \text{ of } \$20 = 0.30 \times 20 = \$6
$$

**Discounted price per ticket:**
$$
\$20 - \$6 = \$14
$$

**Total amount Pam pays:**
$$
5 \times \$14 = \$70
$$

---

### **Step 3: Difference in Payment**

We now find how much more Pam pays compared to Susan:
$$
\text{Difference} = \$70 - \$60 = \$10
$$

---

### **Final Answer**

$$
\boxed{10}
$$

Solution:
We need to calculate the total purchase price that Susan paid and that Pam paid.

Susan bought $4$ tickets with $25\%$ discount: $$4 \times \$20 = \$80.$$With a $25$ percent discount, she paid $\$80 * .75 = \$60.$

Pam bought $5$ tickets with a $30\%$ discount: $$5 \times \$20 = \$100$$With a $30$ percent discount, she paid $\$100 * .70 = \$70.$

Thus Pam paid $\$70 - \$60 = \$\boxed{10}$ more than Susan.