Exterior Angle Theorem worksheet with problems requiring students to solve for variables using triangle angle relationships.

You've uploaded a worksheet titled “Exterior Angle Theorem”. The goal is to solve for the given variable in each triangle using the Exterior Angle Theorem.

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✔ What is the Exterior Angle Theorem?

> The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles (the two angles that are not adjacent to the exterior angle).

Also, remember:
- The sum of the interior angles of any triangle is 180°.
- An exterior angle and its adjacent interior angle form a linear pair, so they add up to 180°.

We’ll use these facts to solve each problem.

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## 🔢 Problem-by-Problem Solutions:

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1) h = ?

Given: Two base angles = 58° each.

This is an isosceles triangle.

Using the Exterior Angle Theorem:
> h = sum of the two remote interior angles = 58° + 58° = 116°

✔ Answer: h = 116

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2) b = ?

Given: Interior angles = 82° and 32°.

b is the exterior angle at the third vertex.

So, by the Exterior Angle Theorem:
> b = 82° + 32° = 114°

✔ Answer: b = 114

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3) k = ?

Given: Right triangle (90°), exterior angle = 147°.

First, find the adjacent interior angle:
> Adjacent interior angle = 180° - 147° = 33°

Now, in the right triangle, angles are: 90°, 33°, and k.

Sum of interior angles = 180°
> k = 180° - 90° - 33° = 57°

✔ Answer: k = 57

*(Alternatively, you could say: k + 90° = 147° → k = 57° — since the exterior angle equals the sum of the two remote interior angles.)*

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4) x = ?

Given: Right triangle with one acute angle = 56°.

x is the exterior angle at the other acute angle.

By Exterior Angle Theorem:
> x = 90° + 56° = 146°

✔ Answer: x = 146

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5) c = ?

Given: One interior angle = 38°, exterior angle = 91°.

c is the other interior angle (remote from the exterior angle).

By Exterior Angle Theorem:
> 91° = 38° + c
→ c = 91° - 38° = 53°

✔ Answer: c = 53

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6) n = ?

Given: Right triangle (90°), exterior angle = 142°.

Find the adjacent interior angle:
> Adjacent interior angle = 180° - 142° = 38°

Then, n = 180° - 90° - 38° = 52°

✔ Answer: n = 52

*(Or directly: n + 90° = 142° → n = 52°)*

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7) f = ?

Given: Right triangle, exterior angle = 153°.

Adjacent interior angle = 180° - 153° = 27°

Then, f = 180° - 90° - 27° = 63°

✔ Answer: f = 63

*(Or: f + 90° = 153° → f = 63°)*

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8) s = ?

Given: Two base angles = 31° each.

s is the exterior angle at the top vertex.

By Exterior Angle Theorem:
> s = 31° + 31° = 62°

✔ Answer: s = 62

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## 📝 Final Answers:

| Problem | Answer |
|---------|--------|
| 1 | 116 |
| 2 | 114 |
| 3 | 57 |
| 4 | 146 |
| 5 | 53 |
| 6 | 52 |
| 7 | 63 |
| 8 | 62 |

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Let me know if you’d like a diagram or step-by-step visuals for any of these! 😊