Exterior Angle Theorem practice worksheet with eight problems involving triangles and angle calculations.

Let's solve each problem using the Exterior Angle Theorem.

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🔷 Exterior Angle Theorem:

> The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.

Also, remember:
- The sum of angles in a triangle = 180°
- A right angle = 90°

We'll go through each problem one by one.

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1)

Given: Two interior angles are 58° and 58°.
Find: $ h^\circ $ — the exterior angle at the top vertex.

Step 1: Use the Exterior Angle Theorem:
$$
h = 58^\circ + 58^\circ = 116^\circ
$$

✔ Answer: $ h = 116^\circ $

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2)

Interior angles: 82° and 32°
Find: $ b^\circ $ — the exterior angle adjacent to the third angle.

Step 1: Find the third interior angle (opposite to $ b $):
$$
\text{Third angle} = 180^\circ - 82^\circ - 32^\circ = 66^\circ
$$

But wait! $ b $ is the exterior angle adjacent to this 66° angle, so:
$$
b = 180^\circ - 66^\circ = 114^\circ
$$

Alternatively, use Exterior Angle Theorem:
$ b $ is formed by extending one side, so it equals the sum of the two non-adjacent interior angles:
$$
b = 82^\circ + 32^\circ = 114^\circ
$$

✔ Answer: $ b = 114^\circ $

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3)

Right triangle, one acute angle is unknown ($ k^\circ $), exterior angle is 147°.

Since it's a right triangle, one angle is 90°, and the exterior angle is given as 147°.

Let’s find the interior angle adjacent to the exterior angle:
$$
\text{Adjacent interior angle} = 180^\circ - 147^\circ = 33^\circ
$$

Now, the triangle has:
- One angle: 90°
- Another: 33°
- So, $ k = 180^\circ - 90^\circ - 33^\circ = 57^\circ $

✔ Answer: $ k = 57^\circ $

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4)

Right triangle, one angle is 56°, find exterior angle $ x^\circ $

First, find the third interior angle:
$$
\text{Third angle} = 180^\circ - 90^\circ - 56^\circ = 34^\circ
$$

Now, $ x $ is the exterior angle at the 34° angle, so:
$$
x = 180^\circ - 34^\circ = 146^\circ
$$

Or, using the Exterior Angle Theorem:
$$
x = 90^\circ + 56^\circ = 146^\circ
$$

✔ Answer: $ x = 146^\circ $

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5)

Triangle with angles: 38° and $ c^\circ $, exterior angle = 91°

The exterior angle (91°) is formed by extending one side. It should be equal to the sum of the two non-adjacent interior angles.

So:
$$
91^\circ = 38^\circ + c^\circ
$$
$$
c = 91^\circ - 38^\circ = 53^\circ
$$

✔ Answer: $ c = 53^\circ $

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6)

Right triangle, exterior angle = 142°, find $ n^\circ $

The exterior angle is adjacent to one of the interior angles. First, find that interior angle:
$$
\text{Interior angle} = 180^\circ - 142^\circ = 38^\circ
$$

Now, the triangle has:
- One angle: 90° (right angle)
- One angle: 38°
- So, $ n = 180^\circ - 90^\circ - 38^\circ = 52^\circ $

✔ Answer: $ n = 52^\circ $

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7)

Right triangle, exterior angle = 153°, find $ f^\circ $

Find the adjacent interior angle:
$$
\text{Interior angle} = 180^\circ - 153^\circ = 27^\circ
$$

Now, triangle has:
- 90° (right angle)
- 27°
- So, $ f = 180^\circ - 90^\circ - 27^\circ = 63^\circ $

✔ Answer: $ f = 63^\circ $

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8)

Triangle with two base angles of 31° each, find $ s^\circ $ — the exterior angle at the top.

Use Exterior Angle Theorem:
$$
s = 31^\circ + 31^\circ = 62^\circ
$$

✔ Answer: $ s = 62^\circ $

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1) | $ h = 116^\circ $ |
| 2) | $ b = 114^\circ $ |
| 3) | $ k = 57^\circ $ |
| 4) | $ x = 146^\circ $ |
| 5) | $ c = 53^\circ $ |
| 6) | $ n = 52^\circ $ |
| 7) | $ f = 63^\circ $ |
| 8) | $ s = 62^\circ $ |

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