Exterior Angle Theorem interactive worksheet - Free Printable

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 is 180°.
- A right angle = 90°.

We'll go through each problem step by step.

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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: Find the third interior angle (opposite $ h $).

Sum of angles in triangle = 180°
So, third angle = $ 180^\circ - 58^\circ - 58^\circ = 64^\circ $

But wait — $ h $ is the exterior angle adjacent to this third angle.

So, since the interior angle is 64°, the exterior angle $ h = 180^\circ - 64^\circ = 116^\circ $

✔ Alternatively, using Exterior Angle Theorem:
$ h = 58^\circ + 58^\circ = \boxed{116^\circ} $

✔️ Answer: $ h = 116^\circ $

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2)

Given: Interior angles = 82° and 32°
Find: $ b^\circ $ — exterior angle at the base

Using Exterior Angle Theorem:
$ b = 82^\circ + 32^\circ = \boxed{114^\circ} $

✔️ Answer: $ b = 114^\circ $

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3)

Right triangle with one acute angle = 147° exterior angle
Find: $ k^\circ $

Note: The exterior angle is 147°, so the interior angle adjacent to it is:
$ 180^\circ - 147^\circ = 33^\circ $

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

Alternatively, use Exterior Angle Theorem:
The exterior angle (147°) equals sum of two non-adjacent interior angles:
- One is 90°, the other is $ k $
So: $ 147^\circ = 90^\circ + k $ → $ k = 147^\circ - 90^\circ = \boxed{57^\circ} $

✔️ Answer: $ k = 57^\circ $

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4)

Right triangle with one angle = 56°
Find: $ x^\circ $ — exterior angle at the bottom

Interior angles:
- Right angle = 90°
- Given = 56°
- So third angle = $ 180^\circ - 90^\circ - 56^\circ = 34^\circ $

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

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

✔️ Answer: $ x = 146^\circ $

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5)

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

The exterior angle (91°) is adjacent to angle $ c $, so:

Using Exterior Angle Theorem:
$ 91^\circ = 38^\circ + c $

Wait — no! The exterior angle equals the sum of the two non-adjacent interior angles.

So, the exterior angle is formed by extending one side. It should be equal to the sum of the two opposite interior angles.

Here, we have:
- One interior angle = 38°
- Another interior angle = $ c $
- Exterior angle = 91°

So: $ 91^\circ = 38^\circ + c $? No — unless $ c $ is the other non-adjacent angle.

Actually, the exterior angle is adjacent to $ c $, so the non-adjacent angles are the other two.

But only one other angle is given (38°). We need to find $ c $.

Let’s suppose the triangle has:
- One angle = 38°
- One angle = $ c $
- Then the third angle is $ 180^\circ - 38^\circ - c $

But the exterior angle is 91°, which is adjacent to $ c $. So the interior angle at that vertex is $ 180^\circ - 91^\circ = 89^\circ $

Wait — that would mean $ c = 89^\circ $? But then:

Then the third angle = $ 180^\circ - 38^\circ - 89^\circ = 53^\circ $

But the exterior angle should equal the sum of the two non-adjacent interior angles:
- 38° and 53° → $ 38 + 53 = 91^\circ $ ✔

So yes: $ c = 89^\circ $

But wait — is $ c $ the interior angle adjacent to the 91° exterior?

Yes — because the exterior angle is outside, so the interior angle is $ 180^\circ - 91^\circ = 89^\circ $

So $ c = 89^\circ $

✔️ Answer: $ c = 89^\circ $

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6)

Right triangle with exterior angle = 142°
Find: $ n^\circ $

Exterior angle = 142° → interior angle adjacent = $ 180^\circ - 142^\circ = 38^\circ $

Triangle has:
- Right angle = 90°
- One angle = 38°
- So $ n = 180^\circ - 90^\circ - 38^\circ = 52^\circ $

Alternatively, using Exterior Angle Theorem:
Exterior angle = sum of two non-adjacent interior angles = $ 90^\circ + n $

So: $ 142^\circ = 90^\circ + n $ → $ n = 142^\circ - 90^\circ = \boxed{52^\circ} $

✔️ Answer: $ n = 52^\circ $

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7)

Right triangle with exterior angle = 153°
Find: $ f^\circ $

Exterior angle = 153° → adjacent interior angle = $ 180^\circ - 153^\circ = 27^\circ $

Triangle has:
- Right angle = 90°
- One angle = 27°
- So $ f = 180^\circ - 90^\circ - 27^\circ = 63^\circ $

Using Exterior Angle Theorem:
$ 153^\circ = 90^\circ + f $ → $ f = 153^\circ - 90^\circ = \boxed{63^\circ} $

✔️ Answer: $ f = 63^\circ $

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8)

Triangle with two angles = 31°, 31°
Find: $ s^\circ $ — exterior angle at the top

This is an isosceles triangle.

Sum of interior angles = 180°
So third interior angle = $ 180^\circ - 31^\circ - 31^\circ = 118^\circ $

Now, $ s $ is the exterior angle at the top vertex (opposite the 118° angle), so:

$ s = 180^\circ - 118^\circ = 62^\circ $

Alternatively, using Exterior Angle Theorem:
$ s = 31^\circ + 31^\circ = \boxed{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 = 89^\circ $ |
| 6) | $ n = 52^\circ $ |
| 7) | $ f = 63^\circ $ |
| 8) | $ s = 62^\circ $ |

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