Geometry Worksheets | Triangle Worksheets - 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.

That is:
> Exterior Angle = Sum of Two Remote Interior Angles

We'll apply this to each problem.

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🔹 Problem 1:

Given:
- One interior angle = 40°
- Another interior angle = 15°
- Exterior angle = $ k^\circ $

Using the theorem:
$$
k = 40^\circ + 15^\circ = 55^\circ
$$
✔ Answer: $ k = 55 $

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🔹 Problem 2:

Given:
- Interior angles: 51° and 89°
- Exterior angle = $ 6r + 2^\circ $

Apply the theorem:
$$
6r + 2 = 51 + 89 = 140
$$
Solve for $ r $:
$$
6r = 138 \Rightarrow r = 23
$$
✔ Answer: $ r = 23 $

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🔹 Problem 3:

Given:
- Exterior angle = 108°
- One interior angle = 53°
- Other interior angle = $ d^\circ $

So:
$$
108 = d + 53 \Rightarrow d = 108 - 53 = 55
$$
✔ Answer: $ d = 55 $

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🔹 Problem 4:

Given:
- Interior angles: $ 3s^\circ $, 20°
- Exterior angle = 74°

So:
$$
74 = 3s + 20 \Rightarrow 3s = 54 \Rightarrow s = 18
$$
✔ Answer: $ s = 18 $

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🔹 Problem 5:

Given:
- Interior angle = 38°
- Exterior angle = 95°
- Other interior angle = $ a^\circ $

So:
$$
95 = 38 + a \Rightarrow a = 95 - 38 = 57
$$
✔ Answer: $ a = 57 $

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🔹 Problem 6:

Given:
- Interior angles: $ e^\circ $, 35°
- Exterior angle = 70°

So:
$$
70 = e + 35 \Rightarrow e = 35
$$
✔ Answer: $ e = 35 $

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🔹 Problem 7:

Given:
- Interior angle = $ 4n + 3^\circ $
- Exterior angle = 166°
- Another interior angle = 139°

Wait — let’s be careful. The triangle has:
- One interior angle = 139°
- Exterior angle = 166°
- But wait: an exterior angle cannot be smaller than an interior angle unless it's not adjacent.

Actually, in this diagram, the exterior angle is 166°, and one interior angle is 139°.

But remember: the exterior angle is formed by extending one side, so it is adjacent to one interior angle, and equals the sum of the other two interior angles.

So the remote interior angles are:
- $ 4n + 3^\circ $
- And the other interior angle (not shown) must be $ 180^\circ - 166^\circ = 14^\circ $? Wait — no.

Wait! Let's clarify:

The exterior angle is 166°, so the adjacent interior angle is:
$$
180^\circ - 166^\circ = 14^\circ
$$

Then the other two interior angles are:
- $ 4n + 3^\circ $
- $ 139^\circ $

Wait — but that can't be, because the sum of all three interior angles must be 180°.

Let’s re-express:

From the diagram:
- One interior angle = $ 4n + 3^\circ $
- Another interior angle = $ 139^\circ $
- The third interior angle (adjacent to the exterior angle) = $ 180^\circ - 166^\circ = 14^\circ $

Now sum of interior angles:
$$
(4n + 3) + 139 + 14 = 180
\Rightarrow 4n + 156 = 180
\Rightarrow 4n = 24
\Rightarrow n = 6
$$

✔ Answer: $ n = 6 $

Alternatively, using the Exterior Angle Theorem:
> Exterior angle = sum of two remote interior angles

So:
$$
166 = (4n + 3) + 139
\Rightarrow 166 = 4n + 142
\Rightarrow 4n = 24 \Rightarrow n = 6
$$

Same result.

✔ Answer: $ n = 6 $

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🔹 Problem 8:

Given:
- Interior angles: 59°, $ 3v + 1^\circ $
- Exterior angle = 110°

Apply the theorem:
$$
110 = 59 + (3v + 1)
\Rightarrow 110 = 60 + 3v
\Rightarrow 3v = 50
\Rightarrow v = \frac{50}{3} \approx 16.67
$$

But let's check if this makes sense.

Wait — is 110° the exterior angle? Yes.

So:
$$
110 = 59 + (3v + 1)
\Rightarrow 110 = 60 + 3v
\Rightarrow 3v = 50
\Rightarrow v = \frac{50}{3}
$$

But that’s a fraction. Is that okay?

Yes, unless specified otherwise.

But let’s double-check the logic.

The two remote interior angles are:
- 59°
- $ 3v + 1^\circ $

Their sum should equal the exterior angle: 110°

So:
$$
59 + 3v + 1 = 110 \Rightarrow 60 + 3v = 110 \Rightarrow 3v = 50 \Rightarrow v = \frac{50}{3}
$$

✔ Answer: $ v = \frac{50}{3} $ or $ 16\frac{2}{3} $

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1) | $ k = 55 $ |
| 2) | $ r = 23 $ |
| 3) | $ d = 55 $ |
| 4) | $ s = 18 $ |
| 5) | $ a = 57 $ |
| 6) | $ e = 35 $ |
| 7) | $ n = 6 $ |
| 8) | $ v = \frac{50}{3} $ |

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📝 Summary of Steps:

For each problem:
1. Identify the exterior angle.
2. Identify the two non-adjacent (remote) interior angles.
3. Set up equation:
$$
\text{Exterior Angle} = \text{Sum of Remote Interior Angles}
$$
4. Solve for the variable.

Let me know if you'd like a visual explanation or help with any specific problem!