Finding Missing Interior and Exterior Angles of Triangles #1 ... - Free Printable

Let's solve each of the six triangle problems step by step, using the triangle angle sum theorem and exterior angle theorem.

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🔷 Triangle Angle Sum Theorem:

The sum of the interior angles in any triangle is:
> $$
> \angle A + \angle B + \angle C = 180^\circ
> $$

🔷 Exterior Angle Theorem:

An exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles:
> $$
> \text{Exterior angle} = \text{Sum of two remote interior angles}
> $$

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Let’s go through each problem:

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1. Find $ m\angle P $

Given:
- One angle: $ 59^\circ $
- Another angle: $ 77^\circ $

We need to find the third interior angle at $ P $.

$$
m\angle P = 180^\circ - 59^\circ - 77^\circ = 180^\circ - 136^\circ = 44^\circ
$$

✔ Answer: $ m\angle P = 44^\circ $

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2. Find $ m\angle T $

Given:
- One angle: $ 118^\circ $
- Another angle: $ 41^\circ $

Wait — this looks like an exterior angle at $ T $? But it's labeled as $ m\angle T $. Let's check the diagram description.

Actually, based on standard notation and layout, if $ 118^\circ $ is outside the triangle at vertex $ T $, then it's an exterior angle, and we use the exterior angle theorem.

But since the problem says “find the measure of each angle represented by the variable”, and $ \angle T $ is likely the interior angle at $ T $, but there's an exterior angle marked as $ 118^\circ $, then:

- Exterior angle at $ T $ = $ 118^\circ $
- Therefore, the interior angle at $ T $ is:
$$
m\angle T = 180^\circ - 118^\circ = 62^\circ
$$

But wait — the triangle also has another interior angle $ 41^\circ $. Let's verify with the triangle sum.

If $ m\angle T = 62^\circ $, and one other angle is $ 41^\circ $, then the third angle would be:
$$
180^\circ - 62^\circ - 41^\circ = 77^\circ
$$

That works.

So, the interior angle at $ T $ is $ 62^\circ $.

✔ Answer: $ m\angle T = 62^\circ $

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3. Find $ m\angle W $

Given:
- One angle: $ 40^\circ $
- Right angle (square symbol): $ 90^\circ $
- Need to find $ m\angle W $

Use triangle sum:
$$
m\angle W = 180^\circ - 90^\circ - 40^\circ = 50^\circ
$$

✔ Answer: $ m\angle W = 50^\circ $

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4. Find $ m\angle P $

Given:
- Right angle at $ P $: $ 90^\circ $
- Another angle: $ 40^\circ $
- Need $ m\angle P $? Wait — if the right angle is at $ P $, then $ m\angle P = 90^\circ $

But let’s double-check.

Looking at the triangle:
- Right angle at $ P $ → $ \angle P = 90^\circ $
- One other angle: $ 40^\circ $
- So the third angle (not labeled) is $ 50^\circ $

But the question asks for $ m\angle P $. Since it's a right angle at $ P $, that’s already given.

✔ Answer: $ m\angle P = 90^\circ $

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5. Find $ m\angle K $

Given:
- Two angles: $ 66.2^\circ $ and $ 8^\circ $
- Need $ m\angle K $

Assuming $ \angle K $ is the third angle in the triangle.

$$
m\angle K = 180^\circ - 66.2^\circ - 8^\circ = 105.8^\circ
$$

✔ Answer: $ m\angle K = 105.8^\circ $

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6. Find $ m\angle G $

Given:
- Interior angle: $ 59.7^\circ $
- Another angle: $ 46^\circ $
- $ \angle G $ is an exterior angle at vertex $ G $

Wait — the diagram shows $ \angle G $ as an exterior angle adjacent to the $ 46^\circ $ angle?

But actually, the angle $ \angle G $ is labeled outside the triangle, so it's an exterior angle.

It’s opposite to the $ 46^\circ $ angle? Or adjacent?

Let’s assume the triangle has angles:
- One angle: $ 59.7^\circ $
- Another: $ 46^\circ $
- Then the third interior angle is:
$$
180^\circ - 59.7^\circ - 46^\circ = 74.3^\circ
$$

Now, $ \angle G $ is the exterior angle at the vertex where the $ 46^\circ $ angle is located? Or at the $ 74.3^\circ $?

Wait — the exterior angle $ \angle G $ is adjacent to the $ 46^\circ $ angle.

Then:
$$
m\angle G = 180^\circ - 46^\circ = 134^\circ
$$

Alternatively, if $ \angle G $ is the exterior angle formed by extending one side, and it’s opposite to the $ 46^\circ $ and $ 59.7^\circ $ angles, then by exterior angle theorem:
$$
m\angle G = 46^\circ + 59.7^\circ = 105.7^\circ
$$

Which interpretation is correct?

Let’s analyze:

In triangle geometry, when an exterior angle is drawn at a vertex, it equals the sum of the two non-adjacent interior angles.

So if $ \angle G $ is an exterior angle at the vertex opposite the $ 46^\circ $ and $ 59.7^\circ $ angles, then yes:
$$
m\angle G = 46^\circ + 59.7^\circ = 105.7^\circ
$$

But the diagram shows $ \angle G $ at the same vertex as the $ 46^\circ $ angle? That doesn't make sense.

Wait — likely, the $ 46^\circ $ is an interior angle, and $ \angle G $ is the exterior angle at that same vertex.

Then:
$$
m\angle G = 180^\circ - 46^\circ = 134^\circ
$$

But that would mean $ \angle G $ is adjacent to $ 46^\circ $, forming a straight line.

Alternatively, maybe $ \angle G $ is the exterior angle at the vertex where the $ 59.7^\circ $ and $ 46^\circ $ meet?

Wait — more likely: the triangle has two known interior angles: $ 59.7^\circ $ and $ 46^\circ $, and $ \angle G $ is the exterior angle at the third vertex.

Then:
$$
\text{Third interior angle} = 180^\circ - 59.7^\circ - 46^\circ = 74.3^\circ
$$

Then the exterior angle at that vertex (opposite the other two) is:
$$
m\angle G = 180^\circ - 74.3^\circ = 105.7^\circ
$$

Or, by exterior angle theorem:
$$
m\angle G = 59.7^\circ + 46^\circ = 105.7^\circ
$$

✔ Answer: $ m\angle G = 105.7^\circ $

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✔ Final Answers:

1. $ m\angle P = 44^\circ $
2. $ m\angle T = 62^\circ $
3. $ m\angle W = 50^\circ $
4. $ m\angle P = 90^\circ $
5. $ m\angle K = 105.8^\circ $
6. $ m\angle G = 105.7^\circ $

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