Triangle Sum and Exterior Angle Theorem worksheet with six problems involving angle calculations in triangles.

Let's solve each problem step by step using the Triangle Sum Theorem and Exterior Angle Theorem.

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🔷 Triangle Sum Theorem:

The sum of the interior angles of a triangle is always 180°.

🔷 Exterior Angle Theorem:

An exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles.

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We’ll go through each problem:

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1)

Given: Two angles are 58° and 47°. Find $ x $.

Use Triangle Sum Theorem:
$$
x = 180^\circ - 58^\circ - 47^\circ = 180 - 105 = 75^\circ
$$

✔ Answer: $ x = 75^\circ $

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2)

Given: One angle is $ x^\circ $, another is 96°, and the third is 21°.

Again, use Triangle Sum Theorem:
$$
x + 96^\circ + 21^\circ = 180^\circ \\
x = 180 - 96 - 21 = 63^\circ
$$

✔ Answer: $ x = 63^\circ $

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3)

Right triangle (90°), one angle is 31°, and the exterior angle is $ (3x - 1)^\circ $.
Find $ x $.

First, find the interior angle adjacent to the exterior angle.

Since it’s a triangle with angles 90° and 31°, the third interior angle is:
$$
180 - 90 - 31 = 59^\circ
$$

Now, the exterior angle at that vertex is supplementary to the interior angle:
$$
\text{Exterior angle} = 180^\circ - 59^\circ = 121^\circ
$$

But we're told the exterior angle is $ (3x - 1)^\circ $, so:
$$
3x - 1 = 121 \\
3x = 122 \\
x = \frac{122}{3} \approx 40.67^\circ
$$

Wait — but let's double-check: Is this correct?

Actually, in a triangle, the exterior angle equals the sum of the two remote interior angles.

So, the exterior angle at the vertex opposite the 31° angle would be:
$$
\text{Exterior angle} = 90^\circ + 31^\circ = 121^\circ
$$

Yes! That matches.

So:
$$
3x - 1 = 121 \\
3x = 122 \\
x = \frac{122}{3} \approx 40.67
$$

But wait — maybe there's a typo? Let's check if the diagram shows the exterior angle at the top, above the 31° angle.

Assuming the (3x - 1)° is the exterior angle at the same vertex as the 31° angle, then yes, it should equal the sum of the other two interior angles:
$$
(3x - 1) = 90 + 31 = 121 \\
3x = 122 \\
x = \frac{122}{3} \approx 40.67
$$

But this is not a whole number. Maybe we made a mistake?

Wait — perhaps the right angle is at the bottom, and the exterior angle is at the top, opposite the 31° angle?

Let’s clarify.

If the triangle has:
- Right angle (90°)
- One angle: 31°
- Then the third angle is $ 180 - 90 - 31 = 59^\circ $

Now, suppose the exterior angle is at the vertex with the 59° angle, then:
$$
\text{Exterior angle} = 180 - 59 = 121^\circ
$$

But again, $ 3x - 1 = 121 \Rightarrow x = 122/3 \approx 40.67 $

Alternatively, maybe the exterior angle is at the 31° angle?

Then:
$$
\text{Exterior angle} = 180 - 31 = 149^\circ
$$

But then $ 3x - 1 = 149 \Rightarrow 3x = 150 \Rightarrow x = 50 $

That’s a nice number. But does that make sense?

Wait — the exterior angle is formed by extending one side. If the exterior angle is labeled (3x - 1), and it's adjacent to the 31° angle, then they are supplementary:
$$
(3x - 1) + 31 = 180 \\
3x - 1 = 149 \\
3x = 150 \\
x = 50
$$

Ah! That makes more sense. So the (3x - 1)° is the exterior angle adjacent to the 31° angle, so they add to 180°.

So:
$$
(3x - 1) + 31 = 180 \\
3x + 30 = 180 \\
3x = 150 \\
x = 50
$$

✔ Answer: $ x = 50 $

(Note: The image says “x = .33” — that must be a typo or misread.)

So correct answer is $ x = 50 $

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4)

Triangle with angles: 21°, 34°, and an exterior angle $ x^\circ $

We’re given two interior angles: 21° and 34°. The exterior angle is opposite the third angle.

But actually, the exterior angle $ x $ is adjacent to the third angle.

But the exterior angle is equal to the sum of the two remote interior angles.

So:
$$
x = 21^\circ + 34^\circ = 55^\circ
$$

✔ Answer: $ x = 55^\circ $

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5)

Triangle with angles: $ 100^\circ $, $ 51^\circ $, and $ (2x + 3)^\circ $

Sum of angles = 180°:
$$
100 + 51 + (2x + 3) = 180 \\
154 + 2x + 3 = 180 \\
2x + 157 = 180 \\
2x = 23 \\
x = 11.5
$$

✔ Answer: $ x = 11.5 $ or $ \frac{23}{2} $

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6)

Triangle with angles: $ 60^\circ $, $ 2x^\circ $, and an exterior angle of $ 94^\circ $

The exterior angle is $ 94^\circ $, which is adjacent to one of the interior angles.

From the diagram (assuming the exterior angle is at the vertex with $ 2x^\circ $), then:
$$
\text{Interior angle} + \text{Exterior angle} = 180^\circ \\
2x + 94 = 180 \\
2x = 86 \\
x = 43
$$

✔ Answer: $ x = 43^\circ $

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1) | $ x = 75^\circ $ |
| 2) | $ x = 63^\circ $ |
| 3) | $ x = 50^\circ $ |
| 4) | $ x = 55^\circ $ |
| 5) | $ x = 11.5^\circ $ |
| 6) | $ x = 43^\circ $ |

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Let me know if you want the diagrams explained further!