Exterior Angle Worksheets (printable, online, answers, examples) - Free Printable

Problem Analysis:

The task involves solving for unknown angles in triangles using the Exterior Angle Theorem. The theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.

We will solve each part step by step.

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Part 1: Find the value of \( x \)

#### Given:
- A triangle with one exterior angle labeled as \( 51^\circ \).
- One interior angle labeled as \( 19^\circ \).
- The unknown angle is labeled as \( x \).

#### Solution:
Using the Exterior Angle Theorem:
\[
\text{Exterior angle} = \text{Sum of the two non-adjacent interior angles}
\]
Here, the exterior angle is \( 51^\circ \), and the two non-adjacent interior angles are \( x \) and \( 19^\circ \). Therefore:
\[
51^\circ = x + 19^\circ
\]
Solve for \( x \):
\[
x = 51^\circ - 19^\circ = 32^\circ
\]

#### Answer:
\[
\boxed{32}
\]

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Part 2: Find the value of \( x \)

#### Given:
- A triangle with one exterior angle labeled as \( x \).
- Two interior angles labeled as \( 32^\circ \) and \( 24^\circ \).

#### Solution:
Using the Exterior Angle Theorem:
\[
\text{Exterior angle} = \text{Sum of the two non-adjacent interior angles}
\]
Here, the exterior angle is \( x \), and the two non-adjacent interior angles are \( 32^\circ \) and \( 24^\circ \). Therefore:
\[
x = 32^\circ + 24^\circ = 56^\circ
\]

#### Answer:
\[
\boxed{56}
\]

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Part 3: Find the values of \( x \) and \( y \)

#### Given:
- A triangle \( \triangle ABC \) with an exterior angle at vertex \( C \) labeled as \( 120^\circ \).
- Another triangle with angles \( 50^\circ \), \( y \), and an exterior angle labeled as \( 92^\circ \).

#### Step 1: Solve for \( x \) in the first triangle.
Using the Exterior Angle Theorem:
\[
\text{Exterior angle} = \text{Sum of the two non-adjacent interior angles}
\]
Here, the exterior angle is \( 120^\circ \), and the two non-adjacent interior angles are \( x \) and \( 60^\circ \) (since \( \triangle ABC \) is equilateral, each interior angle is \( 60^\circ \)). Therefore:
\[
120^\circ = x + 60^\circ
\]
Solve for \( x \):
\[
x = 120^\circ - 60^\circ = 60^\circ
\]

#### Step 2: Solve for \( y \) in the second triangle.
Using the Exterior Angle Theorem:
\[
\text{Exterior angle} = \text{Sum of the two non-adjacent interior angles}
\]
Here, the exterior angle is \( 92^\circ \), and the two non-adjacent interior angles are \( 50^\circ \) and \( y \). Therefore:
\[
92^\circ = 50^\circ + y
\]
Solve for \( y \):
\[
y = 92^\circ - 50^\circ = 42^\circ
\]

#### Answers:
\[
\boxed{60, 42}
\]

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Final Answers:

1. \( \boxed{32} \)
2. \( \boxed{56} \)
3. \( \boxed{60, 42} \)