### Understanding the Exterior Angles of a Polygon**Problem Statement:**In the polygon below, what is the measure of angle $ x^\circ $? Show all work.![Hexagon with one extended line forming an exterior angle](image.png)**Diagram Analysis:**The diagram shows a hexagon (a six-sided polygon) with one of its exterior angles marked. The interior angle adjacent to this exterior angle is given as 55°.**Step-by-Step Solution:**1. **Identify the relevant properties of the polygon:** - The given shape is a regular hexagon (all internal angles are equal).2. **Calculate the interior angle of the hexagon:** - The formula to find the measure of an interior angle of a regular polygon with $ n $ sides is: \[ \text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n} \] - For a hexagon ($ n = 6 $): \[ \text{Interior Angle} = \frac{(6 - 2) \times 180^\circ}{6} = \frac{4 \times 180^\circ}{6} = 120^\circ \]3. **Examine the relationship between an interior and exterior angle:** - An exterior angle is supplementary to the corresponding interior angle, meaning they sum up to 180°. - Therefore, the measure of the exterior angle $ x^\circ $ is: \[ x^\circ = 180^\circ - \text{Interior Angle} \] - Substitute the interior angle value: \[ x^\circ = 180^\circ - 120^\circ = 60^\circ \]**Conclusion:**Angle $ x^\circ $ in the given hexagon is $ 60^\circ $.**Summary:**To find the measure of an exterior angle of any regular polygon, use the supplementary relationship between an interior and exterior angle. For a regular hexagon, the exterior angle measures $ 60^\circ $.

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Description: ### Understanding the Exterior Angles of a Polygon**Problem Statement:**In the polygon below, what is the measure of angle $ x^\circ $? Show all work.![Hexagon with one extended line forming an exterior angle](image.png)**Diagram Analysis:**The di

To find the value of x° in the given polygon.

Answered: In the polygon below, what is the…

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In the polygon below, what is the measure of angle x°? Show all work. 55° X°

Elementary Geometry For College Students, 7e

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ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterP: Preliminary Concepts

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Fundamentals of Trigonometric Identities

Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.

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### Understanding the Exterior Angles of a Polygon

**Problem Statement:**
In the polygon below, what is the measure of angle $ x^\circ $? Show all work.

![Hexagon with one extended line forming an exterior angle](image.png)

**Diagram Analysis:**
The diagram shows a hexagon (a six-sided polygon) with one of its exterior angles marked. The interior angle adjacent to this exterior angle is given as 55°.

**Step-by-Step Solution:**

1. **Identify the relevant properties of the polygon:**
- The given shape is a regular hexagon (all internal angles are equal).

2. **Calculate the interior angle of the hexagon:**
- The formula to find the measure of an interior angle of a regular polygon with $ n $ sides is:
\[
\text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n}
\]
- For a hexagon ($ n = 6 $):
\[
\text{Interior Angle} = \frac{(6 - 2) \times 180^\circ}{6} = \frac{4 \times 180^\circ}{6} = 120^\circ
\]

3. **Examine the relationship between an interior and exterior angle:**
- An exterior angle is supplementary to the corresponding interior angle, meaning they sum up to 180°.
- Therefore, the measure of the exterior angle $ x^\circ $ is:
\[
x^\circ = 180^\circ - \text{Interior Angle}
\]
- Substitute the interior angle value:
\[
x^\circ = 180^\circ - 120^\circ = 60^\circ
\]

**Conclusion:**
Angle $ x^\circ $ in the given hexagon is $ 60^\circ $.

**Summary:**
To find the measure of an exterior angle of any regular polygon, use the supplementary relationship between an interior and exterior angle. For a regular hexagon, the exterior angle measures $ 60^\circ $.

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