Finding Angle Measures in Regular Polygons The trampoline shown is shaped like a regular dodecagon. a. Find the measure of each interior angle. b. Find the measure of each exterior angle. SOLUTION a. Use the Polygon Interior Angles Theorem to find the sum of the measures of the interior angles.

Then find the measure of one interior angle. A regular dodecagon has 12 congruent interior angles. Divide
by 12.

The measure of each interior angle in the dodecagon is
. b. By the Polygon Exterior Angles Theorem, the sum of the measures of the exterior angles, one angle at each vertex, is
. Divide
by 12 to find the measure of one of the 12 congruent exterior angles.

The measure of each exterior angle in the dodecagon is
.

The Correct Answer and Explanation is:

The Correct Answer and Explanation is:

Here is the completed text from the image:

EXAMPLE 6 Finding Angle Measures in Regular Polygons
The trampoline shown is shaped like a regular dodecagon.
a. Find the measure of each interior angle.
b. Find the measure of each exterior angle.

SOLUTION
a. Use the Polygon Interior Angles Theorem to find the sum of the measures of the interior angles.

(n – 2) • 180° = (12 – 2) • 180°
= 1800°

Then find the measure of one interior angle. A regular dodecagon has 12 congruent interior angles. Divide 1800° by 12.

1800° / 12 = 150°

The measure of each interior angle in the dodecagon is 150°.

b. By the Polygon Exterior Angles Theorem, the sum of the measures of the exterior angles, one angle at each vertex, is 360°. Divide 360° by 12 to find the measure of one of the 12 congruent exterior angles.

360° / 12 = 30°

The measure of each exterior angle in the dodecagon is 30°.

Explanation

This example demonstrates how to calculate the measures of the interior and exterior angles of a regular polygon, specifically a regular dodecagon. A dodecagon is a polygon with 12 sides and 12 angles. The term “regular” is important because it signifies that all sides have equal length and all interior angles have equal measures.

a. Finding the Interior Angle

To find the measure of a single interior angle, we first need to determine the sum of all interior angles. This is accomplished using the Polygon Interior Angles Theorem. The formula for this theorem is (n – 2) * 180°, where ‘n’ represents the number of sides of the polygon.

For a dodecagon, n = 12. Substituting this value into the formula gives us:
(12 – 2) * 180° = 10 * 180° = 1800°.
This result, 1800°, is the total sum of the measures of all 12 interior angles combined.

Since the dodecagon is regular, all 12 of its interior angles are congruent, or equal in measure. To find the measure of just one of these angles, we divide the total sum by the number of angles.
1800° / 12 = 150°.
Therefore, each interior angle of a regular dodecagon measures 150°.

b. Finding the Exterior Angle

To find the measure of an exterior angle, we use the Polygon Exterior Angles Theorem. This theorem states that for any convex polygon, the sum of the measures of the exterior angles, taking one at each vertex, is always 360°.

This principle holds true for a triangle, a square, or a dodecagon. The total sum is consistently 360°. A regular dodecagon has 12 congruent exterior angles. To find the measure of a single exterior angle, we simply divide the total sum of 360° by the number of angles, which is 12.
360° / 12 = 30°.
Thus, each exterior angle of a regular dodecagon measures 30°.thumb_upthumb_down

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