Question

Deshawn draws a regular pentagon and rotates it about its center. Which angle measures can Deshawn rotate the regular pentagon through to map it onto itself? Select each correct answer. 36° 72° 144° 180° 216° 288° A regular pentagon is divided into 5 equal parts. The lengths of all the sides are equal.

Answer 1

Deshawn can rotate the regular pentagon through angles of 72°, 144°, and 216° to map it onto itself.

Step-by-step explanation:

A regular pentagon has 5 equal angles, each measuring 108°. When Deshawn rotates the pentagon about its center, he can only map it onto itself through angles that are factors of 360°, which is the total measure of a full rotation. The angles that Deshawn can rotate the pentagon through to map it onto itself are:

• 72°
• 144°
• 216°

The prefix "penta" means five or five-sided, like pentathlon has five events. So a pentagon by definition has five angles and five sides.

The sum of the internal angles in any pentagon is 540 degrees (each internal angle measures 108 degrees). This comes from the formula for calculating the sum of angles in an n-sided polygon: (n - 2) × 180 degrees.

A regular pentagon means all the sides and internal angles are equal. The five sides would each measure the same length, and each of the five internal angles would measure 108 degrees.

Answer 2

Option (2).

Step-by-step explanation:

From the figure of a regular pentagon attached,

Central angle formed at the center of a regular pentagon is given by,

Central angle =
`(360)/(n)` degrees

Since, m∠COD ≅ m∠EOD ≅ m∠AOE ≅ m∠AOB ≅ m∠BOC

Therefore, by the rotation of a central angle about the center (∠EOD or ∠COD) pentagon will map on to itself.

Measure of the central angle of the given regular pentagon,

m∠COD =
`(360)/(5)`

= 72°

Therefore, Option (2) will be the answer.