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°

Explanation:

Step1: Calculate the central angle of a regular pentagon

The formula for the central angle of a regular \(n -\)sided polygon is \(\frac{360^{\circ}}{n}\). For a pentagon, \(n = 5\), so \(\frac{360^{\circ}}{5}=72^{\circ}\).

Step2: Determine the rotation angles that map the pentagon onto itself

A regular pentagon maps onto itself when rotated by an integer multiple of its central angle.

• For \(k = 1\): \(72^{\circ}\times1 = 72^{\circ}\)
• For \(k = 2\): \(72^{\circ}\times2=144^{\circ}\)
• For \(k = 3\): \(72^{\circ}\times3 = 216^{\circ}\) (not in the options)
• For \(k = 4\): \(72^{\circ}\times4=288^{\circ}\) (not in the options)
• For \(k = 5\): \(72^{\circ}\times5 = 360^{\circ}\) (not in the options)

Since \(36^{\circ}=\frac{72^{\circ}}{2}\) (not an integer - multiple of \(72^{\circ}\)) and \(180^{\circ}=\frac{360^{\circ}}{2}\) (not an integer - multiple of \(72^{\circ}\) for a pentagon), the correct angles are \(72^{\circ}\) and \(144^{\circ}\).

Answer:

B. \(72^{\circ}\), C. \(144^{\circ}\)