Question

use the regular pentagon to answer the question. which is a degree of rotational symmetry for the regular pentagon? o a. 36° o b. 45° o c. 60° o d. 120° o e. 144°

Explanation:

Step1: Recall the formula for rotational symmetry of a regular polygon

For a regular polygon with \( n \) sides, the smallest angle of rotational symmetry is given by the formula \( \theta=\frac{360^{\circ}}{n} \).

Step2: Identify the number of sides of a pentagon

A pentagon has \( n = 5 \) sides.

Step3: Calculate the angle of rotational symmetry

Substitute \( n = 5 \) into the formula: \( \theta=\frac{360^{\circ}}{5}=72^{\circ} \)? Wait, no, wait. Wait, the question is about the degree of rotational symmetry. Wait, a regular pentagon has rotational symmetry angles that are multiples of \( \frac{360}{5}=72^{\circ} \)? Wait, no, the options are 36, 45, 60, 120, 144. Wait, maybe I made a mistake. Wait, no, the formula for the angle of rotational symmetry (the smallest angle by which you can rotate the polygon to get the same figure) is \( \frac{360}{n} \). For a pentagon, \( n = 5 \), so \( \frac{360}{5}=72 \). But 72 is not an option. Wait, maybe the question is about the exterior angle or something else? Wait, no, the options include 72? No, the options are A. 36, B.45, C.60, D.120, E.144. Wait, maybe I misread the figure. Wait, the figure is a regular pentagon? Wait, no, maybe it's a regular pentagon? Wait, no, a regular pentagon has 5 sides. Wait, maybe the question is about the central angle? Wait, no, the rotational symmetry angle. Wait, maybe the question is wrong? Wait, no, let's check again. Wait, the formula for rotational symmetry: the order of rotational symmetry is \( n \), and the angle is \( \frac{360}{n} \). For a pentagon, \( n = 5 \), so \( 72^{\circ} \). But 72 is not an option. Wait, maybe the figure is a regular decagon? No, the figure is a pentagon. Wait, maybe the question is about the interior angle? No, interior angle of a regular pentagon is \( 108^{\circ} \). Wait, the options: 36, 45, 60, 120, 144. Wait, 360 divided by 10 is 36, but a pentagon has 5 sides. Wait, maybe the figure is a regular pentagon, but the question is about the angle of rotational symmetry, but maybe the options are wrong? Wait, no, maybe I made a mistake. Wait, the regular pentagon: when you rotate it by \( 72^{\circ} \), it maps onto itself. But 72 is not an option. Wait, the options include 144, which is \( 180 - 36 \)? No, 144 is \( 360 - 216 \)? Wait, no, 144 is \( \frac{360\times2}{5}=144 \)? Wait, no, \( \frac{360}{5}=72 \), so multiples of 72: 72, 144, 216, 288, 360. Ah! 144 is a multiple of 72 (72×2 = 144). So 144° is a degree of rotational symmetry for the regular pentagon. Let's check the options: E is 144°. So that's the answer.

Answer:

E. \( 144^\circ \)