Question

by which angle measures can the square be rotated so it maps onto itself?
select each correct answer.
45°
60°
90°
135°
180°
270°

Explanation:

Step1: Recall rotational symmetry of square

A square has rotational symmetry of order 4. The formula for the angle of rotation is \(\frac{360^{\circ}}{n}\), where \(n\) is the order of symmetry. For a square, \(n = 4\), so \(\frac{360^{\circ}}{4}=90^{\circ}\). Also, multiples of \(90^{\circ}\) (i.e., \(90^{\circ}\times k\), where \(k = 1,2,3,4\)) will map the square onto itself.

Step2: Check each angle

• For \(45^{\circ}\): \(360\div45 = 8\) (not the order of square's symmetry).
• For \(60^{\circ}\): \(360\div60=6\) (not the order of square's symmetry).
• For \(90^{\circ}\): \(360\div90 = 4\) (order of square's symmetry).
• For \(135^{\circ}\): \(360\div135=\frac{8}{3}\) (not an integer).
• For \(180^{\circ}\): \(180 = 90\times2\) (multiple of \(90^{\circ}\)).
• For \(270^{\circ}\): \(270=90\times3\) (multiple of \(90^{\circ}\)).

Answer:

\(90^{\circ}\), \(180^{\circ}\), \(270^{\circ}\)