Chapter 7.4, Problem 14PSC

To determine

To prove : ACDF is a rectangle

Explanation of Solution

Given information : The following information has been given

ABCDEF is a regular hexagon

Formula used : each angle and each side of any regular polygon are always congruent

We know that each angle of a regular hexagon will measure ${120}^0$

Proof : We know that each angle of a regular hexagon will measure $\angle B={120}^0$ , and AB and BC are congruent.

Thus, in triangle ABC, we know that $\angle A=\angle C={30}^0$

Now, $\angle FAC=\angle DCA={120}^0-{30}^0={90}^0$

Similarly, we can say that $\angle AFD=\angle CDF={90}^0$

Also, we know that diagonals AC and FD are congruent, and sides AF and CD are congruent.

Thus, we can say that ACDF is a rectangle.