Chapter 5.7, Problem 15PSB

To determine

To show: $\overline{HF}\cong \overline{DB}$ .

Explanation of Solution

Given information:

A parallelogram is given as below.

  

$A\text{ is the midpoint of }\overline{QR}$ , $\stackrel{\rightharpoonup }{PA}\text{ bisects }\angle QPS$ .

Calculation:

A parallelogram is given as below.

  

$A\text{ is the midpoint of }\overline{QR}$ , $\stackrel{\rightharpoonup }{PA}\text{ bisects }\angle QPS$ .

  $\begin{array}{l}\text{In above figure,}\\ PQRS\text{ is a parallelogram}\text{. }\\ A\text{ is the midpoint of }\overline{QR}\text{ and }\stackrel{\rightharpoonup }{PA}\text{ bisects }\angle QPS.\\ \overline{QA}=\overline{AR}\\ \overline{QP}\cong \overline{RS}\\ \angle PSA\cong \angle RAS\left\{\text{Alternate interior angle of }\stackrel{\rightharpoonup }{QR}\parallel \stackrel{\rightharpoonup }{PS}\right\}\\ \angle RSA\cong \angle RAS\\ \Rightarrow \angle PSA\cong \angle RSA\\ \stackrel{\rightharpoonup }{SA}\text{ bisects }\angle PSR.\end{array}$