Chapter 9.5, Problem 26WE

To determine

To Prove: $PS\left|\right|QR$ is a parallelogram.

Explanation of Solution

Given information:

  $PQ\left|\right|SR$

  

Considering the figure:

  $\square PSXY$ and $\square XQRY$

  $\angle P+\angle Y=180°$

Opposite angle of an cyclic quadrilateral,

Also,

  $\begin{array}{l} \angle Q+\angle n=180°\hfill \\ \angle n+\angle Y=180°\hfill \end{array}$

Equating above equations, it is found that above angles forms linear pair

  $\angle Y=\angle Q$

Therefore, $\angle P$ and $\angle Q$ are complementary.

Similarly, $\angle S$ and $\angle R$ are complementary.

  $\square PQRS$ is a parallelogram

Hence, $PS\left|\right|QR$ is proved.