Chapter 3.4, Problem 32WE

To determine

To Find: The congruent angles of the given figure.

Answer to Problem 32WE

The congruent angles are $\angle 7\cong \angle 8\text{and}\angle 11\cong 12$

Explanation of Solution

Given information:

  $\frac{\overrightarrow{PR}}{PS}\perp \overrightarrow{SQ};\overrightarrow{RQ}\perp \overrightarrow{PQ}$

  

Refer to the figure provide in the text book.

Consider the following data:

PR bisects $\angle SPG$

  $PS\perp SQ$ and $RQ\perp PQ$

Hence, $\angle 7\cong \angle 8$

  $\begin{array}{cc}\angle Q& \cong \angle S\\ & ={90}^{\circ }\end{array}$

In triangles, $\text{Δ}PQR$ and $△PST$

  $\begin{array}{cc}\angle Q& \cong \angle S\\ \angle RPQ& \cong \angle SPT\\ \text{Hence,}& \angle PTS\cong \angle PQR\cong \angle 11\end{array}$

Further, $\angle PTS\cong \angle 11$ and $\angle \text{PTS}\cong \angle 12$

Since they are vertically opposite angles,

Hence, $\angle 11\cong \angle 12$

Therefore, it is obtained that $\angle 7\cong \angle 8\text{and}\angle 11\cong 12$