Question

Given: $\overline{PR}$bisects $\angle SPQ$and $\overline{PS}\perp \overline{SQ};\overline{RQ}\perp \overline{PQ}$.

Which numbered angles must be congruent?

Short answer

The congruent angles are $\angle 7\cong \angle 8$and $\angle 11\cong \angle 12$.

Step-by-step solution

Step 1. Apply the definition of angle bisector.

Angle bisector is a line which divides an angle into two equal parts.

From the figure, it can be observed that $\overline{PR}$bisect $\angle SPQ$. Therefore,

$\angle 7\cong \angle 8$

Step 2. Description of step.

$\overline{PS}\perp \overline{SQ}$and $\overline{RQ}\perp \overline{PQ}$, therefore,

$\begin{array}{c}\angle Q\cong \angle S\\ m\angle Q=90\end{array}$

Step 3. Description of step.

Consider $\Delta PQR\text{and}\Delta PST$, in which $\angle Q\cong \angle S,\angle RPQ\cong \angle SPT$implies that $\angle PTS\cong \angle 11$.

Step 4. Apply definition of vertically opposite angles.

Vertically opposite angles are the opposite angles formed due to intersection of two lines.

Consider $\Delta PQR\text{and}\Delta PST$,

$\angle 11\text{and}\angle 12$are vertically opposite angles, therefore, $\angle 11\cong \angle 12$.

Therefore, the congruent angles are $\angle 7\cong \angle 8$and $\angle 11\cong \angle 12$.