1. given: $ \overline { p s } \cong \overline { p t } , \angle p r s \cong \angle p r t $ prove: $ \angle p r s \perp \angle p r t $ look at the figure. which reason justifies that $ \angle p r s $ and $ \angle p r t $ are right angles? $ \angle p r s $ and $ \angle p r t $ are interior angles of $ \triangle p r s $ and $ \triangle p r t $. $ \angle p r s $ and $ \angle p r t $ are congruent supplementary angles. $ \angle p r s $ and $ \angle p r t $ are vertical angles. $ \angle p r s $ and $ \angle p r t $ are congruent complementary angles.

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# Brief Explanations:
- **Congruent supplementary angles**: If two angles are congruent (equal in measure) and supplementary (sum to $180^{\circ}$), then each angle is $90^{\circ}$. Let $\angle PRS = x$ and $\angle PRT=x$ (since $\angle PRS\cong\angle PRT$). Then $x + x=180^{\circ}$ (because they are supplementary as they form a linear - pair). Solving $2x = 180^{\circ}$ gives $x = 90^{\circ}$.
- **Interior angles**: Just being interior angles of triangles $\triangle PRS$ and $\triangle PRT$ does not imply they are right angles.
- **Vertical angles**: Vertical angles are equal but not necessarily right angles. And $\angle PRS$ and $\angle PRT$ are a linear - pair (not vertical angles).
- **Complementary angles**: Complementary angles sum to $90^{\circ}$. But since $\angle PRS$ and $\angle PRT$ form a linear - pair (sum to $180^{\circ}$), they are not complementary.

# Answer:
$\angle PRS$ and $\angle PRT$ are congruent supplementary angles.

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