Question

use the given information to prove that $\triangle rst\cong\triangle uvt$. given: $\angle rst$ is a right angle $\angle uvt$ is a right angle $\overline{st}\cong\overline{vt}$ prove: $\triangle rst\cong\triangle uvt$

Explanation:

Step1: Identify vertical angles

$\angle RTS$ and $\angle UTV$ are vertical angles. By the Vertical Angles Theorem, $\angle RTS\cong\angle UTV$.

Step2: Use the AAS (Angle - Angle - Side) congruence criterion

We have $\angle RST\cong\angle UVT$ (both are right angles, given), $\angle RTS\cong\angle UTV$ (from Step 1), and $\overline{ST}\cong\overline{VT}$ (given).
By the AAS congruence criterion for triangles, $\triangle RST\cong\triangle UVT$.

Answer:

$\angle RTS\cong\angle UTV$ (Vertical Angles Theorem) and $\triangle RST\cong\triangle UVT$ (AAS)