Question

use the given information to prove that △rst ≅ △vut. given: \\(\overline{rt} \cong \overline{vt}\\), \\(\angle rst \cong \angle vut\\) prove: \\(\triangle rst \cong \triangle vut\\)

Explanation:

Step1: Identify Vertical Angles

$\angle RTS$ and $\angle VTU$ are vertical angles, so $\angle RTS \cong \angle VTU$ (Vertical Angles Theorem).

Step2: List Known Congruences

We know $\overline{RT} \cong \overline{VT}$ (Given), $\angle RST \cong \angle VUT$ (Given), and $\angle RTS \cong \angle VTU$ (from Step 1).

Step3: Apply AAS Congruence

By the Angle - Angle - Side (AAS) Congruence Criterion, if two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, then the triangles are congruent. In $\triangle RST$ and $\triangle VUT$:

• $\angle RST \cong \angle VUT$ (Given)
• $\angle RTS \cong \angle VTU$ (Vertical Angles)
• $\overline{RT} \cong \overline{VT}$ (Given)

So, $\triangle RST \cong \triangle VUT$ by AAS.

Answer:

$\triangle RST \cong \triangle VUT$ is proven by the Angle - Angle - Side (AAS) Congruence Criterion using the given congruences $\overline{RT} \cong \overline{VT}$, $\angle RST \cong \angle VUT$, and the vertical angles $\angle RTS \cong \angle VTU$.