Question

1. given: ∠s ≅ ∠v, t is the midpoint of \\(\overline{sv}\\). prove: \\(\triangle rts \cong \triangle utv\\)

Explanation:

Step1: State given midpoint property

Since $T$ is the midpoint of $\overline{SV}$, $\overline{ST} = \overline{TV}$.

Step2: State given congruent angles

$\angle S \cong \angle V$, so $m\angle S = m\angle V$.

Step3: Identify vertical angles

$\angle RTS$ and $\angle UTV$ are vertical angles, so $\angle RTS \cong \angle UTV$.

Step4: Apply ASA congruence rule

We have two pairs of congruent angles and the included congruent sides: $\angle S \cong \angle V$, $\overline{ST} \cong \overline{TV}$, $\angle RTS \cong \angle UTV$. By the Angle-Side-Angle (ASA) Congruence Postulate, $\triangle RTS \cong \triangle UTV$.

Answer:

$\triangle RTS \cong \triangle UTV$ is proven by the ASA Congruence Postulate.