Question

statement reason 1 $\angle svt \cong \angle quw$ given 2 $\overline{tu} \cong \overline{vw}$ given 3 $\angle q \cong \angle s$ given 4 $uw = vw + uv$ additive property of length 5 $tv = tu + uv$ additive property of length 6 $uw = tu + uv$ 7 $tv = uw$ 8 $\triangle quw \cong \triangle svt$

Explanation:

Step1: Substitute $\overline{TU}\cong\overline{VW}$

From step 2, $\overline{TU}=\overline{VW}$. Substitute into step 4: $UW = TU + UV$

Step2: Set TV and UW equal

From step 5 ($TV = TU + UV$) and step 6 ($UW = TU + UV$), use transitive property: $TV = UW$

Step3: Prove triangle congruence

We have $\angle Q \cong \angle S$, $\angle SVT \cong \angle QUW$, and $TV=UW$. Use AAS congruence: $\triangle QUW \cong \triangle SVT$

Answer:

1. Substitution Property of Equality
2. Transitive Property of Equality
3. Angle-Angle-Side (AAS) Congruence Postulate