Question

complete the proof that $\triangle rvx \cong \triangle twu$.

statement	reason
2 $\overline{uv} \cong \overline{wx}$	given
3 $\angle rvx \cong \angle twu$	given
4 $vx = wx + vw$	additive property of length
5 $uw = uv + vw$	additive property of length
6 $vx = uv + vw$	substitution
7 $uw = vx$	transitive property of equality
8 $\triangle rvx \cong \triangle twu$

Explanation:

Step1: Identify congruent parts

We have $\angle X \cong \angle U$ (Statement 1), $VX \cong UW$ (from Statements 4 - 7, since $VX = UW$ implies $\overline{VX} \cong \overline{UW}$), and $\angle RVX \cong \angle TWU$ (Statement 3).

Step2: Apply AAS congruence

The Angle - Angle - Side (AAS) congruence criterion states that 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 two triangles are congruent. In $\triangle RVX$ and $\triangle TWU$, we have $\angle X \cong \angle U$, $\angle RVX \cong \angle TWU$, and $\overline{VX} \cong \overline{UW}$ (the non - included side between the two angles). So by the AAS (Angle - Angle - Side) Congruence Theorem, $\triangle RVX \cong \triangle TWU$.

Answer:

AAS (Angle - Angle - Side) Congruence Theorem