Question

△uwx ≅ △vxw. complete the proof that △uvw ≅ △vux.

statement\treason
1\t△uwx ≅ △vxw\tgiven
2\tux ≅ vw\tcpctc
3\tuw ≅ vx\tcpctc
4\tuv ≅ uv\treflexive property of congruence
5\t\t

Explanation:

Step1: Identify Congruent Triangles' Sides

We have $\triangle UWX \cong \triangle VXW$ (given). By CPCTC, we get $\overline{UX} \cong \overline{VW}$ (step 2) and $\overline{UW} \cong \overline{VX}$ (step 3). Also, $\overline{UV} \cong \overline{UV}$ (reflexive, step 4).

Step2: Apply SSS Congruence

For $\triangle UVW$ and $\triangle VUX$, we have:

• $\overline{UX} \cong \overline{VW}$ (from step 2)
• $\overline{UW} \cong \overline{VX}$ (from step 3)
• $\overline{UV} \cong \overline{UV}$ (from step 4)

By the Side - Side - Side (SSS) Congruence Postulate, if three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent. So, $\triangle UVW \cong \triangle VUX$ by SSS.

Answer:

$\triangle UVW \cong \triangle VUX$ (by SSS Congruence Postulate)