Question

$\triangle tuw$ is equilateral, $\angle vuw \cong \angle wtx$, and $\overline{uv} \cong \overline{tx}$. complete the proof that $\angle x \cong \angle v$.

Explanation:

Step1: Given

$\triangle TUW$ is equilateral. So, $TW = UW = TU$ (by the definition of an equilateral triangle).

Step2: Given

$\angle VUW\cong\angle WTX$ and $\overline{UV}\cong\overline{TX}$

Step3: SAS (Side - Angle - Side) Congruence

In $\triangle VUW$ and $\triangle WTX$, we have $\overline{UV}\cong\overline{TX}$, $\angle VUW\cong\angle WTX$, and $UW = TW$ (from Step 1). So, $\triangle VUW\cong\triangle WTX$ (by SAS congruence criterion).

Step4: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since $\triangle VUW\cong\triangle WTX$, then $\overline{VW}\cong\overline{WX}$ (CPCTC).

Step5: Given

$\overline{TW}\cong\overline{UW}$ (from Step 1). In $\triangle TWX$ and $\triangle UWV$, $\overline{VW}\cong\overline{WX}$, $\overline{TW}\cong\overline{UW}$, and $\angle VWU+\angle UWT = 180^{\circ}$, $\angle XWT+\angle UWT=180^{\circ}$ (linear pair), so $\angle VWU=\angle XWT$. Then $\triangle TWX\cong\triangle UWV$ (by SAS congruence criterion).

Step6: CPCTC

Since $\triangle TWX\cong\triangle UWV$, then $\angle X\cong\angle V$ (by CPCTC).

Answer:

1. Given (definition of equilateral triangle)
2. Given
3. SAS
4. CPCTC
5. SAS
6. CPCTC