Question

complete the proof that ( \triangle uvx cong \triangle uyw ).

Explanation:

Step1: Identify given information

• $\angle Y\cong\angle V$ (Given)
• $\overline{UW}\cong\overline{UX}$ (Given)
• $\angle VUW\cong\angle XUY$ (Given)

Step2: Analyze angle - addition property

• $m\angle WUY = m\angle WUX + m\angle XUY$ (Angle - addition postulate)
• $m\angle VUX = m\angle VUW + m\angle WUX$ (Angle - addition postulate)

Step3: Substitute equal angles

• Since $\angle VUW\cong\angle XUY$, then $m\angle WUY = m\angle WUX + m\angle VUW$ (Substitution property of equality)
• So, $m\angle VUX = m\angle WUY$ (Transitive property of equality)

Step4: Apply AAS (Angle - Angle - Side) congruence criterion

• In $\triangle UVX$ and $\triangle UYW$:
• $\angle V\cong\angle Y$ (Given)
• $\overline{UX}\cong\overline{UW}$ (Given)
• $\angle VUX\cong\angle WUY$ (From step 3)
• By AAS (Angle - Angle - Side) congruence criterion, $\triangle UVX\cong\triangle UYW$

Answer:

1. Given
2. Given
3. Given
4. Angle - addition postulate
5. Angle - addition postulate
6. Substitution property of equality
7. Transitive property of equality
8. AAS (Angle - Angle - Side) congruence criterion