Question

$\angle vuw \cong \angle xuy$, $\overline{ux} \cong \overline{uw}$, and $\angle uvx \cong \angle uwy$. complete the proof that $\triangle uvx \cong \triangle uyw$.
statement:

1. $\angle vuw \cong \angle xuy$
2. $\overline{ux} \cong \overline{uw}$
3. $\angle uvx \cong \angle uwy$
4. $m\angle wuy = m\angle wux + m\angle xuy$
5. $m\angle vux = m\angle vuw + m\angle wux$
6. $m\angle wuy = m\angle wux + m\angle vuw$
7. $m\angle vux = m\angle wuy$
8. $\triangle uvx \cong \triangle uyw$

Explanation:

Step1: Identify given congruence

$\angle VUW \cong \angle XUY$ (Given)

Step2: Identify given segment congruence

$\overline{UX} \cong \overline{UW}$ (Given)

Step3: Identify given angle congruence

$\angle UXV \cong \angle UWY$ (Given)

Step4: Apply angle addition postulate

$m\angle WUY = m\angle WUX + m\angle XUY$ (Angle Addition Postulate)

Step5: Apply angle addition postulate

$m\angle VUX = m\angle VUW + m\angle WUX$ (Angle Addition Postulate)

Step6: Substitute congruent angles

$m\angle WUY = m\angle WUX + m\angle VUW$ (Substitution, $\angle XUY \cong \angle VUW$)

Step7: Set equal angles equal

$m\angle VUX = m\angle WUY$ (Transitive Property of Equality)

Step8: Apply ASA congruence

$\triangle UVX \cong \triangle UYW$ (ASA Congruence Postulate)

Answer:

1. Given
2. Given
3. Given
4. Angle Addition Postulate
5. Angle Addition Postulate
6. Substitution Property
7. Transitive Property of Equality
8. ASA Congruence Postulate