Question

△uxy ≅ △wvy. complete the proof that △uvy ≅ △wxy.
statement\treason
1\t△uxy ≅ △wvy\tgiven
2\t\\(\overline{uy} \cong \overline{wy}\\)\t
3\t\\(\overline{vy} \cong \overline{xy}\\)\tcpctc
4\t\\(\angle uvy \cong \angle wyx\\)\tvertical angle theorem
5\t△uvy ≅ △wxy\tsas

Explanation:

Step1: Identify given congruence

$\triangle UXY \cong \triangle WVY$ (Given)

Step2: Extract CPCTC side congruences

From $\triangle UXY \cong \triangle WVY$, by CPCTC:
$\overline{UY} \cong \overline{WY}$, $\overline{XY} \cong \overline{VY}$

Step3: Identify vertical angles

$\angle UYV \cong \angle WYX$ (Vertical Angle Theorem)

Step4: Apply SAS congruence

We have $\overline{UY} \cong \overline{WY}$, $\angle UYV \cong \angle WYX$, $\overline{VY} \cong \overline{XY}$. By SAS congruence postulate:
$\triangle UVY \cong \triangle WXY$

Answer:

The completed proof has the following filled-in reasons and statements:

1. $\triangle UXY \cong \triangle WVY$ | Given
2. $\overline{UY} \cong \overline{WY}$ | CPCTC (from step 1)
3. $\overline{VY} \cong \overline{XY}$ | CPCTC (from step 1)
4. $\angle UYV \cong \angle WYX$ | Vertical Angle Theorem
5. $\triangle UVY \cong \triangle WXY$ | SAS (using steps 2, 4, 3)