Question

$overline{vx} cong overline{wy}$, $angle wmy cong angle wmx$, and $overline{vw} parallel overline{xy}$. complete the proof that $\triangle vxy cong \triangle wyx$.

Explanation:

Step1: Identify given congruent sides

Given (stated in problem premise)

Step2: Identify given congruent angles

Given (stated in problem premise)

Step3: Identify given parallel segments

Given (stated in problem premise)

Step4: Use alternate interior angles

$\angle WVX \cong \angle VXY$ (Alternate Interior Angles Theorem, since $\overline{VW} \parallel \overline{XY}$ and $\overline{VX}$ is transversal)

Step5: Substitute congruent angles

$\angle WYX \cong \angle VWY$ (Substitution, using $\angle VWY \cong \angle WVX$ and $\angle WVX \cong \angle VXY$)

Step6: Substitute congruent angles

$\angle VWY \cong \angle VXY$ (Substitution, using $\angle VWY \cong \angle WVX$ and $\angle WVX \cong \angle VXY$)

Step7: Use alternate interior angles

$\angle WYX \cong \angle VXY$ (Alternate Interior Angles Theorem, since $\overline{VW} \parallel \overline{XY}$ and $\overline{XY}$ is transversal)

Step8: Identify reflexive congruent side

$\overline{XY} \cong \overline{XY}$ (Reflexive Property of Congruence)

Step9: Apply ASA congruence

$\triangle VXY \cong \triangle WYX$ (Angle-Side-Angle (ASA) Congruence Postulate, using $\angle VXY \cong \angle WYX$, $\overline{XY} \cong \overline{XY}$, $\angle VYX \cong \angle WXY$ derived from prior steps)

Answer:

Statement	Reason
2. $\angle VWY \cong \angle WVX$	Given
3. $\overline{VW} \parallel \overline{XY}$	Given
4. $\angle WVX \cong \angle VXY$	Alternate Interior Angles Theorem
5. $\angle WYX \cong \angle VWY$	Substitution Property
6. $\angle VWY \cong \angle VXY$	Substitution Property
7. $\angle WYX \cong \angle VXY$	Alternate Interior Angles Theorem
8. $\overline{XY} \cong \overline{XY}$	Reflexive Property of Congruence
9. $\triangle VXY \cong \triangle WYX$	ASA Congruence Postulate