Question

$\overline{xy} \parallel \overline{wz}$, $\overline{wx} \parallel \overline{vy}$, and $\overline{xy} \cong \overline{vw}$. complete the proof that $\angle y \cong \angle v$.

statement	reason
2 $\overline{wx} \parallel \overline{vy}$	given
3 $\overline{xy} \cong \overline{vw}$	given
4 $\overline{wz} \cong \overline{xy}$	parallelograms have congruent opposite sides
5 $\overline{vw} \cong \overline{wz}$	transitive property of congruence
6 $\angle vzw \cong \angle v$	isosceles triangle theorem
7 $\square$	$\square$

Explanation:

Step1: Identify the relationship from parallel lines

Since \(\overline{XY} \parallel \overline{WZ}\) and \(\overline{WX} \parallel \overline{VY}\), quadrilateral \(XYWZ\) is a parallelogram. In a parallelogram, alternate interior angles are congruent. So, \(\angle Y\) and \(\angle VZW\) are alternate interior angles formed by the transversal \(VY\) cutting the parallel lines \(\overline{XY}\) and \(\overline{WZ}\). Thus, \(\angle Y \cong \angle VZW\).

Step2: Use transitive property of congruence

We already have \(\angle VZW \cong \angle V\) (from step 6, Isosceles Triangle Theorem) and \(\angle Y \cong \angle VZW\) (from step 1). By the Transitive Property of Congruence, if \(\angle Y \cong \angle VZW\) and \(\angle VZW \cong \angle V\), then \(\angle Y \cong \angle V\). So the statement for step 7 is \(\angle Y \cong \angle V\) and the reason is Transitive Property of Congruence (or using the two congruences we have).

Answer:

Statement: \(\boldsymbol{\angle Y \cong \angle V}\)
Reason: Transitive Property of Congruence (or Alternate Interior Angles and Transitive Property, but more precisely, since \(\angle Y \cong \angle VZW\) (alternate interior angles from parallelogram) and \(\angle VZW \cong \angle V\) (Isosceles Triangle Theorem), by transitive property, \(\angle Y \cong \angle V\))