Question

statements reasons

1. wxyz is a parallelogram with diagonals (overline{xz}) and (overline{wy}) intersecting at point v. given
2. (overline{wx}paralleloverline{zy}) and (overline{xy}paralleloverline{wz}) definition of a parallelogram
3. (angle wxycongangle yzw) (angle wxzcongangle yzx)
4. (overline{xy}congoverline{zw}) same - side interior angles theorem alternate exterior angles theorem alternate interior angles theorem vertical angles theorem

Explanation:

Step1: Recall parallelogram properties

In a parallelogram, opposite - sides are parallel. Since \(WXYZ\) is a parallelogram (\(\overline{WX}\parallel\overline{ZY}\) and \(\overline{XY}\parallel\overline{WZ}\)), and we consider \(\triangle WXZ\) and \(\triangle YZX\).

Step2: Identify angle - angle - side (AAS) or side - angle - side (SAS) conditions

We know that \(\angle WXY\cong\angle YZW\) (opposite angles of a parallelogram are congruent) and \(\angle WXZ\cong\angle YZX\) (alternate interior angles as \(\overline{WX}\parallel\overline{ZY}\) and \(XZ\) is a transversal). Also, \(XZ = ZX\) (reflexive property).

Step3: Prove triangle congruence

By the angle - side - angle (ASA) congruence criterion, \(\triangle WXZ\cong\triangle YZX\).

Step4: Use congruent - triangle properties

If \(\triangle WXZ\cong\triangle YZX\), then corresponding sides are congruent. So, \(\overline{XY}\cong\overline{ZW}\) because they are corresponding sides of congruent triangles. The reason for \(\overline{XY}\cong\overline{ZW}\) is the alternate interior angles theorem which helps in establishing the congruence of the triangles that lead to the congruence of these sides.

Answer:

Alternate interior angles theorem