Question

complete the proof
given: (overline{wx} parallel overline{zy}); (overline{xy} parallel overline{wz})
prove: (\triangle wxz cong \triangle yzx)

(image of a parallelogram wxyz with diagonal xz)

statements	reasons
2. (angle wxz cong angle yzx); (angle wzx cong angle yxz)	2.
3. (overline{xz} cong overline{xz})	3.
4. (\triangle wxz cong \triangle yzx)	4.

(reasons options: alternate interior angles are congruent; alternate exterior angles are congruent; corresponding angles are congruent; symmetric property; transitive property; side - side - side (sss) theorem; side - angle - side (sas) postulate; reflexive property; angle - side - angle (asa) postulate)

Explanation:

Step1: Justify Statement 2

Since \( \overline{WY} \parallel \overline{XZ} \) and \( \overline{XY} \parallel \overline{WZ} \), when a transversal (like \( \overline{YZ} \) or \( \overline{WX} \)) cuts parallel lines, alternate interior angles are congruent. For \( \angle WYZ \) and \( \angle XZY \), and \( \angle WZY \) and \( \angle XYZ \), the reason is "Alternate interior angles are congruent".

Step2: Justify Statement 3

\( \overline{YZ} \) is a common side to both triangles \( \triangle WYZ \) and \( \triangle XZY \), so by the Reflexive Property, \( \overline{YZ} \cong \overline{YZ} \).

Step3: Justify Statement 4

We have two angles and the included side congruent (from Step 2: two pairs of alternate interior angles, and Step 3: included side \( \overline{YZ} \)). So by the Angle - Side - Angle (ASA) Postulate, \( \triangle WYZ \cong \triangle XZY \).

Answer:

1. Alternate interior angles are congruent
2. Reflexive Property
3. Angle - Side - Angle (ASA) Postulate