Question

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. $angle wxycongangle yzw$, $angle wxzcongangle yzx$. alternate interior angles theorem
4. $overline{xy}congoverline{zw}$. opposite sides of a parallelogram are congruent
5. $\triangle wzvcong\triangle yxv$
6. $overline{wv}congoverline{yv}$
7. $overline{xz}$ bisects $overline{wy}$

asa congruency criteria
sss congruency criteria
sas congruency criteria
ssa congruency criteria

Explanation:

Step1: Recall congruence criteria

We have angle - side - angle information from previous steps. In $\triangle WZV$ and $\triangle YXV$, we have $\angle WXZ\cong\angle YZX$ (alternate - interior angles), $\overline{XY}\cong\overline{ZW}$ (opposite sides of parallelogram), and $\angle WZV\cong\angle YXV$ (alternate - interior angles).

Step2: Select congruence criterion

The ASA (Angle - Side - Angle) congruence criterion states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. Here, we have two pairs of congruent angles and the included sides are congruent, so we use ASA to prove $\triangle WZV\cong\triangle YXV$.

Step3: Use triangle congruence to get side congruence

Since $\triangle WZV\cong\triangle YXV$, corresponding parts of congruent triangles are congruent. So, $\overline{WV}\cong\overline{YV}$. And if $\overline{WV}\cong\overline{YV}$, then $\overline{XZ}$ bisects $\overline{WY}$.

Answer:

1. ASA congruency criteria
2. Corresponding parts of congruent triangles are congruent
3. Definition of a bisector