Question

$\overline{wx} \parallel \overline{vy}$. complete the proof that $\triangle wxz \cong \triangle yvz$.

(image of two triangles with vertices v, y, z and w, x, z, sharing vertex z. table with statements and reasons:

1. $\overline{wx} \cong \overline{vy}$ (given)
2. $\overline{wx} \parallel \overline{vy}$ (given)
3. $\angle vzy \cong \angle wzx$ (reason blank)
4. $\angle y \cong \angle w$ (reason blank)
5. $\triangle wxz \cong \triangle yvz$ (reason blank))

Explanation:

Step1: Analyze ∠VZY and ∠WZX

∠VZY and ∠WZX are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent. So the reason for ∠VZY ≅ ∠WZX is "Vertical Angles Theorem".

Step2: Analyze ∠Y and ∠W

Since \( \overline{WX} \parallel \overline{VY} \), ∠Y and ∠W are alternate interior angles. By the Alternate Interior Angles Theorem, alternate interior angles are congruent when lines are parallel. So the reason for ∠Y ≅ ∠W is "Alternate Interior Angles Theorem".

Step3: Prove Triangle Congruence

We have \( \overline{WX} \cong \overline{VY} \) (given), ∠Y ≅ ∠W (alternate interior angles), and ∠VZY ≅ ∠WZX (vertical angles). So by the AAS (Angle - Angle - Side) Congruence Criterion, \( \triangle WXZ \cong \triangle YVZ \). The reason for \( \triangle WXZ \cong \triangle YVZ \) is "AAS Congruence Criterion".

Answer:

1. Reason: Vertical Angles Theorem
2. Reason: Alternate Interior Angles Theorem
3. Reason: AAS (Angle - Angle - Side) Congruence Criterion