Question

complete the proof that δuvz ≅ δxyw.
(image of two triangles with vertices u, v, z and x, y, w, and a table with statements and reasons:

1. ∠wxy ≅ ∠vuz (given)
2. vw ≅ yz (given)
3. ∠uzv ≅ ∠xwy (given)
4. vz = yz + vy (additive property of length)
5. wy = vw + vy (additive property of length)
6. vz = vw + vy (substitution)
7. wy = vz (transitive property of equality)
8. blank blank)

Explanation:

Step1: Identify Congruent Triangles

We have two triangles, \( \triangle UVZ \) and \( \triangle XYW \). We know \( \angle WXY \cong \angle VUZ \) (given), \( \angle UZV \cong \angle XWY \) (given), and we need to show a side is congruent. From steps 4 - 7, we found \( WY \cong VZ \).

Step2: Apply AAS Congruence

By the Angle - Angle - Side (AAS) congruence criterion, if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, the triangles are congruent. Here, \( \angle VUZ\cong\angle WXY \), \( \angle UZV\cong\angle XWY \), and \( VZ\cong WY \). So, \( \triangle UVZ\cong\triangle XYW \) by AAS.

Answer:

\( \triangle UVZ\cong\triangle XYW \) (by AAS Congruence Criterion)