Question

$$\overline{xy}\cong\overline{vw},\angle s\cong\angle u$$, and $$\angle uxv\cong\angle swy$$. complete the proof that $$\triangle uvx\cong\triangle syw$$.

statement	reason
2	$$\angle s\cong\angle u$$	given
3	$$\angle uxv\cong\angle swy$$	given
4	$$wy = xy + wx$$	additive property of length
5	$$vx = vw + wx$$	additive property of length
6	$$wy = vw + wx$$	substitution
7	$$vx = wy$$	transitive property of equality
8	$$\triangle uvx\cong\triangle syw$$

Explanation:

Step1: Recall the AAS (Angle - Angle - Side) Congruence Theorem

The AAS congruence theorem states that 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, then the two triangles are congruent.

Step2: Identify the corresponding parts

We are given that \(\angle S\cong\angle U\) (one pair of angles), \(\angle UXV\cong\angle SWY\) (another pair of angles), and from step 7 we have \(VX = WY\) (a non - included side).

Answer:

By the AAS (Angle - Angle - Side) Congruence Theorem.