Question

\\( \overline { w x } \cong \overline { t u } \\), \\( \angle s x t \cong \angle u w v \\), and \\( \angle s t x \cong \angle v u w \\). complete the proof that \\( \triangle u v w \cong \triangle t s x \\).

Explanation:

Step1: Substitute \( \overline{WX}\cong\overline{TU} \)

Since \( \overline{WX}\cong\overline{TU} \), then \( WX = TU \). In statement 4, \( TX=WX + TW \), substituting \( WX = TU \) gives \( TX=TU + TW \).

Step2: Use the Angle - Angle - Side (AAS) Congruence Criterion

We know that \( \angle SXT\cong\angle UWV \), \( \angle STX\cong\angle VUW \), and from step 1, \( TX = UW \) (because \( TX=TU + TW\) and \( UW=TU + TW\)).
By the AAS (Angle - Angle - Side) congruence criterion for triangles, if two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, then the two triangles are congruent.

Answer:

\( \triangle U V W\cong\triangle T S X\) by AAS