Question

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

Explanation:

Step1: Definition of midpoint

Since \(T\) is the midpoint of \(\overline{SU}\), by the definition of a midpoint, \(\overline{ST}\cong\overline{TU}\).

Step2: SSS Congruence Criterion

We know \(\overline{UW}\cong\overline{TV}\), \(\overline{SV}\cong\overline{TW}\), and \(\overline{ST}\cong\overline{TU}\).
By the Side - Side - Side (SSS) congruence criterion, \(\triangle STV\cong\triangle TUW\).

Step3: Corresponding parts of congruent triangles

If \(\triangle STV\cong\triangle TUW\), then by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, \(\angle U\cong\angle STV\).

Answer:

1. Given
2. Given
3. Given
4. Definition of midpoint
5. SSS (Side - Side - Side) Congruence
6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)