Question

1. given: u is the midpoint of \\( \overline { s t } \\). \\( \overline { s u } \cong \overline { t u } \\). \\( \overline { v u } \cong \overline { w u } \\) prove: \\( \angle s v u \cong \angle t w u \\)

Explanation:

Step1: Definition of midpoint

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

Step2: Given information

We are given \(SU = TU\), \(VU=WU\) (from the problem statement).

Step3: Vertical angles

\(\angle SUV\) and \(\angle T UW\) are vertical angles. By the vertical - angles theorem, \(\angle SUV\cong\angle T UW\).

Step4: SAS (Side - Angle - Side) criterion

In \(\triangle SUV\) and \(\triangle TUW\), we have \(SU = TU\) (from Step 1), \(\angle SUV\cong\angle T UW\) (from Step 3), and \(VU = WU\) (given). So, by the SAS (Side - Angle - Side) congruence criterion, \(\triangle SUV\cong\triangle TUW\).

Step5: Corresponding parts of congruent triangles

Since \(\triangle SUV\cong\triangle TUW\), by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, \(\angle SVU\cong\angle T WU\).

Answer:

The statements and reasons are as follows:

Statements	Reasons
2. \(SU\cong TU\)	2. Definition of midpoint
3. \(VU\cong WU\)	3. Given
4. \(\angle SUV\cong\angle T UW\)	4. Vertical - angles theorem
5. \(\triangle SUV\cong\triangle TUW\)	5. SAS (Side - Angle - Side) criterion
6. \(\angle SVU\cong\angle T WU\)	6. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)