Question

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

Explanation:

Step1: Given

The problem states that \(V\) is the midpoint of \(\overline{TU}\), so this is given.

Step2: Given

From the diagram, the marks on \(\overline{SU}\) and \(\overline{ST}\) indicate they are congruent, so this is given.

Step3: Definition of midpoint

Since \(V\) is the midpoint of \(\overline{TU}\), by the definition of a midpoint, \(TV = UV\), so \(\overline{TV}\cong\overline{UV}\).

Step4: Reflexive property of congruence

Any segment is congruent to itself, so \(\overline{SV}\cong\overline{SV}\) by the reflexive property.

Step5: SSS (Side - Side - Side) Congruence Theorem

We have \(\overline{SU}\cong\overline{ST}\), \(\overline{TV}\cong\overline{UV}\), and \(\overline{SV}\cong\overline{SV}\). So, by the SSS congruence theorem, \(\triangle STV\cong\triangle SUV\).

Step6: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle STV\cong\triangle SUV\), their corresponding angles \(\angle U\) and \(\angle T\) are congruent by CPCTC.

Answer:

1. Given
2. Given
3. Definition of midpoint
4. Reflexive property of congruence
5. SSS Congruence Theorem
6. CPCTC