Question

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

Explanation:

Step1: Given

Statement 1: \( V \) is the mid - point of \( \overline{RT} \) (Given)
Statement 2: \( V \) is the mid - point of \( \overline{SU} \) (Given)

Step2: Mid - point definition

Since \( V \) is the mid - point of \( \overline{RT} \), by the definition of a mid - point, \( \overline{RV}=\overline{TV} \) (Definition of mid - point)
Since \( V \) is the mid - point of \( \overline{SU} \), by the definition of a mid - point, \( \overline{SV}=\overline{UV} \) (Definition of mid - point)

Step3: Vertical angles

\( \angle RVU\) and \( \angle TVS \) are vertical angles. So, \( \angle RVU\cong\angle TVS \) (Vertical angles are congruent)

Step4: SAS (Side - Angle - Side)

In \( \triangle RVU \) and \( \triangle TVS \), we have \( \overline{RV}=\overline{TV} \), \( \angle RVU\cong\angle TVS \), \( \overline{SV}=\overline{UV} \). So, \( \triangle RVU\cong\triangle TVS \) (SAS congruence criterion)

Step5: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \( \triangle RVU\cong\triangle TVS \), then \( \angle T\cong\angle R \) (CPCTC)

Answer:

1. Given
2. Given
3. Definition of mid - point
4. Definition of mid - point
5. Vertical angles are congruent
6. SAS (Side - Angle - Side)
7. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)