Question

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

statement
1 v is the midpoint of \\( \overline { s u } \\)
2 v is the midpoint of \\( \overline { r t } \\)
3 \\( \overline { s v } \cong \overline { u v } \\)
4 \\( \overline { r v } \cong \overline { t v } \\)
5 \\( \angle r v u \cong \angle s v t \\)
6 \\( \triangle r u v \cong \triangle t s v \\)
7 \\( \angle u \cong \angle s \\)
reason
given
given
definition of midpoint
i

Explanation:

Step1: Definition of vertical angles

Vertical angles are equal. So, \(\angle RVU\cong\angle SVT\) because they are vertical angles.

Step2: SAS (Side - Angle - Side) congruence criterion

We have \(SV = UV\) (from the definition of mid - point of \(SU\)), \(RV=TV\) (from the definition of mid - point of \(RT\)) and \(\angle RVU\cong\angle SVT\). By the SAS congruence criterion, \(\triangle RVU\cong\triangle TVS\).

Step3: Corresponding parts of congruent triangles

If \(\triangle RVU\cong\triangle TVS\), then their corresponding parts are congruent. So, \(\angle U\cong\angle S\) as they are corresponding angles of congruent triangles \(\triangle RVU\) and \(\triangle TVS\).

Answer:

\(\angle U\cong\angle S\)