Question

1. \\( \overleftrightarrow { q t } \\) bisects \\( \angle r q s \\). complete the proof that \\( \triangle q s t \cong \triangle q r t \\).

statement
1 \\( \overleftrightarrow { q t } \\) bisects \\( \angle r q s \\)
2 \\( \overline { q r } \cong \overline { q s } \\)
3 \\( \angle r q t \cong \angle s q t \\)
4
reason
given
given
definition of angle bisector

Explanation:

Step1: Identify the common side

$\overline{QT}\cong\overline{QT}$ (Reflexive property of congruence)

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

In $\triangle QST$ and $\triangle QRT$:

• $\overline{QR}\cong\overline{QS}$ (Given)
• $\angle RQT\cong\angle SQT$ (From the definition of angle bisector as $\overleftrightarrow{QT}$ bisects $\angle RQS$)
• $\overline{QT}\cong\overline{QT}$ (Reflexive property)

By the SAS congruence criterion, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Answer:

$\triangle QST\cong\triangle QRT$ (by SAS congruence criterion)