Question

given: qs bisects ∠tqr; tq ≅ rq. prove: △qrs ≅ △qts. complete the missing parts of the paragraph proof. we know that segment qs bisects angle tqr because bisector, angle tqs is congruent to angle. by the definition of angle. we see that segment qs is congruent to segment sq by. therefore, we can conclude that triangles qrs and qts are congruent by

Explanation:

Step1: Identify angle bisector result

Since $\overline{QS}$ bisects $\angle TQR$, $\angle TQS \cong \angle RQS$ (angle bisector definition).

Step2: Note given congruent sides

Given $\overline{TQ} \cong \overline{RQ}$.

Step3: Recognize common side

$\overline{QS} \cong \overline{QS}$ (reflexive property).

Step4: Apply SAS congruence

$\triangle QRS \cong \triangle QTS$ by SAS (two sides and included angle congruent).

Answer:

$\triangle QRS \cong \triangle QTS$ by SAS congruence criterion.