Question

$\overleftrightarrow{qt}$ bisects $\angle rtv$. complete the proof that $\triangle qtu \cong \triangle qts$.

	statement	reason
2	$\angle s \cong \angle u$	given
3	$\angle utv \cong \angle rts$	vertical angle theorem
4	$\angle qtv \cong \angle qtr$	definition of angle bisector
5	$m\angle qtu = m\angle qtv + m\angle utv$	additive property of angle measure
6	$m\angle qts = m\angle qtr + m\angle rts$	additive property of angle measure
7	$m\angle qtu = m\angle qtr + m\angle rts$	substitution
8	$m\angle qts = m\angle qtu$	transitive property of equality
9	$\overline{qt} \cong \overline{qt}$	reflexive property of congruence
10

Explanation:

Step1: Identify congruent parts

We have $\angle S \cong \angle U$ (given), $\angle QTS \cong \angle QTU$ (from step 8), and $\overline{QT} \cong \overline{QT}$ (reflexive property). So by AAS (Angle - Angle - Side) congruence criterion, $\triangle QTU \cong \triangle QTS$.

Step2: State the congruence

Using the AAS congruence postulate, since two angles and a non - included side of $\triangle QTU$ are congruent to two angles and the corresponding non - included side of $\triangle QTS$ ($\angle U\cong\angle S$, $\angle QTU\cong\angle QTS$, $\overline{QT}\cong\overline{QT}$), we can conclude that $\triangle QTU\cong\triangle QTS$.

Answer:

$\triangle QTU \cong \triangle QTS$ by the AAS (Angle - Angle - Side) Congruence Postulate (Statement: $\triangle QTU \cong \triangle QTS$; Reason: AAS Congruence Postulate)