Question

$\angle rqs \cong \angle tqu$, $\angle qsu \cong \angle qtr$, and $\overline{qs} \cong \overline{qt}$. complete the proof that $\triangle qrt \cong \triangle qus$.

	statement	reason
2	$\angle qsu \cong \angle qtr$	given
3	$\overline{qs} \cong \overline{qt}$	given
4	$m\angle squ = m\angle sqt + m\angle tqu$	additive property of angle measure
5	$m\angle rqt = m\angle rqs + m\angle sqt$	additive property of angle measure
6	$m\angle squ = m\angle sqt + m\angle rqs$	substitution
7	$m\angle rqt = m\angle squ$
8	$\triangle qrt \cong \triangle qus$

Explanation:

Step1: Justify angle equality

From steps 5 and 6, we use the Transitive Property of Equality: if $m\angle RQT = m\angle RQS + m\angle SQT$ and $m\angle SQU = m\angle SQT + m\angle RQS$, then $m\angle RQT = m\angle SQU$.

Step2: Justify triangle congruence

We have: $\angle QTR \cong \angle QSU$ (given), $\overline{QS} \cong \overline{QT}$ (given), and $\angle RQT \cong \angle SQU$ (proven). This matches the ASA (Angle-Side-Angle) Congruence Postulate.

Answer:

1. Reason: Transitive Property of Equality
2. Reason: ASA (Angle-Side-Angle) Congruence Postulate