Question

$\angle qsr \cong \angle sqt$, $\overline{rs} \perp \overline{qr}$, and $\overline{qt} \perp \overline{st}$. complete the proof that $\triangle qst \cong \triangle sqr$.

	statement	reason
2	$\overline{rs} \perp \overline{qr}$	given
3	$\overline{qt} \perp \overline{st}$	given
4	$\angle r \cong \angle t$	all right angles are congruent
5	$\overline{qs} \cong \overline{qs}$
6	$\triangle qst \cong \triangle sqr$

Explanation:

Step1: Analyze Statement 5

The segment $\overline{QS}$ is common to both $\triangle QST$ and $\triangle SQR$. The reason for $\overline{QS} \cong \overline{QS}$ is the Reflexive Property of Congruence, which states that any segment is congruent to itself.

Step2: Analyze Statement 6

We have $\angle QSR \cong \angle SQT$ (given), $\angle R \cong \angle T$ (all right angles are congruent), and $\overline{QS} \cong \overline{QS}$ (reflexive property). This satisfies the AAS (Angle - Angle - Side) Congruence Criterion for triangles. So the reason for $\triangle QST \cong \triangle SQR$ is AAS Congruence Criterion.

Answer:

For statement 5, the reason is "Reflexive Property of Congruence". For statement 6, the reason is "AAS (Angle - Angle - Side) Congruence Criterion".