Question

$overline{r s} perp overline{r t}, overline{q t} perp overline{q s}$, and $overline{q u} cong overline{r u}$. complete the proof that $\triangle r s u cong \triangle q t u$.

statement	reason
2 $overline{q t} perp overline{q s}$	given
3 $overline{q u} cong overline{r u}$	given
4 $angle q cong angle r$
5 $angle q u t cong angle r u s$
6 $\triangle r s u cong \triangle q t u$

Explanation:

Step1: Determine the reason for \( \angle Q\cong\angle R \)

Since \( \overline{RS}\perp\overline{RT} \) and \( \overline{QT}\perp\overline{QS} \), \( \angle R\) and \( \angle Q \) are right angles. By the definition of right angles, all right angles are congruent. So the reason for \( \angle Q\cong\angle R \) is "All right angles are congruent".

Step2: Determine the reason for \( \angle QUT\cong\angle RUS \)

\( \angle QUT \) and \( \angle RUS \) are vertical angles. By the vertical angles theorem, vertical angles are congruent. So the reason for \( \angle QUT\cong\angle RUS \) is "Vertical angles theorem".

Step3: Determine the reason for \( \triangle RSU\cong\triangle QTU \)

We have \( \angle Q\cong\angle R \), \( \overline{QU}\cong\overline{RU} \), and \( \angle QUT\cong\angle RUS \). By the Angle - Side - Angle (ASA) congruence criterion, \( \triangle RSU\cong\triangle QTU \). So the reason for \( \triangle RSU\cong\triangle QTU \) is "ASA (Angle - Side - Angle) congruence criterion".

Answer:

1. All right angles are congruent
2. Vertical angles theorem
3. ASA (Angle - Side - Angle) congruence criterion