Question

1. given: \\(\overline{pq} \cong \overline{rs}\\)\\(\overline{pq} \perp \overline{qs}\\)\\(\overline{rs} \perp \overline{qs}\\)prove: \\(\overline{ps} \cong \overline{rq}\\)\\(\

$$\begin{array}{|c|c|}\\hline \\text{statements} & \\text{reasons} \\\\ \\hline 1. \\ \\overline{pq} \\cong \\overline{rs}, \\ \\overline{pq} \\perp \\overline{qs}, \\ \\overline{rs} \\perp \\overline{qs} & 1. \\text{given} \\\\ \\hline \\end{array}$$

\\)

Explanation:

Step1: Identify Right Angles

Since \(\overline{PQ} \perp \overline{QS}\) and \(\overline{RS} \perp \overline{QS}\), \(\angle PQS = 90^\circ\) and \(\angle RSQ = 90^\circ\). So, \(\triangle PQS\) and \(\triangle RSQ\) are right triangles.

Step2: Identify Common Side

\(\overline{QS}\) is a common side to both \(\triangle PQS\) and \(\triangle RSQ\), so \(\overline{QS} \cong \overline{SQ}\) (reflexive property).

Step3: Apply SAS Congruence

We know \(\overline{PQ} \cong \overline{RS}\) (given), \(\angle PQS \cong \angle RSQ\) (both right angles), and \(\overline{QS} \cong \overline{SQ}\) (reflexive). By the Side - Angle - Side (SAS) congruence criterion, \(\triangle PQS \cong \triangle RSQ\).

Step4: Corresponding Parts of Congruent Triangles

Since \(\triangle PQS \cong \triangle RSQ\), their corresponding sides \(\overline{PS}\) and \(\overline{RQ}\) are congruent. So, \(\overline{PS} \cong \overline{RQ}\).

Now, filling in the proof table:

Statements	Reasons
2. \(\angle PQS = 90^\circ\), \(\angle RSQ = 90^\circ\)	2. Definition of perpendicular lines
3. \(\angle PQS \cong \angle RSQ\)	3. All right angles are congruent
4. \(\overline{QS} \cong \overline{SQ}\)	4. Reflexive property of congruence
5. \(\triangle PQS \cong \triangle RSQ\)	5. SAS (Side - Angle - Side) congruence criterion
6. \(\overline{PS} \cong \overline{RQ}\)	6. Corresponding parts of congruent triangles are congruent (CPCTC)

Answer:

The proof is completed by showing \(\triangle PQS \cong \triangle RSQ\) using SAS and then using CPCTC to conclude \(\overline{PS} \cong \overline{RQ}\), with the steps filled in the table as above.