Question

$overline{qr} cong overline{rs}$ and $overline{ps} cong overline{pq}$. complete the proof that $\triangle prs cong \triangle prq$.
(image of a quadrilateral with vertices s, p, q, r and diagonal pr, and a table with statement and reason columns, rows: 1. $overline{qr} cong overline{rs}$; 2. $overline{ps} cong overline{pq}$; 3. $overline{pr} cong overline{pr}$; 4. $\triangle prs cong \triangle prq$)

Explanation:

Step1: Identify Given Information

We are given \(\overline{QR} \cong \overline{RS}\) (Statement 1) and \(\overline{PS} \cong \overline{PQ}\) (Statement 2). Also, \(\overline{PR}\) is a common side to both \(\triangle PRS\) and \(\triangle PRQ\), so \(\overline{PR} \cong \overline{PR}\) (Statement 3) by the Reflexive Property of Congruence.

Step2: Determine Congruence Criterion

For triangles \(\triangle PRS\) and \(\triangle PRQ\), we have three sides:

• \(RS = QR\) (from Statement 1)
• \(PS = PQ\) (from Statement 2)
• \(PR = PR\) (from Statement 3)

By the Side - Side - Side (SSS) Congruence Postulate, if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. So, we can conclude that \(\triangle PRS \cong \triangle PRQ\) by SSS.

Now, let's fill in the reasons:

• Reason for Statement 1: Given
• Reason for Statement 2: Given
• Reason for Statement 3: Reflexive Property of Congruence
• Reason for Statement 4: SSS (Side - Side - Side) Congruence Postulate

Answer:

1. Reason for \(\overline{QR} \cong \overline{RS}\): Given
2. Reason for \(\overline{PS} \cong \overline{PQ}\): Given
3. Reason for \(\overline{PR} \cong \overline{PR}\): Reflexive Property of Congruence
4. Reason for \(\triangle PRS \cong \triangle PRQ\): SSS (Side - Side - Side) Congruence Postulate