Question

$\overline{rs} \cong \overline{pq}$ and $\overline{ps} \cong \overline{qr}$. complete the proof that $\angle prq \cong \angle rps$.
(image of a quadrilateral pqrs with diagonal pr, and a table with statements 1 - 5 and empty reason columns)

Explanation:

Step1: Identify Given Information

We know \(\overline{RS} \cong \overline{PQ}\) and \(\overline{PS} \cong \overline{QR}\), and \(\overline{PR}\) is a common side.

Step2: Prove Triangle Congruence

For \(\triangle PQR\) and \(\triangle RSP\), we have:

• \(\overline{RS} \cong \overline{PQ}\) (Given)
• \(\overline{PS} \cong \overline{QR}\) (Given)
• \(\overline{PR} \cong \overline{PR}\) (Reflexive Property of Congruence)

By the SSS (Side - Side - Side) Congruence Criterion, \(\triangle PQR \cong \triangle RSP\).

Step3: Corresponding Angles of Congruent Triangles

Since \(\triangle PQR \cong \triangle RSP\), their corresponding angles are congruent. The angle \(\angle PRQ\) in \(\triangle PQR\) corresponds to \(\angle RPS\) in \(\triangle RSP\), so \(\angle PRQ \cong \angle RPS\).

Now, let's fill in the reasons for each statement:

1. Statement: \(\overline{RS} \cong \overline{PQ}\)

• Reason: Given

1. Statement: \(\overline{PS} \cong \overline{QR}\)

• Reason: Given

1. Statement: \(\overline{PR} \cong \overline{PR}\)

• Reason: Reflexive Property of Congruence

1. Statement: \(\triangle PQR \cong \triangle RSP\)

• Reason: SSS Congruence Criterion (since we have three pairs of congruent sides)

1. Statement: \(\angle PRQ \cong \angle RPS\)

• Reason: Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

Answer:

1. Reason: Given
2. Reason: Given
3. Reason: Reflexive Property of Congruence
4. Reason: SSS Congruence Criterion
5. Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)