Question

complete the proofs below using the most appropriate method, sss or sas.

1. given: \\(\overline{pq} \cong \overline{sr}\\), \\(\angle pqr \cong \angle srq\\)

prove: \\(\delta pqr \cong \delta srq\\)
(there is a diagram of a quadrilateral pqrs with q and r connected, forming two triangles pqr and srq)
statements | reasons

1. \\(\square\\) | \\(\square\\)
2. \\(\square\\) | \\(\square\\)
3. \\(\square\\) | \\(\square\\)
4. \\(\square\\) | \\(\square\\)

(options below: given, definition of midpoint, reflexive property, vertical angles, definition of angle bisector, alternate interior angles, alternate exterior angles, corresponding angles, sss, sas, \\(qr \cong qr\\), \\(pq \cong sr\\), \\(\delta pqr \cong \delta srq\\), \\(\angle pqr \cong \angle srq\\))

Explanation:

Step1: Identify Given Information

We know from the problem that \( \overline{PQ} \cong \overline{SR} \) and \( \angle PQR \cong \angle SRQ \). Also, \( QR \) is a common side to both triangles \( \triangle PQR \) and \( \triangle SRQ \), so by the Reflexive Property, \( \overline{QR} \cong \overline{QR} \).

Step2: Match Statements and Reasons

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

Reason 1: Given (since this is one of the given congruences in the problem).

1. Statement 2: \( \angle PQR \cong \angle SRQ \)

Reason 2: Given (this is the other given congruence).

1. Statement 3: \( \overline{QR} \cong \overline{QR} \)

Reason 3: Reflexive Property (a side is congruent to itself).

1. Statement 4: \( \triangle PQR \cong \triangle SRQ \)

Reason 4: SAS (we have two sides and the included angle: \( PQ \cong SR \), \( \angle PQR \cong \angle SRQ \), and \( QR \cong QR \), so Side - Angle - Side congruence applies).

Answer:

1. \( \boldsymbol{\overline{PQ} \cong \overline{SR}} \); Given
2. \( \boldsymbol{\angle PQR \cong \angle SRQ} \); Given
3. \( \boldsymbol{\overline{QR} \cong \overline{QR}} \); Reflexive Property
4. \( \boldsymbol{\triangle PQR \cong \triangle SRQ} \); SAS