Question

$\angle r \cong \angle q$. complete the proof that $\overline{pr} \cong \overline{pq}$.

	statement	reason
2	$\overline{qr} \cong \overline{qr}$
3	$\triangle pqr \cong \triangle prq$
4	$\overline{pr} \cong \overline{pq}$

Explanation:

Step1: Identify Given Information

The problem states $\angle R \cong \angle Q$, so the reason for statement 1 is "Given".

Step2: Identify Reflexive Property

For $\overline{QR} \cong \overline{QR}$, the reason is the "Reflexive Property of Congruence" (a segment is congruent to itself).

Step3: Identify Congruence Criterion

To prove $\triangle PQR \cong \triangle PRQ$, we use the "ASA" (Angle - Side - Angle) congruence criterion? Wait, no, let's check. We have $\angle R \cong \angle Q$ (given), $\overline{QR} \cong \overline{QR}$ (reflexive), and also $\angle Q \cong \angle R$? Wait, actually, the triangles $\triangle PQR$ and $\triangle PRQ$: angle at $R$ in $\triangle PQR$ is $\angle R$, angle at $Q$ in $\triangle PRQ$ is $\angle Q$ (which are congruent), side $QR$ is common. Wait, maybe it's "ASA" or "AAS"? Wait, actually, the correct congruence criterion here is "ASA" if we consider the angles and the included side, but since $\angle R \cong \angle Q$, $\overline{QR} \cong \overline{QR}$, and also $\angle Q \cong \angle R$? Wait, no, let's re - express. The triangles $\triangle PQR$ and $\triangle PRQ$: $\angle R \cong \angle Q$ (given), $\overline{QR} \cong \overline{QR}$ (reflexive), and $\angle QRP \cong \angle RQP$? Wait, maybe it's "ASA" (Angle - Side - Angle) or "AAS" (Angle - Angle - Side). Wait, actually, the correct congruence reason is "ASA" because we have two angles and the included side. Wait, no, the side $QR$ is between $\angle R$ and $\angle P$ in $\triangle PQR$ and between $\angle Q$ and $\angle P$ in $\triangle PRQ$. Since $\angle R \cong \angle Q$, $\overline{QR} \cong \overline{QR}$, and $\angle P$ is common (wait, $\angle P$ is the same in both triangles). So $\triangle PQR \cong \triangle PRQ$ by "ASA" (Angle - Side - Angle) congruence criterion ( $\angle R \cong \angle Q$, $\overline{QR} \cong \overline{QR}$, $\angle P \cong \angle P$ (reflexive for angles)).

Step4: Corresponding Parts of Congruent Triangles

For $\overline{PR} \cong \overline{PQ}$, the reason is "Corresponding Parts of Congruent Triangles are Congruent" (CPCTC), since $\triangle PQR \cong \triangle PRQ$, so their corresponding sides $PR$ and $PQ$ are congruent.

Answer:

1. Reason for $\angle R \cong \angle Q$: Given
2. Reason for $\overline{QR} \cong \overline{QR}$: Reflexive Property of Congruence
3. Reason for $\triangle PQR \cong \triangle PRQ$: ASA (Angle - Side - Angle) Congruence Criterion
4. Reason for $\overline{PR} \cong \overline{PQ}$: Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

If we are to fill in the table:

1. Reason: Given
2. Reason: Reflexive Property of Congruence
3. Reason: ASA (Angle - Side - Angle) Congruence
4. Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)