Question

given: $overline{nq}$ is the bisector of $angle mnp$ and $angle nmq cong angle npq$
prove: $\triangle mnq cong \triangle pnq$
triangle diagram with n at top, m and p at base, q on base between m and p

statements	reasons
2. $angle mnq cong angle pnq$	2.
3. $angle nmq cong angle npq$	3. given
4. $overline{nq} cong overline{nq}$	4.
5. $\triangle mnq cong \triangle pnq$	5.

matching: statement 2, statement 4, statement 5, segment mn is congruent to segment pn with options: sas, definition of angle bisector, aas, reflexive property, asa, cpctc

Explanation:

Step1: Analyze Statement 2

Since \(\overline{NQ}\) is the bisector of \(\angle MNP\), by the definition of an angle bisector, it divides \(\angle MNP\) into two congruent angles. So \(\angle MNQ\cong\angle PNQ\) because of the "Definition of angle bisector".

Step2: Analyze Statement 4

The segment \(\overline{NQ}\) is congruent to itself. This is due to the "Reflexive Property" (a segment is congruent to itself).

Step3: Analyze Statement 5

We have \(\angle MNQ\cong\angle PNQ\) (from step 1), \(\angle NMQ\cong\angle NPQ\) (given), and \(\overline{NQ}\cong\overline{NQ}\) (from step 2). This is the AAS (Angle - Angle - Side) congruence criterion, so \(\triangle MNQ\cong\triangle PNQ\) by "AAS".

Step4: Analyze Segment MN and PN

If \(\triangle MNQ\cong\triangle PNQ\), then by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), segment \(MN\) is congruent to segment \(PN\).

Answer:

• Statement 2: Definition of angle bisector
• Statement 4: Reflexive Property
• Statement 5: AAS
• Segment \(MN\) is congruent to segment \(PN\): CPCTC