Question

1. given: (overline{mp}) bisects (angle lmn) (overline{lm}congoverline{nm}) prove: (overline{lp}congoverline{np}) 1. (overline{mp}) bisects (angle lmn) 1. 2. 2. definition of bisect 3. 3. given 4. (overline{mp}congoverline{mp}) 4. 5. (\triangle lmpcong\triangle nmp) 5. 6. 6. cpctc

Explanation:

Step1: <Identify the given information>

1. $\overline{MP}$ bisects $\angle LMN$ (Given)

Step2: <Use the definition of angle bisector>

1. $\angle LMP=\angle NMP$ (Definition of bisect)

Step3: <Identify the given congruent segments>

1. $\overline{LM}\cong\overline{NM}$ (Given)

Step4: <Use the reflexive property>

1. $\overline{MP}\cong\overline{MP}$ (Reflexive property of congruence)

Step5: <Prove triangle congruence>

1. $\triangle LMP\cong\triangle NMP$ (SAS - Side - Angle - Side congruence criterion: $\overline{LM}\cong\overline{NM}$, $\angle LMP=\angle NMP$, $\overline{MP}\cong\overline{MP}$)

Step6: <Use CPCTC (Corresponding Parts of Congruent Triangles are Congruent)>

1. $\overline{LP}\cong\overline{NP}$ (CPCTC)

Answer:

$\overline{LP}\cong\overline{NP}$