Question

given: △lmp, lm ≅ lp
prove: ∠m ≅ ∠p
proof:
statements\treasons

1. let n be the midpoint of mp\t1. every segment has exactly one midpoint
2. draw an auxiliary segment ln\t2. two points determine a line
3. mn ≅ pn\t3.

4.\t4. reflexive property of congruence

1. lm ≅ lp\t5.
2. △lmn ≅ △lpn\t6. sss
3. ∠m ≅ ∠p\t7. cpctc

Explanation:

Step1: Reason for midpoint congruence

Definition of midpoint: A midpoint divides a segment into two congruent parts, so $MN \cong PN$.

Step2: Statement for reflexive property

Reflexive property applies to a segment congruent to itself: $\overline{LN} \cong \overline{LN}$.

Step3: Reason for given congruence

$\overline{LM} \cong \overline{LP}$ is the given information from the problem.

Answer:

1. Definition of midpoint
2. $\overline{LN} \cong \overline{LN}$
3. Given