Question

1. given: lp bisects mo, lo || mp

prove: △lno ≅ △mnp

1. given
2. given
3. ln ≅ pn
4. alternate interior
5. vertical angles
6. asa

Explanation:

Step1: Identify the given information

We are given that \( \overline{LP} \) bisects \( \overline{MO} \), so \( \overline{LN}\cong\overline{MN} \) (by the definition of a bisector). Also, \( \overline{LO}\parallel\overline{MP} \).

Step2: Use the properties of parallel lines

Since \( \overline{LO}\parallel\overline{MP} \), \( \angle LON\cong\angle MPN \) (alternate interior angles).

Step3: Consider vertical angles

\( \angle LNO\cong\angle MNP \) (vertical angles are congruent).

Step4: Apply the ASA (Angle - Side - Angle) congruence criterion

In \( \triangle LNO \) and \( \triangle MNP \), we have \( \overline{LN}\cong\overline{MN} \) (given, from the bisector), \( \angle LNO\cong\angle MNP \) (vertical angles), and \( \angle LON\cong\angle MPN \) (alternate interior angles). So, by ASA, \( \triangle LNO\cong\triangle MNP \).

Answer:

\( \triangle LNO\cong\triangle MNP \) by the ASA (Angle - Side - Angle) congruence criterion.