Question

1. given: (overline{lp}) bisects (overline{mo}), (overline{lo}paralleloverline{mp}) prove: (\triangle lnocong\triangle mnp) 1. given 2. given 3. (overline{ln}congoverline{pn}) 4. alternate interior 5. vertical angles 6. asa

Explanation:

Step1: Use the property of parallel lines

Since \(\overline{LO}\parallel\overline{MP}\), by the Alternate - Interior Angles Theorem, \(\angle LON=\angle MPN\) (alternate interior angles are congruent).

Step2: Use the vertical angles property

\(\angle LNO\) and \(\angle MNP\) are vertical angles. By the Vertical Angles Theorem, \(\angle LNO\cong\angle MNP\).

Step3: Use the given side - congruence

We are given that \(\overline{LN}\cong\overline{PN}\).

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

In \(\triangle LNO\) and \(\triangle MNP\), we have \(\angle LON=\angle MPN\) (from step 1), \(\overline{LN}\cong\overline{PN}\) (given), and \(\angle LNO\cong\angle MNP\) (from step 2). By the ASA (Angle - Side - Angle) congruence criterion, \(\triangle LNO\cong\triangle MNP\).

Answer:

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