Question

prove that \\( \overline { n p } \\) and \\( \overline { m o } \\) bisect each other. drag the correct reason into each blank to complete the proof.\\( \overline { m n } \\) is congruent and parallel to \\( \overline { o p } \\) based off the properties of a parallelogram.\\( \angle n m o \cong \angle p o m \\) because of the alternate interior angles theorem.\\( \angle m n p \cong \angle o p n \\) because of the alternate interior angles theorem.\\( \triangle m n q \cong \triangle o p q \\) because of\\( \overline { n q } \cong \overline { p q } \\) and \\( \overline { m q } \cong \overline { o q } \\) because corresponding parts of congruent triangles are congruent.therefore, \\( \overline { n p } \\) and \\( \overline { m o } \\) bisect each other.alternate interior angles theoremalternate exterior angles theoremcorresponding angle theoremside angle angle congruenceangle side angle congruence

Explanation:

Step1: Find the congruent triangles

Since \(\overline{MN}\parallel\overline{OP}\), we have \(\angle NMO\cong\angle POM\) (Alternate Interior Angles Theorem) and \(\angle MNP\cong\angle OPN\) (Alternate Interior Angles Theorem). Also, \(\overline{MN}\cong\overline{OP}\) (property of a parallelogram). So, \(\triangle MNQ\cong\triangle OPQ\) by Angle - Side - Angle Congruence.

Step2: Use the properties of congruent triangles

Because \(\triangle MNQ\cong\triangle OPQ\), their corresponding parts are congruent. So, \(\overline{NQ}\cong\overline{PQ}\) and \(\overline{MQ}\cong\overline{OQ}\) (Corresponding Parts of Congruent Triangles are Congruent).

Step3: Conclude the bisection

Since \(\overline{NQ}\cong\overline{PQ}\) and \(\overline{MQ}\cong\overline{OQ}\), \(\overline{NP}\) and \(\overline{MO}\) bisect each other.

Answer:

\(\triangle MNQ\cong\triangle OPQ\) because of Angle - Side - Angle Congruence. \(\overline{NQ}\cong\overline{PQ}\) and \(\overline{MQ}\cong\overline{OQ}\) because corresponding parts of congruent triangles are congruent.