Question

in quadrilateral mnop, mp is congruent to on and mp is parallel to on. anne has written a proof to show that mnop is a parallelogram. fill in the blanks to complete the proof.
since mp is parallel to ____, alternate interior angles __ and nom are congruent. mo is congruent to __ since segments are congruent to themselves. along with the given information that mp is congruent to on, triangle pmo is congruent to __ by the __ triangle congruence. since the triangles are congruent, all pairs of corresponding angles are congruent, so angle pom is congruent to__. since those alternate interior angles are congruent, mn must be parallel to ____. since we define a parallelogram as a quadrilateral with both pairs of opposite sides parallel, mnop is a parallelogram.

Explanation:

Step1: Identify parallel - side related angles

Since \(MP\) is parallel to \(ON\), alternate - interior angles \(\angle PMO\) and \(\angle NOM\) are congruent.

Step2: Consider self - congruent segment

\(MO\) is congruent to \(MO\) since segments are congruent to themselves.

Step3: Determine congruent triangles

Along with the given information that \(MP\) is congruent to \(ON\), triangle \(PMO\) is congruent to triangle \(NOM\) by the \(SAS\) (Side - Angle - Side) Triangle Congruence.

Step4: Find corresponding congruent angles

Since the triangles are congruent, all pairs of corresponding angles are congruent, so angle \(\angle POM\) is congruent to \(\angle NMO\).

Step5: Prove parallel sides

Since those alternate interior angles are congruent, \(MN\) must be parallel to \(PO\). Since we define a parallelogram as a quadrilateral with both pairs of opposite sides parallel, \(MNOP\) is a parallelogram.

Answer:

Since \(MP\) is parallel to \(ON\), alternate - interior angles \(\angle PMO\) and \(\angle NOM\) are congruent. \(MO\) is congruent to \(MO\). Triangle \(PMO\) is congruent to triangle \(NOM\) by \(SAS\). Angle \(\angle POM\) is congruent to \(\angle NMO\). \(MN\) is parallel to \(PO\), so \(MNOP\) is a parallelogram.