Question

a partial proof was constructed given that mnop is a parallelogram. by the definition of a parallelogram, mn ∥ po and mp ∥ no. using mp as a transversal, ∠m and ∠p are same - side interior angles, so they are supplementary. using no as a transversal, ∠n and ∠o are same - side interior angles, so they are supplementary. using op as a transversal, ∠o and ∠p are same - side interior angles, so they are supplementary. therefore, ________ because they are supplements of the same angle. which statement should fill in the blank in the last line of the proof? ∠m is supplementary to ∠o ∠n is supplementary to ∠p ∠m ≅ ∠p ∠n ≅ ∠p

Explanation:

1. Recall the property: If two angles are supplements of the same angle, then they are congruent (the Congruent Supplements Theorem).
2. Analyze the angles:

• From the proof, ∠N and ∠O are supplementary (using \(\overline{NO}\) as transversal), and ∠O and ∠P are supplementary (using \(\overline{OP}\) as transversal). So ∠N and ∠P are both supplements of ∠O.
• Also, ∠M and ∠P are supplementary (using \(\overline{MP}\) as transversal), and ∠M and ∠N could be related, but let's check the options:
• Option 1: ∠M supplementary to ∠O – Not supported by "supplements of the same angle" logic here.
• Option 2: ∠N supplementary to ∠P – We need congruence, not supplementary (since they are supplements of same angle, they should be congruent).
• Option 3: ∠M ≅ ∠P – ∠M and ∠P are same - side interior angles (supplementary), not congruent (unless it's a rectangle, but we only know it's a parallelogram).
• Option 4: ∠N ≅ ∠P – ∠N is supplementary to ∠O, ∠P is supplementary to ∠O. By Congruent Supplements Theorem, ∠N ≅ ∠P.

Answer:

∠N ≅ ∠P (the fourth option: ∠N ≅ ∠P)