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, __ and _ because they are supplements of the same angle.

Which statements should fill in the blanks in the last line of the proof?

∠M is supplementary to ∠N; ∠M is supplementary to ∠O
∠M is supplementary to ∠O; ∠N is supplementary to ∠P
∠M ≅ ∠P; ∠N ≅ ∠O
∠M ≅ ∠O; ∠N ≅ ∠P

A diagram of parallelogram MNOP is attached below

We have side MN || side OP and side MP || NO

Using the rule of angles in parallel lines, ∠M and ∠P are supplementary as well as ∠M and ∠N.

Since ∠M+∠P = 180° and ∠M+∠N=180°, we can conclude that ∠P and ∠N are of equal size.

∠N and ∠O are supplementary by the rules of angles in parallel lines
∠O and ∠P are supplementary by the rules of angles in parallel lines

∠N+∠O=180° and ∠O+∠P=180°

∠N and ∠P are of equal size

we deduce further that ∠M and ∠O are of equal size

Hence, the correct statement to complete the proof is

∠M ≅ ∠O; ∠N ≅ ∠P

Martebi Martebi · Answer 2 · 2022-04-25T19:04:25+02:00

The last line of the proof for the angle is that , ∠M ≅ ∠O and ∠N ≅ ∠P because they are supplements of the same angle.

What are the supplements of the same angle?

The supplements of the same angle is known to occur when two angles are said to be congruent angles. Note that the Congruent Complements Theorem state that when two angles are complements of the same angle, one can say therefore that the two angles are congruent in nature.

Note that from the image of parallelogram MNOP, we can see the sides MN || OP and also MP || NO.

Note also that ∠M and ∠P are supplementary and also ∠M and ∠N and they are all =180°.

Due to the above, we can say that we can conclude that ∠M ≅ ∠O; ∠N ≅ ∠P is the best fill up o the blanks.

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