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, __________ 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

Answer 1

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.

<M + <P = 180° .......(1)

Using NO as a transversal, ∠N and ∠O are same-side interior angles, so they are supplementary.

<N + <O = 180° ......(2)

Using OP as a transversal, ∠O and ∠P are same-side interior angles, so they are supplementary.

<O + <P = 180° .......(3)

From (1) , (2) & (3) we have

< M + <P = < P+ < O = < O + <N

We get

<M = <O and <P = <N

Therefore, <M = <O and < P = <N because they are supplements of the same angle.

Answer 2

Your answer is [d] ∠M ≅ ∠O; ∠N ≅ ∠P