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

From the above information we are told that MNOP is a parallelogram with <M supplementary to <N, <P supplementary to <M and  <N supplementary to <O. We can therefore deduce that <O is supplementary to <P if OP will act as the transversal to MP and NO also, <O-=<M

Answer 2

∠M is supplementary to ∠N and ∠M ≅ ∠O and ∠N ≅ ∠P .

Explanation:

We have been given an parallelogram MNOP, MN ∥ PO and MP ∥ NO.

Since we know that consecutive angles of parallelogram are supplementary and opposite angles of parallelogram are congruent.

We have been given that ∠M and ∠P,∠N and ∠O and ∠O and ∠P are same-side interior angles, so they are supplementary.

Upon taking MN as a transversal we can see that ∠M and ∠N are same-side interior angles, so they are also supplementary.

We know that opposite angles of parallelogram are not supplementary so the statement that ∠M is supplementary to ∠O is wrong. Statements ∠M ≅ ∠P and ∠N ≅ ∠O are wrong as well (∠M+∠P=180 degrees).

Therefore, our statement in last line should be ∠M is supplementary to ∠N and ∠M ≅ ∠O; ∠N ≅ ∠P .