Question

use the image to answer the question. chloe wrote a paragraph proof to prove that the diagonals of parallelogram mnop bisect each other. drag and drop the statements and reasons of her proof to put them in the correct order, starting with the first step at the top. (2 points) - i am given that quadrilateral mnop is a parallelogram. i know that opposite sides of a parallelogram are congruent because of the definition of a parallelogram, so $overline{mn}paralleloverline{po}$ and $overline{mp}paralleloverline{no}$. - because of the definition of a parallelogram, i know that opposite sides are congruent. i can say that $overline{mn}congoverline{op}$. i now have enough information to say that $\triangle mnlcong\triangle opl$ by the asa congruence theorem. - when i think of the diagonals as transversals cutting parallel lines, i can see that $angle mopcongangle omn$ and $angle npocongangle pnm$ because of the alternate interior angles theorem. - by the cpctc theorem, i know that $ml = ol$ and $nl = pl$. mo and np bisect each other because this is the definition of a bisector.

Explanation:

Step1: State given and property of parallelogram

Given that $MNOP$ is a parallelogram. By definition of parallelogram, opposite - sides are parallel, so $\overline{MN}\parallel\overline{PO}$ and $\overline{MP}\parallel\overline{NO}$.

Step2: Use alternate - interior angles theorem

When the diagonals are considered as transversals cutting parallel lines, $\angle MOP=\angle OMN$ and $\angle NPO = \angle PNM$ because of the Alternate Interior Angles Theorem.

Step3: Use congruence of opposite sides

Since opposite sides of a parallelogram are congruent, $\overline{MN}=\overline{OP}$.

Step4: Prove triangle congruence

We have enough information to say that $\triangle MNL\cong\triangle OPL$ by the ASA (Angle - Side - Angle) Congruence Theorem.

Step5: Use CPCTC

By the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) Theorem, $\overline{ML}=\overline{OL}$ and $\overline{NL}=\overline{PL}$. Since this is the definition of a bisector, the diagonals $MP$ and $NO$ bisect each other.

Answer:

The correct order of statements and reasons for the proof that the diagonals of parallelogram $MNOP$ bisect each other is:

1. Given that quadrilateral $MNOP$ is a parallelogram. I know that opposite sides of a parallelogram are congruent because of the definition of a parallelogram, so $\overline{MN}\parallel\overline{PO}$ and $\overline{MP}\parallel\overline{NO}$.
2. When I think of the diagonals as transversals cutting parallel lines, I can see that $\angle MOP=\angle OMN$ and $\angle NPO=\angle PNM$ because of the Alternate Interior Angles Theorem.
3. Because of the definition of a parallelogram, I know that opposite sides are congruent. I can say that $\overline{MN}=\overline{OP}$. I now have enough information to say that $\triangle MNL\cong\triangle OPL$ by the ASA Congruence Theorem.
4. By the CPCTC Theorem, I know that $\overline{ML}=\overline{OL}$ and $\overline{NL}=\overline{PL}$. $MP$ and $NO$ bisect each other because this is the definition of a bisector.