Question

use the image to answer the question.
olivia 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)
by the cpctc theorem, i know that \\( \overline { m l } \cong \overline { o l } \\) and \\( \overline { n l } \cong \overline { p l } \\). \\( \overline { m o } \\) and \\( \overline { n p } \\) bisect each other because this is the definition of a bisector.
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 { m n } \parallel \overline { p o } \\) and \\( \overline { m p } \parallel \overline { n o } \\).
when i think of the diagonals as transversals cutting parallel lines, i can see that \\( \angle m o p \cong \angle o m n \\) and \\( \angle n p o \cong \angle p n m \\) because of the alternate interior angles theorem.
because of the definition of a parallelogram, i know that opposite sides are congruent. i can say that \\( \overline { m n } \cong \overline { o p } \\). i now have enough information to say that \\( \triangle m n l \cong \triangle o p l \\) by the asa congruence theorem.

Explanation:

Step1: Start with given information

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 \(\overline{MN}\parallel\overline{PO}\) and \(\overline{MP}\parallel\overline{NO}\).

Step2: Use alternate - interior angles

When I think of the diagonals as transversals cutting parallel lines, I can see that \(\angle MOP\cong\angle OMN\) and \(\angle NPO\cong\angle PNM\) because of the Alternate Interior Angles Theorem.

Step3: Use parallelogram side - congruence and ASA

Because of the definition of a parallelogram, I know that opposite sides are congruent. I can say that \(\overline{MN}\cong\overline{OP}\). I now have enough information to say that \(\triangle MNL\cong\triangle OPL\) by the ASA Congruence Theorem.

Step4: Use CPCTC

By the CPCTC Theorem, I know that \(\overline{ML}\cong\overline{OL}\) and \(\overline{NL}\cong\overline{PL}\). \(\overline{MO}\) and \(\overline{NP}\) bisect each other because this is the definition of a bisector.

Answer:

1. 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 \(\overline{MN}\parallel\overline{PO}\) and \(\overline{MP}\parallel\overline{NO}\).
2. When I think of the diagonals as transversals cutting parallel lines, I can see that \(\angle MOP\cong\angle OMN\) and \(\angle NPO\cong\angle PNM\) 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 \(\overline{MN}\cong\overline{OP}\). 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 \(\overline{ML}\cong\overline{OL}\) and \(\overline{NL}\cong\overline{PL}\). \(\overline{MO}\) and \(\overline{NP}\) bisect each other because this is the definition of a bisector.