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. (4 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 parallelogram, so \\( \overline { m n } \parallel \overline { p o } \\) and \\( \overline { m p } \parallel \overline { n o } \\). when i think of diagonal as transversal cutting parallel line, 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 asa congruence theorem. 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 bisector.

Explanation:

Step1: Given parallelogram

I am given that quadrilateral MNOP is a parallelogram. I know that opposite sides of a parallelogram are congruent because of the definition of parallelogram, so \( \overline{MN}\parallel\overline{PO} \) and \( \overline{MP}\parallel\overline{NO} \).

Step2: Alternate interior angles

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

Step3: Prove triangle congruence

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 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 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 parallelogram, so \( \overline{MN}\parallel\overline{PO} \) and \( \overline{MP}\parallel\overline{NO} \).
2. When I think of diagonal as transversal cutting parallel line, 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 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 bisector.