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.
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 } \\).
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.
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.
by the cpctc theorem, i know that \\( \overline { m l } \cong \overline { o l } \\) and \\( \overline { n l } \cong \overline { p l } \\). \\( \overline { mo } \\) and \\( \overline { np } \\) bisect each other because this is the definition of a bisector

Explanation:

Step1: Use the definition of a parallelogram

A parallelogram has opposite sides parallel. So, for parallelogram \(MNOP\), \( \overline{MN}\parallel\overline{PO}\) and \( \overline{MP}\parallel\overline{NO}\).

Step2: Use the Alternate Interior Angles Theorem

Since \( \overline{MN}\parallel\overline{PO}\) and \( \overline{MO}\) is a transversal, \( \angle MOP\cong\angle OMN\). Also, since \( \overline{MP}\parallel\overline{NO}\) and \( \overline{NP}\) is a transversal, \( \angle NPO\cong\angle PNM\).

Step3: Prove triangle congruence

In \( \triangle MNL\) and \( \triangle OPL\), we have \( \overline{MN}\cong\overline{OP}\) (opposite sides of a parallelogram are congruent), \( \angle MNL\cong\angle OPL\) (alternate interior angles), and \( \angle MLN\cong\angle OLP\) (vertical angles). So, \( \triangle MNL\cong\triangle OPL\) by the \(ASA\) (Angle - Side - Angle) Congruence Theorem.

Step4: Use \(CPCTC\) (Corresponding Parts of Congruent Triangles are Congruent)

From \( \triangle MNL\cong\triangle OPL\), we get \( \overline{ML}\cong\overline{OL}\) and \( \overline{NL}\cong\overline{PL}\). So, \( \overline{MO}\) and \( \overline{NP}\) bisect each other (by 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.