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Question

Parallelogram upper M upper N upper O upper P is shown with base upper O upper P larger than height upper N upper O. Diagonals are drawn from upper M to upper O and upper N to upper P to meet at upper L.

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)
Arrange responses in the correct order to answer the question. Select a response, navigate to the desired position and insert response at that position. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can be moved using the up and down arrow keys or by dragging with a mouse.
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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.
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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 MN¯¯¯¯¯¯¯¯¯¯∥PO¯¯¯¯¯¯¯¯ and MP¯¯¯¯¯¯¯¯¯∥NO¯¯¯¯¯¯¯¯.
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Because of the definition of a parallelogram, I know that opposite sides are congruent. I can say that MN¯¯¯¯¯¯¯¯¯¯≅OP¯¯¯¯¯¯¯¯. I now have enough information to say that △MNL≅△OPL by the ASA Congruence Theorem.
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When I think of the diagonals as transversals cutting parallel lines, I can see that ∠MOP≅∠OMN and ∠NPO≅∠PNM because of the Alternate Interior Angles Theorem.

Answer 1

To arrange Olivia's paragraph proof in the correct order, we can follow the logical steps of proving that the diagonals of parallelogram MNOP bisect each other. Here is the correct sequence:

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 MN || PO and MP || NO.

2. When I think of the diagonals as transversals cutting parallel lines, I can see that ∠MOP ≅ ∠OMN and ∠NPO ≅ ∠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 MN ≅ OP. I now have enough information to say that △MNL ≅ △OPL by the ASA Congruence Theorem.

4. 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.

This order logically follows the progression of the proof, starting from the definition of the parallelogram, using properties of the angles and sides, establishing congruent triangles, and concluding with the fact that the diagonals bisect each other.