olivia wrote a paragraph proof that the diagonals of parallelogram mnop bisect each other drag and drop the statement and reasons of her proofs to put them in the correct order starting with the first step at the top

Question

I am given that quadrilateral MNOPMNOPMNOP is a parallelogram. I know that opposite sides of a parallelogram are congruent because of the definition of a parallelogram, so MN¯¯¯¯¯¯¯¯¯¯∥PO¯¯¯¯¯¯¯¯MN¯¯¯¯¯¯¯¯¯¯∥PO¯¯¯¯¯¯¯¯MN¯∥PO¯ and MP¯¯¯¯¯¯¯¯¯∥NO¯¯¯¯¯¯¯¯MP¯¯¯¯¯¯¯¯¯∥NO¯¯¯¯¯¯¯¯MP¯∥NO¯.

Because of the definition of a parallelogram, I know that opposite sides are congruent. I can say that MN¯¯¯¯¯¯¯¯¯¯≅OP¯¯¯¯¯¯¯¯MN¯¯¯¯¯¯¯¯¯¯≅OP¯¯¯¯¯¯¯¯MN¯≅OP¯. I now have enough information to say that △MNL≅△OPL△MNL≅△OPL△MNL≅△OPL by the ASA Congruence Theorem.

When I think of the diagonals as transversals cutting parallel lines, I can see that ∠MOP≅∠OMN∠MOP≅∠OMN∠MOP≅∠OMN and ∠NPO≅∠PNM∠NPO≅∠PNM∠NPO≅∠PNM because of the Alternate Interior Angles Theorem.

By the CPCTC Theorem, I know that ML¯¯¯¯¯¯¯¯¯≅OL¯¯¯¯¯¯¯ML¯¯¯¯¯¯¯¯¯≅OL¯¯¯¯¯¯¯ML¯≅OL¯ and NL¯¯¯¯¯¯¯¯≅PL¯¯¯¯¯¯¯NL¯¯¯¯¯¯¯¯≅PL¯¯¯¯¯¯¯NL¯≅PL¯. MO¯¯¯¯¯¯¯¯¯MO¯¯¯¯¯¯¯¯¯MO¯ and NP¯¯¯¯¯¯¯¯NP¯¯¯¯¯¯¯¯NP¯ bisect each other because this is the definition of a bisector.

Answer 1

Here is the correct order for Olivia's paragraph proof that the diagonals of parallelogram MNOP bisect each other, along with the corresponding reasons:

1. Statement: Quadrilateral MNOP is a parallelogram.
   Reason: Given.

2. Statement: Opposite sides of a parallelogram are congruent, so \( MN \parallel PO \) and \( MP \parallel NO \).
   Reason: Definition of a parallelogram.

3. Statement: Since \( MN \parallel PO \) and \( MP \parallel NO \), we can say that \( MN \cong OP \).
   Reason: Opposite sides of a parallelogram are congruent.

4. Statement: By the ASA Congruence Theorem, \( \triangle MNL \cong \triangle OPL \).
   Reason: If two pairs of sides and the included angle are congruent, the triangles are congruent.

5. Statement: By Alternate Interior Angles Theorem, \( \angle MOP \cong \angle OMN \) and \( \angle NPO \cong \angle PNM \).
   Reason: Transversal cutting parallel lines.

6. Statement: By CPCTC, \( ML \cong OL \) and \( NL \cong PL \).
   Reason: Corresponding Parts of Congruent Triangles are Congruent.

7. Statement: Therefore, \( MO \) and \( NP \) bisect each other.
   Reason: Definition of a bisector.

This logical sequence establishes that the diagonals of the parallelogram bisect each other effectively.