given: parallelogram mvop with diagonals mo and vp. prove: mo and vp bisect each other. statements: parallelogram mvop with diagonals mo and vp, mr ≅ or, pr ≅ vr, mo and vp bisect each other, mv ≅ op, mv ∥ op, pm ≅ no, pm ∥ no, ∠rop ≅ ∠rmn, ∠rpo ≅ ∠rnm, ∠rpm ≅ ∠rno, △rpm ≅ △rno, △rop ≅ △rmn, ∠rmp ≅ ∠ron. reasons: given, definition of parallelogram, alternate interior angles formed by parallel lines are congruent, opposite sides of a parallelogram are congruent, asa, corresponding parts of congruent triangles are congruent, definition of bisector

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Accepted Answer

# Answer:
| Statements | Reasons |
|--|--|
| Parallelogram $MNOP$ with diagonals $\overline{MO}$ and $\overline{NP}$ | Given |
| $\overline{MN}\parallel\overline{OP}$ | Definition of parallelogram |
| $\angle ROP\cong\angle RMN$, $\angle RPO\cong\angle RNM$ | Alternate - interior angles formed by parallel lines are congruent |
| $\overline{MN}\cong\overline{OP}$ | Opposite sides of a parallelogram are congruent |
| $\triangle ROP\cong\triangle RMN$ (ASA) | ASA (Angle - Side - Angle) |
| $\overline{MR}\cong\overline{OR}$, $\overline{PR}\cong\overline{NR}$ | Corresponding parts of congruent triangles are congruent |
| $\overline{MO}$ and $\overline{NP}$ bisect each other | Definition of bisector |

# Explanation:
## Step1: State the given
We are given parallelogram $MNOP$ with diagonals $\overline{MO}$ and $\overline{NP}$.
## Step2: Use parallelogram properties
By the definition of a parallelogram, $\overline{MN}\parallel\overline{OP}$.
## Step3: Identify congruent angles
Since $\overline{MN}\parallel\overline{OP}$, the alternate - interior angles $\angle ROP\cong\angle RMN$ and $\angle RPO\cong\angle RNM$.
## Step4: Recall side - congruence
In a parallelogram, opposite sides are congruent, so $\overline{MN}\cong\overline{OP}$.
## Step5: Prove triangle congruence
Using the ASA (Angle - Side - Angle) postulate, $\triangle ROP\cong\triangle RMN$.
## Step6: Get congruent segments
Corresponding parts of congruent triangles are congruent, so $\overline{MR}\cong\overline{OR}$ and $\overline{PR}\cong\overline{NR}$.
## Step7: Conclude bisection
By the definition of a bisector, $\overline{MO}$ and $\overline{NP}$ bisect each other.

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QUESTION IMAGE

Question

Explanation:

Step1: State the given

Step2: Use parallelogram properties

Step3: Identify congruent angles

Step4: Recall side - congruence

Step5: Prove triangle congruence

Step6: Get congruent segments

Step7: Conclude bisection

Answer:

Statements	Reasons
\(\overline{MN}\parallel\overline{OP}\)	Definition of parallelogram
\(\angle ROP\cong\angle RMN\), \(\angle RPO\cong\angle RNM\)	Alternate - interior angles formed by parallel lines are congruent
\(\overline{MN}\cong\overline{OP}\)	Opposite sides of a parallelogram are congruent
\(\triangle ROP\cong\triangle RMN\) (ASA)	ASA (Angle - Side - Angle)
\(\overline{MR}\cong\overline{OR}\), \(\overline{PR}\cong\overline{NR}\)	Corresponding parts of congruent triangles are congruent
\(\overline{MO}\) and \(\overline{NP}\) bisect each other	Definition of bisector