MP ≅ ∆ONQ (SAS Criterion). MN and PQ Bisect Each Other at O.

To prove that ∆MP and ∆ONQ are congruent, we can use the Side-Angle-Side (SAS) congruence criterion.

Given: MP = NQ

We need to prove: 1) ∆MP ≅ ∆ONQ 2) MN and PQ bisect each other at O

Proof: 1) ∆MP ≅ ∆ONQ (using SAS congruence criterion)

• Side MP = Side NQ (given)
• Angle MPN = Angle QNO (corresponding angles)
• Side MN = Side OQ (corresponding sides)

2) MN and PQ bisect each other at O

• Since ∆MP ≅ ∆ONQ, we know that corresponding sides and angles are congruent.
• Therefore, MN = OQ and PQ = NO.
• This shows that MN and PQ bisect each other at O.

Thus, we have proven that ∆MP and ∆ONQ are congruent, and MN and PQ bisect each other at O.