Solve the following problem. Write the complete proof in your paper homework and for online (only) complete the probing statement (if any) that is a part of your proof or related to it.

Given: AB ∥ CD and BC ∥ AD BD ∩ AC = O, O ∈ MN, M ∈ BC, N∈ AD Prove: OM = ON

OM = ON by using concurrency of Δs BOM and DON

Step-by-step explanation:

In the figure ABCD:

∵ AB / / CD

∵ CB / / AD

∵ In any quadrilateral If every two sides are parallel then

it will be a parallelogram

∴ ABCD is a parallelogram

∴ AC and BD bisects each other at O ⇒ (properties of parallelogram)

∴ OD = OB ⇒ (1)

∵ BC / / AD

∴ m∠CBD = m∠ADB ⇒ alternate angles

∵ M ∈ BC, N ∈ AD, O ∈ BD

∴ m∠MBO = m∠NDO ⇒ (2)

∵ BD intersects MN at O

∴ m∠MOB = m∠NOD ⇒ (3) (vertically opposite angles)

From (1), (2) and (3)

∴ ΔBOM ≅ ΔDON

∴ OM = ON