Question

Two lines AB and CD intersect at O such that BC is equal and parallel to AD. Prove that the lines AB and CD bisect at O.

Answer 1

Proved.

•GIVEN: AD = BC

AD || BC

•TO PROVE: AB Bisects CD at O

i.e. DO =CO

•PROOF: In triangle AOD and triangle

BOC

• AD=BC (GIVEN)

• <OAD=<OCB ( Alternate

• <ODA =<OBC interior angles) •Triangle AOD is congruent to

triangle BOC

• OD = OC (CPCT)

• AB bisects CD at O

hence proved

Answer 2

Lines AB and CD bisect at O. (Proved)

Step-by-step explanation:

See the attached figure.

Now, BC = AD and BC ║ AD.

So, between the triangles Δ COB and Δ AOD,

(i) ∠ AOD = ∠ COB {Vertically opposite angles}

(ii) ∠ ADO = ∠ OCB and

(iii) AD = BC

{Since BC ║ AD and line CD is transverse line and ∠ ADO, and ∠ OCB are alternate angles}

Therefore, by Angle-Angle-Side i.e. AAS criteria Δ COB ≅ Δ AOD.

Hence, AO = OB and CO = OD {Corresponding sides}

Therefore, lines AB and CD bisect at O. (Proved)