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

Hint – Draw the figure using the information given in the question and then using the properties of the congruent triangle, prove that the lines AB and CD bisect at O.
Complete step-by-step answer:
Given in the question-
Two lines AB and CD intersect at O such that BC is equal and parallel to AD.
This implies, AD = BC and AD || BC.
To prove – AB and CD bisect at O.
Refer to the figure given below-

First, we have to prove that triangle AOD and BOC are congruent-
$\angle OCB = \angle ODA$ [AD||BC and CD is transversal]
$AD = BC$ [given in question]
$\angle OBC = \angle OAD$ [AD || BC and AB is transversal]
By ASA (Angle Side Angle) criterion, we can say-
$\vartriangle AOD \cong \vartriangle BOC$
Now, we can also say, OA = OB and OD = OC (By Corresponding Parts of the Congruent Triangle).
Therefore, AB and CD bisect each other at O {Since, OA = OB and OD = OC} [Hence Proved]
Note – Whenever such types of questions appear then write the information given in the question. Then prove the triangles AOD and BOC are congruent by using the given conditions in the question. After proving the two triangles congruent, we can write OA = OB and OD = OC, which proves that AB and CD bisect each other at O.