Question

in the adjoining figure two lines AB and CD intersect each other at the point O such that BC||DA and BC=DA

Answer 1

Heya Dear ☺

For providing O is the middle point we need to Prove ∆COB & ∆AOD must be congurent

Let's start

In ∆COB & ∆AOD

<COD= <AOD (VERTICALLY OPPOSITE BECAUSE BC//DA)
AD=BC (GIVEN)
<OAD<OBC (ALTERNATE INTERIOR ANGLES BECAUSE BC//DA)

Hence , ∆COB congurent to ∆AOD by ASA congurent condition

Now. by cpct , AO=OB
OD=OC
If this would be possible then it means. O is the mid point

Hence , proved

Hope it helps you ☺

Answer 2

Given,
AD || BC
AD = BC

Since, AD || BC and AB is their transversal,
m<BAD = m<ABC ..........( 1 )

Similarly,
m<CDA = m<DCB ...........( 2 )

In triangle AOD and triangle BOD,
m<BAD = m<ABC.......( from equation 1 )
AD = BC...........…..........( Given )
m<CDA = m<DCB.......( from equation 2 )
So, by ASA Congruency Theorem,
triangle AOD is congruent to triangle BOD
Therefore,
AO = OB ............( By CPCT )........ ( 3 )
DO = OC ............( By CPCT )....... ( 4 )

From equation 3 and 4,
We can say that,
O is the midpoint of both the line segments AB and CD

Hence Proved.