Question

10. Prove that diagonals
of a parallelogram
bisect each other.
(Hint. Prove that
AOB and COD are
congruent by ASA)
Fig. 12.35​

Answer 1

Question :- The Diagonals of Parallelogram bisect each other.

Solution:-

Given :- ABCD is a parallelogram with AC and BD diagonals & O is the point of intersection of AC & BD.

To prove :- OA = OC & OB = OD

To Proof:-

We know that Opposite sides of parallelogram are parallel

So, AD || BC (With Transversal BD)

∠ODA = ∠OBC (Alternate Interior Angles) -------------eq(1)

Same as

AD ||BC ( With Transversal AC)

∠OAD = OCB ( Alternate interior angles)--------eq(2)

Now, In ∆AOD & ∆BOC

• ∠OAD = ∠OBC ---( From EQ(2)
• AD = CB ---(opposite sides of parallelogram are equal)
• ∠ODA = ∠OBC ---(From eq(1)

• ∆AOD ≅ ∆BOC ---( By ASA (Angle Side Angle Congruency Rule)

So, OA = OC & OB = OD ---(by CPCT)

Hence Proved