Given that abcd is a parallelogram, what must be proven to prove that the diagonals bisect each other?

a) δacd≅δabd
b) ab||cd
c) ad≅cb
d) ao≅od

explain why and choose from the awnser choices given

Answer with explanation:

To prove that , the diagonals of Parallelogram ,A B CD,Bisect each other

That is, 1. A O=OD

   2. BO=O C

We need to prove ,that either of two triangles

1. ΔA OB ≅ Δ DOC

or

2.  ΔA O C ≅ Δ DOB

We will prove ,ΔA OB ≅ Δ DOC , in the following way.

1.∠A OB ≅ ∠ DOC→→→[Vertically Opposite angles]

2. AB=CD →→→[Opposite sides of parallelogram]

3. ∠BAD=∠C DA→[As, AB║CD,so Alternate interior angles are equal.]

⇒ΔA OB ≅ Δ DOC→→→ [A A S]

So, A O=OD→→[C PCT]

and, CO=OD→→[C PCT]

Similarly,we can prove that, ΔA O C ≅ Δ DOB,and get

   A O=OD

and, CO=OD

To prove that,diagonals bisect each other of a Parallelogram,we need to prove

D)  A O≅OD