We have to prove Δ ABO ≅ Δ CDO or, Δ DAO ≅ Δ BCO.
Let us assume that ABCD is a parallelogram having diagonals AC and BD.
We have to prove that in a parallelogram the diagonals bisect each other.
Assume that the diagonals of ABCD i.e. AC and BD intersect at point O.
Therefore, to prove that the diagonals AC and BD bisect each other, we have to first prove that Δ ABO and Δ CDO are congruent or Δ DAO and Δ BCO are congruent.
In symbol, we have to prove Δ ABO ≅ Δ CDO or, Δ DAO ≅ Δ BCO. (Answer)
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
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
To prove that,diagonals bisect each other of a Parallelogram,we need to prove
D) A O≅OD