Answer:
Given: Two line segments AB and CD bisect each other at point O.
To prove: ∆ AC D≅ ∆ B DC
Construction:Join AC, AD ,BC and B D.Quadrilateral ABCD is formed.
Proof: In Quadrilateral ACBD,
Point of intersection of AB and CD is O.
also, A O=O B and CO=OD
As AB and CD are diagonals of quadrilateral A BC D, and they bisect each other .
∴Δ A O C≅ Δ B OD→ [SA S]∴{∠AOC=∠BOD,AO=OB,CO=OD}
Similarly, Δ A O D≅ Δ B O C→ [SA S]∴{∠AOD=∠BOC ,AO=OB,CO=OD}
AC=BD,∠ACD=∠BDC,→ [ CPCT ]
AD=BC,∠ADC=∠B CD→ [ C P CT ]
But ∠A CD and ∠B DC, ∠ADC and ∠B CD are alternate angles of Quadrilateral AC B D.
⇒AC║B D and AD║BC [If alternate angles are equal then lines are parallel]
∴ Quadrilateral A C B D is a parallelogram.→If in a quadrilateral opposite sides are equal and parallel it is a parallelogram]
In Δ A CD and Δ B DC
AC = B D [proved above]
AD= B C [proved above]
C D=C D [common]
∆A CD ≅ ∆B DC [ SSS] S→Side
Hence proved.
∆A CD ≅ ∆B DC [ Diagonal of a parallelogram divides it into two congruent triangles]
