Question

Segments AB and CD bisect each other at point O. Prove that ∆ACD is congruent to ∆BDC.

Answer 1

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]