Question

؈ABCD is a parallelogram. Prove that ΔABD and ΔBDC are similar.

Answer 1

For showing similarity we have to give at-least 3 conditions.

So, we will use opposite angle and parallel sides.

Answer 2

Given ABCD is a parallelogram
To prove : - Δ ABD ∼ Δ BDC
Proof :- For the correspondence ABD ↔ CDB between ΔABD and Δ BDC
∠ BAD ≅ ∠ DCB (opposite angles of parallelogram)
∠ABD ≅ ∠CDB (alternate angles )
∠ADB ≅ ∠CDB (alternate angles )
AB = CD (opposite sides of a  parallelogram ABCD )
∴ AB/CD = 1 ......(i)
Also AD = CB (opposite sides of a parallelogram ABCD)
∴ AD/CB  = 1.......(ii)
And BD = DB (common)
∴ BD/DB = 1.......(iii)
from equations (i), (ii) and (iii)
AB/CD = AD/BC = BD/DB = 1
Hence, for the correspondence, ABD ↔ CDB between ΔABD and ΔBDC
The corresponding angles are congruent and the lengths of its corresponding sides are in proportion.
∴for the correspondence ABD ↔ CDB, ΔABD and ΔBDC are similar.