Question

In triangle abc if d is a point on ac such that ad=cd=bd,then prove that triangle abc is a right angled triangle

Answer 1

in triangle  abd , ad = bd 
∴∠bac = ∠abd
similarly in triangle cbd
∠bca = ∠cbd
now , ∠abc = ∠abd + ∠cbd
∴∠abc = ∠bca + ∠bac
now by a,s,p of a triangle,
∠abc + ∠bca + ∠bac = 180
∴∠abc + ∠abc = 180 (as ∠abc = ∠bac + ∠bca)
∴2 * ∠abc = 180
∠abc = 90
therefore abc is a right angled triangle right angled at ∠b
hence proved

Answer 2

Answer:

Step-by-step explanation:

in triangle abd , ad = bd

∴∠bac = ∠abd

similarly in triangle cbd

∠bca = ∠cbd

now , ∠abc = ∠abd + ∠cbd

∴∠abc = ∠bca + ∠bac

now by a,s,p of a triangle,

∠abc + ∠bca + ∠bac = 180

∴∠abc + ∠abc = 180 (as ∠abc = ∠bac + ∠bca)

∴2 * ∠abc = 180

∠abc = 90

therefore abc is a right angled triangle right angled at ∠b

hence proved