Question

abc is a triangle the bisector of the angle bca meets bca at x

Answer 1

Answer:

The question is incomplete. The original Question is:

In Δ ABC, the bisector of angle BCA meets AB in X. A point Y lies on CX such that AX=AY. Prove that ∠ CAY= ∠ ABC.

Proof:

Refer to the attachment for diagram.

Consider Δ ABC. Here, CX is acting as the bisector. Due to this,

=> ∠ ACY = ∠ BCX   ...( 1 )

Also it is given that, AX = AY. This implies,

=> ∠ AYX = ∠ AXY  ( Angles opposite to equal sides are equal )   ...( 2 )

We know that, Angle made by a straight line is always 180°.

=> Angle made by CX = AB = 180°

CX = ∠ AYX + ∠ AYC

AB = ∠ AXY + ∠ BXY

Now equating them we get,

=> ∠ AYX + ∠ AYC = ∠ AXY + ∠ BXY

We know that, ∠ AYX = ∠ AXY ( From ( 2 ) ). Hence they get cancelled.

=> ∠ AYC = ∠ BXC   ...( 3 )

Now consider Δ AYC and Δ BXC,

Angle sum property of two triangles would be equal to 180°. Hence we can write,

=> ∠ AYC + ∠ ACY + ∠ CAY = ∠ BXC + ∠ BCX + ∠ XBC

Using ( 1 ) and ( 3 ), terms get cancelled. Hence we get,

=> ∠ CAY = ∠ XBC

∠ XBC can be also written as ∠ ABC.

Hence ∠ CAY = ∠ ABC

Hence Proved !