Question

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

Answer 1

In ΔABC, CX is the angle bisector of ∠C

∠ACY = ∠BCX --- (1)

and AX = AY

So in Δ AXY

∠AXY = ∠AYX these are the angles which are opposite to equal sides -- (2)

Now ∠XYC = ∠AXB = 180° this is a straight line

∠AYX + ∠AYC = ∠AXY + ∠BXY

∠AYC + ∠BXY this is taken from 2 and now this will become the equation 3

Also in Δ AYC and ΔBXC

∠AYC + ∠YCA + ∠CAY = ∠BXC + ∠BCX + ∠XBC = 180°

∠CAY = ∠XBC this is taken from 1 and 3

∠CAY = ∠ABC

Answer 2

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