the side BC of triangle abc is produced such that D is on the ray BC the bisector of /_A - Maths - Triangles

Question

the side BC of triangle abc is produced such that D is on the
ray BC the bisector of /_A meets BC IN L prove that angle abc+angle
acd =2angle ALC

Shivam Mishra · Accepted Answer

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since AL is the angle bisector of ∠A
let ∠BAC=2x
therefore ∠BAL=∠LAC=x
∠ACD=∠ABC+BAC [exterior angle is equal to the sum of
opposite interior angles]
∠ACD=∠ABC+2x (1)
therefore
∠ABC+∠ACD=∠ABC+∠ABC+2x
=2∠ABC+2x
=2(∠ABC+x) (2)
∠ALC=∠ABC+∠BAL=∠ABC+x (3) [since ABL is a
triangle and ALC is an exterior angle,exterior angle is equal to
the sum of opposite interior angles]
by (2) and (3)
∠ABC+∠ACD=2∠ALC

the side BC of triangle abc is produced such that D is on the ray BC the bisector of /_A meets BC IN L prove that angle abc+angle acd =2angle ALC