in the figure, the side BC of triangle ABC is produced to D. the bisector of angle A meets BC at L. prove that angle ABC+ angleACD= 2 angle ALC

The side BC of a triangle ABC is produced to D. The bisector of angle A meets BC in L. Prove that angle ABC + angle ACD = 2 angle ALC.

Triangle ABL.
Let angle BAL = a and angle LBA = b.
Exterior angle ALC = angle BAL + angle LBA
= a + b.

Triangle ABC.
As AL bisects angle BAC, angle LAC = a.
Exterior angle ACD = angle BAC + angle ABC
= 2a + b.

Angle ABC + angle ACD
= b + (2a + b)
= 2(a + b)
= 2 angle ALC 

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