Question

In triangle ABC,angleB=2(angleC).AD is the angle bisector such that DC=AB.then the measure of angleA is

Answer 1

see diagram.    I hope we can apply Sine and Cos trigonometric ratios here.

we are given DC = AB.       angle B = 2C.    AD is bisector of angle A.

AB / Sin C  = AC / Sin B             Sine of angles rule in ΔABC
                 = AC / Sin 2C

=>  AC / AB = 2 Sin C Cos C/ Sin C  = 2 Cos C

But Cos C = CF / CD = CD / AB      in triangle CFD.
 =>  AF = CF        =>  AF = CF = AC / 2

now DF is the perpendicular bisector for the side AC.

    =>  angle ADF = 3C/2 = angle FDC = 90 -C

     =>  C = 36 deg.    B = 72 deb.  A = 72 deg.