Question

show that the internal angle bisector of the vertical angle of an isosceles triangle bisects the base at right angle

Answer 1

To prove - AD perpendicularly bisects base BC
Tr. ABC IS ISOSCELES
 SO, AB=AC AND  angle b = angle c

In triangle's ABD AND ACD
AB=AC
Ang BAD = CAD ( AD IS INTERNAL BISECTOR)
AD=AD
Tr's ABD IS CONGRUENT TO AGD ( BY SAS)
 ANG. ADB = ADC (CPCT)    --- 1)
           BD = DC (CPCT)
 ANGLE(A + B + C = 180)
 ANGLE'S( BAD+DAC+ABD+ACD=180)  (A = BAD+DAC; BAD=DAC AD IS                                                                                                       BISECTOR)
 2DAC+2ACD = 180 (ACB = ABC TR. IS ISOSCELES)
 DAC + ACD = 90          - 2)
 
  IN TRIANGLE ADC,
 ANGLE'S(ADC+DAC+ACD) = 180 (ANGLE SUM PROPERTY)
ADC+90=180 (FROM EQ. 2)
 ADC=90=ADB (FROM EQ. 1) 

HENCE, PROVED