Question

in a given figure ab and ac are the bisectors of angle A and angle C respectively if angle abc is equal to 90 degree find angle ADC +angle AEC

Answer 1

in triangle ABC, angle ABC = 90
so, angle BAC + angle BCA = 90 (angle sum property)

Now, AD is the bisector of angle BAC, so angle OAC = 1/2 angle BAC
and CE is the bisector of angle BCA, so angle OCA = 1/2 angle BCA

In triangle AOC,
angle AOC + angle OAC + angle OCA = 180
or angle AOC + 1/2 angle BAC + 1/2 angle BCA = 180
or angle AOC + 1/2 (angle BAC + angle BCA) = 180
or angle AOC + 1/2 * 90 = 180
or angle AOC + 45 = 180
or angle AOC = 180 - 45 = 135