Question

AP and BP are the bisetors of two adjacent angles A and B of quadrilateral ABCD. prove that 2 angleAPB =angleC+angleD

Answer 1

In tri ABP
angle PAB + angle PBA + angle APB = 180
1/2 angle A + 1/2 angle B + angle P = 180
Angle P = 180 - angle A - angle B .... 1

In quad ABCD
angle A + angle B + angle C + angle D = 360
1/2 angle A + 1/2angle B + 1/2angle C + 1/2angle D = 180 (dividing by 2 both sides)
1/2 (angle C + angle D) = 180 - angle A -angle B
Angle C + angle D = 2 * angle APB (from 1)
Hence proved
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