ABCD is a quadrilateral A and B bisect a and bABCD is a quadrilateral A and B if bisect A and B prove that angle b is equal to half and into angle C + angle d
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BP is the angle bisector of angle ABC,
So angle ABP = angle CBP
DQ is the angle bisector of angle CDA,
So angle CDQ = angle ADQ
In ΔADQ,
ADQ + AQD + DAQ = 180
⇒ 0.5 ADC+ Q + A = 180 -----------------(1)
In ΔCBP
CBP + CPB + BCP = 180
⇒ 0.5 CBA + P + C = 180 ------------------(2)
adding (1) and (2);
0.5 ADC+ Q + A + 0.5 CBA + P + C = 180+180
⇒ C + A + 0.5 ABC + 0.5 ADC + P + Q = 360 ---------------(3)
we know that sum of all angles of a quadrilateral = 360
or A+C+ ABC + ADC = 360 ----------------------(4)
from (3) and (4)
A+C+ ABC + ADC = C + A + 0.5 ABC + 0.5 ADC + P + Q
⇒ ABC + ADC = 0.5 ABC + 0.5 ADC + P + Q
⇒ P + Q = ABC - 0.5 ABC + ADC - 0.5 ADC
⇒ P+Q = 0.5 ABC + 0.5 ADC
⇒ P+Q = 0.5(ABC + ADC)
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