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CBSE Class 8 Answered

ABCD is a parallelogram. Side ap bisects angle A and side CQ bisects angle C. P lies on Side CD and Q lies on Side AB. show that-(1)AP||CQ (2)AQCP is a parallelogram. 
Asked by aj072263 | 28 Dec, 2018, 12:04: AM
answered-by-expert Expert Answer
Given :- ABCD is a parallelogram. AP bisects begin mathsize 12px style angle end styleA and CQ bisects begin mathsize 12px style angle end styleC
 
In the given parallelogram, begin mathsize 12px style angle end styleDAB = begin mathsize 12px style angle end styleDCB. let each of angle measures α
begin mathsize 12px style angle end styleB + begin mathsize 12px style angle end styleDAB = begin mathsize 12px style angle end styleB + α = 180 ...............(1)
 
since AP and CQ are angle bisectors, we have begin mathsize 12px style angle end styleDAP = begin mathsize 12px style angle end stylePAQ = begin mathsize 12px style angle end stylePCQ = begin mathsize 12px style angle end styleBCQ = α/2
 
begin mathsize 12px style angle end styleAQC = begin mathsize 12px style angle end styleB+begin mathsize 12px style angle end styleBCQ ( exterior angle equals sum of interior angles in ΔQBC
hence begin mathsize 12px style angle end styleAQC + begin mathsize 12px style angle end stylePAQ = begin mathsize 12px style angle end styleB+begin mathsize 12px style angle end styleBCQ+begin mathsize 12px style angle end stylePAQ = begin mathsize 12px style angle end styleB + α = 180° ( using eqn.(1)
Sum of interior angles is 180°, hence AP || QC
 
also we have PC || AQ,  hence AQCP is parallelogram
Answered by Thiyagarajan K | 28 Dec, 2018, 10:38: AM

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