Question

In the parallelogram abcd of the given figure, paq ia an obtuse angle. Two equilateral triangle abp and adq are drawn outside the paralleigram. Prove that cpq ia also an euilateral triabgle

Answer 1

Take △PBC and △CDQ:

PB=AB=CD

BC=AD=QD 

∠QDC=r+60=∠PBC 

Thus, by SAS △PBC ≅△CDQ

So, PC=QC

Now, ∠PAQ=360−∠QAD−∠PAB−∠BAD=360−60−60−(180−r)

⇒∠PAQ=60+r=∠PBC

Also, QA=BC and PA=PB

So, by SAS, △PAQ≅△PBC≅△CDQ

So, PC=QC=PQ

Hence proved.