Question

In parallelogram ∠A is an obtuse angle. Two equilateral triangles ABP and ADQ are drawn which are exterior to the parallelogram. Prove that ΔCPQ is an equilateral triangle

Answer 1

In trianglePBC and triangleCDQ
PB = AB
AB = CD 
PB = CD
Also
BC = AD
AD = QD
BC = QD
angle QDC = angle ABC + 60 degree
trianglePBC congruent to triangleCDQ (SAS congruence rule)
PC = QC (CPCT)

 anglePAQ = 360 - angleQAD - anglePAB - angleBAD
                  = 360 - 60 - 60 - (180 - angleABC)
anglePAQ = 60 + angleABC
                 = anglePBC
Also QA = BC and PA = PB
trianglePAQ congruent to trianglePBC (SAS congruence rule)
Also congruent to triangleCDQ (SAS congruence rule)
 so, PC = QC = PQ (CPCT).
This implies triangleCPQ is equilateral as all sides are equal.