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

Angle PAQ + angle DAP + Angle BAD + Angle BAQ = 360 degrees
Angle PAQ + 60 degrees + (180 degrees - Angle CDA)  + 60 degrees = 360 Degrees
Angle PAQ = 360 Degrees - 60 degrees (180 Degrees - Angle CDA) - 60 Degrees
Angle PAQ = 360 Degrees - 60 Degrees - 180 Degrees + Angle CDA
Angle PAQ = 60 Degrees + Angle CDA
Angle CDA + Angle ADP = Angle CDP
Angle CDA +60 Degrees = Angle CDP

Angle PAQ = Angle CDA

In triangle CDP and triangle AQP

AQ = CD (AQ = AB = CD)

AP = DP (sides  of equailateral triangle)
Angle PAQ = Angle CDA

Triangle CDP = Triangle AQP

Triangle PQ = CP 

CQ = CP = PQ

Triangle C'PQ is an equailtaral triangle.