Explain why triangle pqs is isosceles

Answer    

By using the exterior angle property

The measure of exterior angle is equal to the sum of two opposite interior angles.  

In ∠QSR is exterior angle.

∠SPQ and ∠SQP are the opposite  interior angle of the exterior angle ∠QSR .

Thus

∠QSR  =  ∠SPQ + ∠SQP

As given in the diagram

∠QSR  =  2x  , ∠SQP = x

Put in the above

2x = x + ∠SQP

∠SQP = 2x - x

∠SQP = x

Thus

∠SPQ =  ∠SQP = x

By using the property

if two angles are equal in a triangle than their opposite sides are also equal .

Thus

PS = SQ

Two sides of triangle are equal .

therefore Δ PQS is a isosceles triangle .