Question

A transversal EF of line AB and line CD intersects the line at point P and Q respectively. Ray PR and ray QS are parallel and bisectors of angle BPQ and angle PQC respectively.
Prove that line AB ll line CD.

Answer 1

as angle SPQ =angle QPR (alternate angles)
and also ray SQ and rayRP bisect angle CQP and angleBPQ
So 2×angle SQP=2×angleRPQ so angle CQP=angle BPQ then CD//AB.

Answer 2

angle SQP = angle QPR (alternate angles)

Ray SQ and Ray RP bisect angle CQP and angle BPQ (given)

2×angle CQR = 2×angle BPQ

angle CQR = angle BPQ

Ifa pair of alternate angles formed by a transversal of two lines is congruent then the two lines are parallel (by alternate angle test)

therefore, line AB is parallel to line CD