Question

a transversal ef of line ab and line cd intersects thr lines 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 parallel to line cd

Answer 1

line ab and line cd are parallel by corrwsponding angles

Answer 2

Answer:

line ab || line cd

Step-by-step explanation:

let angle BPR = angle QPR = x       ( ray PR bisects BPQ) [1]

angle CQS = angle PQS = y      ( ray QS bisects PQC) [2]

∴angle BPQ = angle BPR + angle QPR    (angle addition)

∴angle BPQ = x + x   (from 1)

∴angle BPQ = 2x        [3]

∴angle PQC = angle PQS + angle CQS   (angle addition)

∴angle PQC = y + y     (from 2)

∴angle PQC = 2y        [4]

ray PR || Ray QS   ( || means 'is parallel to')

∴on transversal PQ

∴angle QPR = angle PQS  (alt. angles theorem)

∴ x = y   (from 1 and 2)

∴ 2x = 2y    (multiplying both by 2)

∴angle BPQ = angle PQC  (from 3 and 4)

∴ line AB || line CD  (by alt. angles test)