Question

E
P
A
B.
R
. A transversal EF of line AB and
line CD intersects the lines at point P
and Q respectively. Ray PR and ray QS
are parallel and bisectors of ZBPQ and
ZPQC respectively.
Prove that line AB | line CD.
SS
>
C С
Q
D​

Answer 1

Given:-

A tr ansversal EF of line AB and line CD in tersects the lines at point P and Q respectively. Ray PR and ray QS are parallel and bisectors of ZBPQ and ZPQC respectively. Prove that line AB | line CD.

SolutioN:-

Given: Ray PR || ray QS  Ray PR and ray QS are the bisectors of ∠BPQ and ∠PQC respectively.  To prove: line AB || line CD Proof:  Ray PR || ray QS and seg PQ is their transversal. ∠RPQ = ∠SQP ….(i) [Alternate angles]  ∠RPQ = (1/2) ∠BPQ …. (ii) [Ray PR bisects ∠BPQ]  ∠SQP = (1/2) ∠PQC [Ray QS bisects ∠PQC]  ∴ (1/2) ∠BPQ = (1/2) ∠PQC ∴ ∠BPQ = ∠PQC  But, ∠BPQ and ∠PQC are alternate angles on lines AB and CD when line EF is the transversal.  ∴ line AB || line CD [Alternate angles test]...