How would I prove the following (With a formal mathematical proof)?:

Question

If a transversal intersects two parallel lines, then the bisectors of two corresponding angles are parallel.

Answer 1

let AB and CD be parallel lines with EF a transversal cutting AB at P and CD at Q

by properties of parallel lines (PPL)
<EPB = <PQD = a
<BPQ = <DQF = b

let PO be the bisector of <EPB and
let QR be the bisector of <PQD

then <OPB = a/2 and
<RQD = a/2

<FQR = b + a/2
<QPO = b + a/2

therefore <FQR = <QPO (but they are corresponding angles)

so PO || QR