(In the text below, some words are omitted. These have been underlined. Click just above the line. See what happens.)

Angles BAP and CAQ are and hence equal. Triangles BAP and CAQ are with equal base angles at A. The other pair of the angles are also equal. I.e., ABP = ACQ. These two angles are internal to the lines BP and CQ and transversal BC. Since the internal angles are equal, the lines are parallel: PB || QC.