Given: In circle O, arc AB = arc CD, OP ⟂ AB, and OQ ⟂ CD

Prove: AP = CQ

Algebra
AB = AP + PB and CD = CQ + QD
Definition
OP ⟂ AB and OQ ⟂ CD
Substitution
AP + PB = CQ + QD
AP + AP = CQ + CQ
Equal arcs have equal chords
Substitution
If a diameter is perpendicular to a chord, it bisects the chord
AP = CQ
OP bisects AB and OQ bisects CD
AP = PB and CQ = QD
AB = CD
Given
Segment addition theorem