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

Prove: AP = CQ

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