Given: In circle O, arc AB = arc CD, OP is perpendicular to AB, and OQ is perpendicular to CD

Prove: AP = CQ

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