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

Prove: AP = CQ

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