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

Prove: AP = CQ

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