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1.AP⎯⎯⎯⎯⎯≅BQ⎯⎯⎯⎯⎯ (Given)2.AP⎯⎯⎯⎯⎯⊥AB⎯⎯⎯⎯⎯,BQ⎯⎯⎯⎯⎯⊥AB ⎯⎯⎯⎯⎯ (Given)3.m∠A=m∠B=180∘ (Def. of ⊥)4.∠A≅∠B (Supp. ∠s ≅ Thm.)5.∠AOP≅∠BOQ (Adj. ∠s Thm.)6.△OAP≅△OBQ (AAS Steps 4, 5, 1)7.AO⎯⎯⎯⎯⎯≅OP⎯⎯⎯⎯⎯,BO⎯⎯⎯⎯⎯≅OQ⎯⎯⎯⎯⎯ ⎯ (CPCTC)8.AO=OP, BO=OQ (Def. of ≅)9.O is the midpoint of AB⎯⎯⎯⎯⎯ and PQ⎯⎯⎯⎯⎯.(Def. of midpoint)

1.AP⎯⎯⎯⎯⎯≅BQ⎯⎯⎯⎯⎯ (Given)2.AP⎯⎯⎯⎯⎯⊥AB⎯⎯⎯⎯⎯,BQ⎯⎯⎯⎯⎯⊥AB ⎯⎯⎯⎯⎯ (Given)3.m∠A=m∠B=180∘ (Def. of ⊥)4.AO⎯⎯⎯⎯⎯≅OB⎯⎯⎯⎯⎯ (Supp. ∠s ≅ Thm.)5.△AOP≅△BOQ (SAS Steps 1, 5, 4)6.AO⎯⎯⎯⎯⎯≅OP⎯⎯⎯⎯⎯,BO⎯⎯⎯⎯⎯≅OQ⎯⎯⎯⎯⎯ ⎯ (CPCTC)7.AO=OP, BO=OQ (Def. of ≅)8.O is the midpoint of AB⎯⎯⎯⎯⎯ and PQ⎯⎯⎯⎯⎯.(Def. of midpoint)

1.AP⎯⎯⎯⎯⎯≅BQ⎯⎯⎯⎯⎯ (Given)2.AP⎯⎯⎯⎯⎯⊥AB⎯⎯⎯⎯⎯,BQ⎯⎯⎯⎯⎯⊥AB ⎯⎯⎯⎯⎯ (Given)3.m∠A=m∠B=90∘ (Def. of ⊥)4.∠A≅∠B (Rt. ∠s ≅ Thm.)5.∠AOP≅∠BOQ (Vert. ∠s Thm.)6.△OAP≅△OBQ (AAS Steps 4, 5, 1)7.AO⎯⎯⎯⎯⎯≅BO⎯⎯⎯⎯⎯,PO⎯⎯⎯⎯⎯≅OQ⎯⎯⎯⎯⎯ ⎯ (CPCTC)8.AO=BO, PO=OQ (Def. of ≅)9.O is the midpoint of AB⎯⎯⎯⎯⎯ and PQ⎯⎯⎯⎯⎯.(Def. of midpoint)