Given that AP¯¯¯¯¯⊥AB¯¯¯¯¯, BQ¯¯¯¯¯⊥AB¯¯¯¯¯, and AP¯¯¯¯¯≅BQ¯¯¯¯¯, which of the following proves that AO¯¯¯¯¯≅BO¯¯¯¯¯ and PO¯¯¯¯¯≅OQ¯¯¯¯¯? A.
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)

B.
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)

C.
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)

D. 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)