Question

prove: ∠pcq is complementary to ∠abc. what is the missing step in the given proof? a. ∠pcq and ∠acp are supplementary by the linear pair theorem. b. for parallel lines cut by a transversal, corresponding angles are congruent, so ∠acb≅∠pcq. c. ∠ocp≅∠bcd by the vertical angles theorem. d. for parallel lines cut by a transversal, corresponding angles are congruent, so ∠ocp≅∠abc. e. for parallel lines cut by a transversal, corresponding angles are congruent, so ∠oca≅∠cbd. proof: since ac⊥cd, m∠ocq = 90° by the definition of perpendicular lines. by angle - addition, we can say m∠ocq = m∠ocp + m∠pcq. but since m∠ocq = 90°, m∠ocp + m∠pcq = 90° by the transitive property of equality. missing step by the definition of congruent angles, m∠ocp = m∠abc. this leads to ∠abc + m∠pcq = 90° by the transitive property of equality. so, based on the definition of complementary angles, ∠pcq is complementary to ∠abc.

Explanation:

Step1: Analyze the proof - goal

We want to show ∠PCQ is complementary to ∠ABC. We know ∠OCQ = 90° and ∠OCQ=∠OCP + ∠PCQ. We need to relate ∠OCP and ∠ABC.

Step2: Recall angle - congruence rules

For parallel lines cut by a transversal, corresponding angles are congruent. Looking at the figure and the proof steps, we need to find the correct pair of corresponding angles that will help us relate ∠OCP and ∠ABC.

Step3: Evaluate options

Option D states that for parallel lines cut by a transversal, corresponding angles are congruent, so ∠OCP≅∠ABC. This is the missing step as it allows us to substitute ∠OCP with ∠ABC in the angle - addition and complementary - angle relationships.

Answer:

D. For parallel lines cut by a transversal, corresponding angles are congruent, so ∠OCP≅∠ABC.