Question

Given: AC||BD , AB||CD, and AC _|_ CD

Prove: ∠PCQ is complementary to ∠ABC.

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 m∠ABC + m∠PCQ = 90° by the Transitive Property of Equality. So, based on the definition of complementary angles, ∠PCQ is complementary to ∠ABC. What is the missing step in the given proof?

A. ∠PQC 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.

Answer 1

answer is D

Explanation:

m∠OCP + m∠PCQ = 90° by the transitive property of equality to the definition of congruent angles, m∠OCP= m∠ABC, letter D which states that For parallel lines cut by a transversal, corresponding angles are congruent, so ∠OCP ≅ ∠ABC is the only statements that fits to what we need to show.

Answer 2

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

Explanation:

Since we need to show the connection of the proof from m∠OCP + m∠PCQ = 90° by the transitive property of equality to the definition of congruent angles, m∠OCP= m∠ABC, letter D which states that For parallel lines cut by a transversal, corresponding angles are congruent, so ∠OCP ≅ ∠ABC is the only statements that fits to what we need to show.