Given: , , and .

prove: ∠pcq is complementary to ∠abc.

proof: since , 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.