Which statement expresses the Congruent Complements Theorem?

• A. Complements of the same angle, or congruent angles, are congruent.
• B. Congruent supplements are always vertical.
• C. Vertical angles are congruent.
• D. If two angles are supplementary, their measures are equal.

Answer: (A)

The statement captured by the Congruent Complements Theorem is about what happens when two angles are both linked to the same angle by a complementary relationship. Remember, two angles are complementary if their measures add up to 90 degrees. The theorem says: if two angles each complement the same angle, or each complement angles that are themselves congruent, then those two angles are congruent.

Why this works is through simple arithmetic with angles. If angle A and angle B both complement angle C, then A + C = 90 and B + C = 90. Subtract C from both equations to get A = B, so the two angles are congruent. If instead they complement angles that are congruent to each other, say C ≅ D, then A + C = 90 and B + D = 90. Since C ≅ D, subtract to conclude A ≅ B as well. Either way, the two angles must have equal measures.

The other statements describe different ideas. Vertical angles are congruent in a separate result about angles formed by intersecting lines. The claim about congruent supplements being always vertical mixes separate truths and isn’t a correct general rule. And saying that supplementary angles have equal measures is false in general because two angles can sum to 180 without being equal (they’re only equal if both are 90 degrees).