Which statement expresses the Congruent Supplements Theorem?

Which statement expresses the Congruent Supplements Theorem?

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

Answer: (C)

The idea tested is how supplementary angles relate when they share a common angle. If two angles are each supplementary to the same angle, or to congruent angles, they must have the same measure and thus be congruent. This is exactly what the statement says: supplements of the same angle, or congruent angles, are congruent. Why this works: if two angles B and C are both supplementary to a given angle A, then their measures are m(B) = 180° − m(A) and m(C) = 180° − m(A). Those are equal, so B and C are congruent. The same logic applies if B and C are supplements to angles that are congruent to each other. Why the other ideas don’t fit: vertical angles being congruent is a different result about angles formed by intersecting lines; the statement about complements concerns angles whose measures add to 90°, not 180°; and two angles could be supplementary without being equal unless both are 90°, so “two angles that are supplementary are equal” isn’t true in general. So the correct notion is that supplements of the same angle (or of congruent angles) are congruent.

The idea tested is how supplementary angles relate when they share a common angle. If two angles are each supplementary to the same angle, or to congruent angles, they must have the same measure and thus be congruent. This is exactly what the statement says: supplements of the same angle, or congruent angles, are congruent.

Why this works: if two angles B and C are both supplementary to a given angle A, then their measures are m(B) = 180° − m(A) and m(C) = 180° − m(A). Those are equal, so B and C are congruent. The same logic applies if B and C are supplements to angles that are congruent to each other.

Why the other ideas don’t fit: vertical angles being congruent is a different result about angles formed by intersecting lines; the statement about complements concerns angles whose measures add to 90°, not 180°; and two angles could be supplementary without being equal unless both are 90°, so “two angles that are supplementary are equal” isn’t true in general.

So the correct notion is that supplements of the same angle (or of congruent angles) are congruent.