Which statement expresses the Vertical Angles Theorem?

• A. Vertical angles are congruent.
• B. Vertical angles are supplementary.
• C. Vertical angles share a vertex.
• D. Vertical angles lie on opposite rays.

Answer: (A)

When two straight lines cross, the angles that are opposite each other—the vertical angles—have equal measures. This happens because the two hands of the intersection create pairs of opposite rays, so the rotation from one ray to the other in one angle is mirrored in the opposite angle. If one of these angles has measure x, its adjacent angle on the same line is 180 − x, and the angle opposite that adjacent angle is also x, giving congruent opposite angles. That is exactly what Vertical Angles Theorem states: vertical angles are congruent. The other statements don’t capture this defining property: vertical angles aren’t described as generally supplementary (that can only happen in the special case of perpendicular lines), though they do share a vertex, and saying they lie on opposite rays is not the right way to describe the relationship between the two angles.