Angle Relationships

A handful of angle pairs come up constantly.

• Complementary: two angles adding to 90°.
• Supplementary: two angles adding to 180°.
• Linear pair: two adjacent angles forming a straight line. Always supplementary.
• Vertical angles: the opposite pairs formed when two lines cross. Always congruent.

Angles also add like segments do. If a ray sits inside an angle, the two smaller angles sum to the whole one. An angle bisector splits an angle into two congruent halves.

In problems these relationships are what supply the equation. Vertical angles give you "set them equal." A linear pair gives you "set the sum to 180."

The Angle Addition Postulate states that if D lies in the interior of ∠ABC, then m∠ABD + m∠DBC = m∠ABC.

The Linear Pair Postulate states that two angles forming a linear pair are supplementary. The Vertical Angles Theorem — that vertical angles are congruent — is then a theorem rather than a postulate, proved in one line: both angles are supplementary to the same angle, and supplements of the same angle are congruent.

That proof pattern, the Congruent Supplements Theorem, generalises: two angles supplementary to the same angle (or to congruent angles) are congruent, and likewise for complements. It is often the missing step in a two-column proof.

Precision of notation matters in proofs. ∠ABC names an angle, an object, while m∠ABC names its measure, a number. Angles are congruent; measures are equal. Writing ∠A = ∠B where ∠A ≅ ∠B is meant is the single most common notation error in the course.