3. Complementary and Supplementary Angles

Two special pairs come up constantly because they let you find a missing angle by subtraction.

The two relationships are
\[ m\angle A+m\angle B=90^\circ \qquad\text{or}\qquad m\angle A+m\angle B=180^\circ. \]

• Complementary angles add up to \(90^\circ\).
• Supplementary angles add up to \(180^\circ\).

To find the missing angle, subtract from 90 or 180.

Worked example (complementary): if one angle is \(35^\circ\), its complement is \(90 - 35 = 55^\circ\).

Worked example (supplementary): if one angle is \(110^\circ\), its supplement is \(180 - 110 = 70^\circ\).

Case study - a board cut to lean against a wall: the cut angle and the angle it makes with the floor are often complementary, adding to \(90^\circ\).

Common mistake: mixing up the two. Complementary "C" goes with the corner (\(90^\circ\)); Supplementary "S" goes with the straight line (\(180^\circ\)).

Turn the relationship into an equation

Let (x) represent the missing angle. For complements, write (x + a = 90). For supplements, write (x + a = 180). Subtracting (a) from both sides gives the missing measure. Writing the equation prevents you from subtracting from the wrong total.

Worked example: one angle is (3x) and its complement is (x + 10). Because complementary angles total (90^°),
\[ 3x + (x+10)=90,\qquad 4x=80,\qquad x=20. \]
The actual angles are (60^°) and (30^°). Check: (60+30=90).

Important idea: complementary and supplementary angles do not have to touch. These words describe their sum. A diagram may show separated angles and still state that they are complementary or supplementary.