Complementary, supplementary and vertical angles

Cross two straight sticks and you get four angles but only two different sizes. The opposite angles always match.

Adjacent angles

Two angles are adjacent when they share a vertex and one side and do not overlap, like two slices of a pie cut next to each other. Adjacent angles are where angle arithmetic starts, because their sizes simply add: if ∠ABD = 30° and ∠DBC = 25° sit side by side, the whole angle ∠ABC is 55°.

Pairs that add to 90° and to 180°

Two angles are complementary if their measures add up to 90°, and supplementary if they add up to 180°. The angles do not have to be next to each other; the words describe the sum, not the position. Each angle of a complementary pair is called the complement of the other: the complement of 35° is 90 − 35 = 55°. In the same way the supplement of 120° is 180 − 120 = 60°.

When two adjacent angles together fill a straight line, they are called a linear pair, and a linear pair is always supplementary. OpenStax uses this to find an unknown angle: if one angle on a straight line is 35°, the angle next to it must be 180 − 35 = 145°. Only acute angles have complements, since an angle of 90° or more leaves nothing to add. Every angle smaller than 180° has a supplement.

Vertical angles

Where two straight lines cross, four angles meet at the point. Call them a, b, c and d going round. The angles opposite each other, a and c, are vertical angles, and so are b and d. They are always equal, and the reason takes two lines. Angles a and b sit along one straight line, so a + b = 180°. Angles b and c sit along the other line, so b + c = 180°. Both sums equal 180° and both contain b, so a must equal c.

Solving for an unknown

The pair rules turn into equations. Suppose two complementary angles measure 2x + 10 and x + 20 degrees. Their sum is 90, so 3x + 30 = 90 and x = 20. The angles are 50° and 40°, which do add to 90°. Always put the value of x back in and check the sum; it catches sign slips straight away.

These pairs come back in every later unit. In a right triangle the two smaller angles always add to 90°, so they are complementary, and that fact is behind the cofunction rule of trigonometry. Notice too that the supplement of an acute angle is always obtuse and the supplement of an obtuse angle is always acute, while a right angle is its own supplement.