Special pairs of angles

Two angles standing alone is one thing. Two angles standing together is when geometry becomes really interesting. Some pairs of angles add up to special totals, and learning to spot them makes many problems easy.

Concept

Two angles are called complementary if their measures add to $90°$. For example, $30°$ and $60°$ are complementary, and so are $25°$ and $65°$. We say "$\angle A$ is the complement of $\angle B$" when $\angle A + \angle B = 90°$.

Two angles are called supplementary if their measures add to $180°$. For example, $40°$ and $140°$ are supplementary, and so are $90°$ and $90°$. Supplementary angles fit together to make a straight line.

There is a neat memory trick: Complementary corner = Corner ($90°$). Supplementary straight = Straight ($180°$).

When two lines (or two rays starting at the same point) cross, four angles are formed around the meeting point. Three useful facts follow:

1. Angles on a straight line sum to $180°$. If a ray stands on a straight line, the two angles it forms on either side add to a straight angle.

2. Angles at a point sum to $360°$. All the angles around a single vertex, taken together, fill one full turn.

3. Vertically opposite angles are equal. When two straight lines cross, the angles directly across from each other are the same size. So if one of the four angles is $50°$, the angle opposite it is also $50°$, and the other two are each $130°$.

These three rules are like little detective clues. Combined with what we know about complementary and supplementary angles, they let us find missing angles without measuring.

A subtle but important point: an adjacent pair of angles shares an arm and a vertex but does not overlap. Two adjacent angles whose non-shared arms together form a straight line are called a linear pair, and any linear pair is supplementary.

Worked examples

Example 1. Find the complement of $35°$.

• Complement $= 90° - 35° = 55°$.

Example 2. Find the supplement of $112°$.

• Supplement $= 180° - 112° = 68°$.

Example 3. Two lines cross. One of the four angles is $73°$. Find the other three.

• Opposite to $73°$: same as $73°$ (vertically opposite).
• The other two: each is $180° - 73° = 107°$ (linear pair with $73°$).
• So the four angles are $73°, 107°, 73°, 107°$.

Example 4. Three angles around a point are $80°$, $145°$, and the third. Find the third.

• Angles at a point sum to $360°$.
• Third $= 360° - 80° - 145° = 135°$.

Example 5. Two complementary angles are in the ratio $2 : 3$. Find them.

• Let the angles be $2x$ and $3x$. Then $2x + 3x = 90°$, so $5x = 90°$, $x = 18°$.
• Angles: $2x = 36°$ and $3x = 54°$.

Try it yourself

1. Find the complement of $48°$.
2. Find the supplement of $25°$.
3. Two supplementary angles are equal. What is each?
4. Two lines cross. One angle is $40°$. Find the other three.
5. Three angles around a point are $120°$, $100°$ and what?
6. Find two complementary angles in the ratio $1 : 4$.
7. Can an obtuse angle have a complement? Why or why not?
8. Are two right angles supplementary? Are they complementary?

Activity

Cross-paper puzzle. Take a square piece of paper. With a ruler, draw two straight lines crossing at the centre, making any angles you like. Use your protractor to measure all four angles formed. Verify two things: (a) opposite angles are equal, and (b) the four angles add up to exactly $360°$. Try a few different angles and see that these two facts always hold.