In Grade 4 you learned that angles that fit together add. Some pairs of angles come up so often that they have names. Knowing them lets you find a missing angle without a protractor.
Key idea
Complementary angles add to $90^\circ$ (they fill a right angle). Supplementary angles add to $180^\circ$ (they make a straight line). Vertical angles, across from each other where two lines cross, are equal. Angles all the way around a point add to $360^\circ$.
Why it works
Why are vertical angles equal? Call the angles around a crossing $\angle 1$, $\angle 2$, $\angle 3$ and $\angle 4$, in order. $\angle 1$ and $\angle 2$ make a straight line, so $\angle 1 + \angle 2 = 180^\circ$. So do $\angle 2$ and $\angle 3$: $\angle 2 + \angle 3 = 180^\circ$. Both $\angle 1$ and $\angle 3$ are "$180^\circ$ minus $\angle 2$," so they are equal.
Worked examples
Worked example: A complement
An angle measures $35^\circ$. What is its complement?
- Complementary angles add to $90^\circ$.
- $90 - 35 = 55$.
Answer: $55^\circ$
Worked example: A supplement
An angle measures $112^\circ$. What is its supplement?
- Supplementary angles add to $180^\circ$.
- $180 - 112 = 68$.
Answer: $68^\circ$
Worked example: Crossing lines
Two lines cross, and one angle is $48^\circ$. Find the other three angles.
- The angle across from it is a vertical angle: $48^\circ$.
- The angles next to it make a straight line with it: $180 - 48 = 132$, so both are $132^\circ$.
- Check: $48 + 132 + 48 + 132 = 360$ ✓.
Answer: $48^\circ$, $132^\circ$ and $132^\circ$
Common mistake
Mixing up complementary ($90^\circ$) and supplementary ($180^\circ$). One way to remember: Complementary makes a Corner, and Supplementary makes a Straight line. Also, C comes before S, and 90 comes before 180.