Math Core

If you cut a pizza slice into two smaller slices, the two small slices together are the same as the whole slice. Angles work the same way. That simple fact lets you find angle measures without a protractor.

Splitting an angle

The ray $BD$ below splits $\angle ABC$ into two smaller angles that don't overlap: $\angle CBD$ measures $30^\circ$ and $\angle DBA$ measures $50^\circ$ .

Ray BD splits angle ABC into a 30° angle and a 50° angle.

The whole angle is made of the $30$ one-degree angles in the first part plus the $50$ one-degree angles in the second part. So

30^\circ + 50^\circ = 80^\circ.

$\angle ABC$ measures $80^\circ$ .

Angle measures add

When an angle is split into parts that don't overlap, the measure of the whole angle is the sum of the measures of the parts.

\text{part} + \text{part} = \text{whole}

Worked example: Find the whole

An angle is split into two parts that measure $25^\circ$ and $40^\circ$ . What does the whole angle measure?

Add the parts: $25 + 40 = 65$ . The whole angle measures $65^\circ$ .

Finding a missing part

If you know the whole and one part, subtract to find the other part. You can write an equation with a $?$ for the unknown angle.

Worked example: Inside a right angle

A right angle is split into two parts. One part is $35^\circ$ . What is the other part?

A right angle split into a 35° part and an unknown part.

A right angle is $90^\circ$ , so $35 + {?} = 90$ .

Subtract: $90 - 35 = 55$ . The missing part is $55^\circ$ .

Check: $35 + 55 = 90$ . ✓

Worked example: On a straight line

A ray comes out of a point on a straight line. It makes a $125^\circ$ angle on one side. What is the angle on the other side?

A straight angle split into a 125° part and an unknown part.

The two angles together make a straight angle, $180^\circ$ . So $125 + {?} = 180$ .

Subtract: $180 - 125 = 55$ . The other angle is $55^\circ$ .

Tip

Remember three "wholes" you'll use often: a right angle is $90^\circ$ , a straight line is $180^\circ$ , and a full circle is $360^\circ$ .

Worked example: Three equal parts

Three equal angles fit together exactly to make a straight angle. What does each angle measure?

Together they make $180^\circ$ . Split into 3 equal parts: $180 \div 3 = 60$ . Each angle measures $60^\circ$ .

Common mistake

Before you add or subtract, check what the whole is. If the picture shows a square corner, the whole is $90^\circ$ . If the sides make a straight line, the whole is $180^\circ$ . Using the wrong whole gives the wrong answer.

Practice

Practice 1

What does the whole angle measure, in degrees?

Type a number.

Practice 2

A right angle is split into two parts. One part is $62^\circ$ . What is the other part, in degrees?

Type a number.

Practice 3

Two angles sit side by side on a straight line. One is $110^\circ$ . What is the other, in degrees?

Type a number.

Practice 4

A sprinkler turns $65^\circ$ . Then it turns another $48^\circ$ in the same direction. How many degrees has it turned in all?

Type a number.

Practice 5

Two equal angles fit together exactly to make a right angle. What does each angle measure, in degrees?

Type a number.

Practice 6

A $130^\circ$ angle is split into two parts. One part is $45^\circ$ . Which equation can you use to find the other part?

Practice 7

A $150^\circ$ angle is split into three parts. Two of the parts measure $40^\circ$ and $55^\circ$ . What does the third part measure, in degrees?

Type a number.

Practice 8

Four angles fit around a point to make a full circle. Three of them measure $95^\circ$ , $110^\circ$ and $70^\circ$ . What does the fourth angle measure, in degrees?

Type a number.