Naming an angle with three letters
An angle can be named by three points. Here the arms are BA and BC, and they meet at B, so B is the vertex. The angle at B is written ∠ABC and read "angle ABC".
The vertex letter always goes in the middle. ∠CBA names the same angle, because B is still in the middle. ∠BAC puts A in the middle, so it names an angle at A, which is a different corner.
The arms BA and BC meet at the vertex B: this is ∠ABC.
Measuring with a protractor
A protractor is a half circle marked in degrees, from 0 to 180. To measure an angle, put the center point of the protractor exactly on the vertex. Then turn the protractor so that its zero line lies along one arm, called the base arm.
Now follow the other arm out to the scale and read the number where it crosses. Here it crosses at 50, so the angle is 50°.
The center is on the vertex and the zero is along the base arm. The other arm crosses the scale at 50: the angle is 50°.
Read the scale that starts at the base arm
A protractor has two scales. The outer one counts up from 0 on the right, and the inner one counts up from 0 on the left. So every arm crosses two numbers, and the two numbers always add up to 180. This arm crosses at 130 and at 50.
Read the scale whose 0 sits on the base arm. Here the base arm runs to the right, and the scale that starts at 0 on the right reads 130 at the other arm. The angle is 130°.
Check with your eyes. This angle has turned past a right angle, so it is obtuse and must be more than 90°. 130° fits, and 50° cannot be right.
The arm crosses 130 on one scale and 50 on the other. Counting from the base arm’s zero gives 130°.
Drawing an angle
To draw an angle of 60°, first draw a straight line with a ruler for the base arm, and mark the vertex at one end. Put the center of the protractor on the vertex, with the zero line along the base arm.
Find 60 on the scale that starts at 0 on the base arm, and make a small dot beside it. Take the protractor away and join the vertex to the dot with the ruler. The two arms now make an angle of 60°.
The usual mistake is to mark 60 on the other scale. That dot makes an angle of 180° − 60° = 120°, which is obtuse, not the acute angle asked for.
Arm A crosses 60 on the scale that starts at the base arm, so it makes 60°. Arm B crosses 60 on the other scale, and it makes 120°.
Worked example: A Turn Made in Two Parts
Question A toy robot turns on the spot. (a) The robot has to make a quarter turn, which is a right angle. It turns 25° and stops. How many more degrees must it turn? (b) Next the robot has to make a half turn, so that it faces the opposite way along a straight line. It turns 130° and stops. How many more degrees must it turn?
1.A quarter turn is a right angle, and a right angle is 90°.
A quarter turn is a right angle, and a right angle is 90°. 2.(a) The robot has turned 25° of the 90°. It must turn 90° − 25° = 65° more. Check: 25° + 65° = 90°.
(a) 90° − 25° = 65° more. 3.A half turn makes a straight line, and the angle on a straight line is 180°.
A half turn makes a straight line, and the angle on a straight line is 180°. 4.(b) The robot has turned 130° of the 180°. It must turn 180° − 130° = 50° more. Check: 130° + 50° = 180°.
(b) 180° − 130° = 50° more.
Answer: (a) 65°; (b) 50°
Common mistakes
- Subtracting from 100° instead of 90° in part (a), which gives 75°. A right angle is 90°, not 100°.
- Using 90° again in part (b). A half turn is two right angles, which is 180°, so the 130° is taken away from 180°.
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