Geometry and Measurement · Shape and Space · Lesson 1
Measuring and Classifying Angles
Angles are everywhere: the corner of your exercise book, the opening of a gate, the hands of a clock and the slope of a roof. In this lesson you will learn to name every kind of angle, measure angles accurately with a protractor, and draw angles of a given size, including the big ones larger than 180°.
By the end you will be able to
- ✔Classify an angle as acute, right, obtuse, straight, reflex or a full turn from its size.
- ✔Measure an angle correctly with a protractor, choosing the right scale.
- ✔Draw angles such as 30°, 45°, 60°, 75°, 90°, 120°, 150°, 270° and 300°.
- ✔Find a reflex angle from the angle measured inside it.
Key words
- Angle
- The amount of turning between two lines (arms) that meet at a point called the vertex. It is measured in degrees (°).
- Acute angle
- An angle bigger than 0° but less than 90°.
- Right angle
- An angle of exactly 90°, like the corner of a page. It is marked with a small square.
- Obtuse angle
- An angle bigger than 90° but less than 180°.
- Reflex angle
- An angle bigger than 180° but less than 360°.
- Protractor
- A flat, usually half-circle, instrument with a scale from 0° to 180° used to measure and draw angles.
Curriculum reference
Content standard: B7.3.1.1 Demonstrate understanding of angles including adjacent, vertically opposite, complementary, supplementary and use them to solve problems.
Indicator: Measure and classify angles according to their measured sizes – right, acute, obtuse and reflex.
Starter
The gate at the school compound
When the school gate in Koforidua is shut, the two leaves lie flat along a straight line. As the watchman Mr Mensah pushes one leaf open, it swings round its hinge. A little way open, the gap is narrow and sharp. Half way, it makes a perfect corner. Opened wide, it nearly lies flat again. Each position is an angle, and mathematicians give every one a name and a number. By the end of this lesson you will be able to read that number with a protractor.
1. What is an angle, and how do we measure it?
An angle is formed when two lines or rays meet at one point. The meeting point is the vertex and the two lines are the arms. The size of an angle tells us how far one arm has turned away from the other. It does not depend on how long the arms are drawn. A tiny angle drawn with long arms is still a tiny angle.
We measure turning in degrees, written with the small circle °. One complete turn, where an arm goes all the way round and comes back to where it started, is 360°. Half a turn is 180° and a quarter of a turn is 90°. The number 360 was chosen long ago because it divides neatly by so many numbers: 2, 3, 4, 5, 6, 8, 9, 10, 12 and more.
| Turn | Fraction of a full turn | Size |
|---|---|---|
| Full turn | 1 | 360° |
| Three-quarter turn | 3/4 | 270° |
| Half turn | 1/2 | 180° |
| Quarter turn | 1/4 | 90° |
| One-third turn | 1/3 | 120° |
▶ Watch the protractor: the arm BC lies along 0°, and arm BA points to 65°. Notice that 65° is less than 90°.
2. The names of angles
We classify an angle by comparing its size with 90°, 180° and 360°. Learn this ladder; you will use it all through junior high school and in the BECE.
| Name | Size | What to picture |
|---|---|---|
| Acute | more than 0° and less than 90° | a sharp corner, like the tip of a pencil |
| Right | exactly 90° | the corner of your book or a door frame |
| Obtuse | more than 90° and less than 180° | a wide-open laptop screen |
| Straight | exactly 180° | a flat, straight line |
| Reflex | more than 180° and less than 360° | the big 'outside' of a corner |
| Full turn (revolution) | exactly 360° | one complete spin |
Notice the word exactly in right, straight and full-turn angles. An angle of 89° is acute, even though it looks almost like a right angle. An angle of 91° is obtuse. We always decide by the number, not by how the angle looks in a quick sketch. The diagrams in your textbook or in an exam are often not drawn to scale, so a number written on the angle is more reliable than your eyes.
▶ This angle is 130°. It is bigger than 90° but smaller than 180°, so it is obtuse.
3. Using a protractor correctly
A protractor has two scales, an inner one and an outer one, running in opposite directions. Many mistakes come from reading the wrong scale. Follow these steps every time and you will always be right.
- 1Place the small hole or cross at the centre of the protractor exactly on the vertex of the angle.
- 2Turn the protractor so that the 0° line (the baseline) lies exactly along one arm of the angle.
- 3Look at the scale that starts with 0° on the arm you chose. Follow that scale round to where the other arm crosses it.
- 4Read the number. If the arms are short, extend them with a ruler first so that the arm reaches the scale.
- 5Check with common sense: if the angle looks acute, your answer must be below 90°; if obtuse, between 90° and 180°.
The check in the last step catches most errors. If you read 115° on an angle that is obviously sharp, you used the wrong scale. The correct reading is 180° − 115° = 65°.
A protractor only goes up to 180°. To measure a reflex angle, measure the smaller angle on the other side and subtract it from 360°. For example, if the inside angle is 80°, the reflex angle on the outside is 360° − 80° = 280°.
▶ A reflex angle of 230°. Notice it is more than a half turn (180°). We can find it as 360° − 130° = 230°.
4. Drawing angles, including reflex angles
To draw an angle of a given size, for example 75°, first draw a straight line with your ruler. Mark the vertex on it. Place the protractor with its centre on the vertex and its baseline on the line. Find 75° on the scale that starts at 0° on your line, make a small dot, remove the protractor and join the vertex to the dot with a ruler. Label the angle and write 75° with a small arc.
To draw a reflex angle such as 300°, remember that it is bigger than 180°, so it cannot be drawn directly with a half-circle protractor. Work out the missing part: 360° − 300° = 60°. Draw an angle of 60°, then mark the large outside region with an arc and write 300°. Likewise 270° is the outside of a 90° angle, because 360° − 270° = 90°.
| Angle to draw | Type | What to draw with the protractor |
|---|---|---|
| 30° | acute | 30° directly |
| 45° | acute | 45° directly |
| 90° | right | 90° directly |
| 120° | obtuse | 120° directly |
| 150° | obtuse | 150° directly |
| 270° | reflex | 90°, then mark the outside |
| 300° | reflex | 60°, then mark the outside |
Clocks give a lovely way to practise. The minute hand moves 360° in 60 minutes, which is 6° every minute. The gap from one hour mark to the next is 360° ÷ 12 = 30°. So at 3 o'clock the hands make 3 × 30° = 90°, a right angle.
Worked examples
Try each one yourself first, then open the solution and compare every step.
Example 1
Classify the angles
Show the solution
- Step 1Compare each size with 90°, 180° and 360°.
- Step 2(a) 47° is less than 90°, so it is acute.
- Step 3(b) 90° is exactly a quarter turn, so it is a right angle.
- Step 4(c) 143° is between 90° and 180°, so it is obtuse.
- Step 5(d) 180° is exactly a half turn, so it is a straight angle.
- Step 6(e) 312° is between 180° and 360°, so it is reflex.
Answer: (a) acute, (b) right, (c) obtuse, (d) straight, (e) reflex
Example 2
Choosing the right scale
▶ Reading from the 0° side gives 40°, an acute angle.
Show the solution
- Step 1The scale that starts at 0° on our arm is the inner scale.
- Step 2Read the inner scale: 40°.
- Step 3Check: the angle looks sharp, and 40° < 90°, which agrees. (140° is the reading from the other scale.)
- Step 440° is less than 90°, so it is an acute angle.
Answer: 40°, acute
Example 3
Finding a reflex angle
Show the solution
- Step 1Full turn = 360°.
- Step 2Reflex angle = 360° − 65°.
- Step 3360° − 65° = 295°.
- Step 4Check: 295° is between 180° and 360° ✓ and 295° + 65° = 360° ✓.
Answer: 295°
Example 4
Angle between the clock hands
Show the solution
- Step 1The gap between two neighbouring numbers is 360° ÷ 12 = 30°.
- Step 2At 5 o'clock the minute hand points to 12 and the hour hand to 5, so the angle is 5 × 30° = 150°.
- Step 3150° is between 90° and 180°, so it is obtuse.
- Step 4The reflex angle is 360° − 150° = 210°.
Answer: 150° (obtuse); the reflex angle is 210°
Example 5
Planning to draw a 300° angle
▶ The 300° angle is the large outside of a 60° angle.
Show the solution
- Step 1300° is bigger than 180°, so it is a reflex angle and cannot be read directly.
- Step 2Find the part left over from a full turn: 360° − 300° = 60°.
- Step 3Kofi draws a 60° angle with his protractor.
- Step 4Then he marks the large region outside the 60° angle with an arc and writes 300° on it.
- Step 5Check: 60° + 300° = 360° ✓.
Answer: He measures 60° and marks the outside of it as 300°.
Summary
- An angle measures turning; a full turn is 360°, a half turn 180°, a quarter turn 90°.
- Acute: below 90°. Right: exactly 90°. Obtuse: 90° to 180°. Straight: exactly 180°. Reflex: 180° to 360°.
- On a protractor, read the scale that starts at 0° on the arm lying along the baseline, and check with common sense.
- A reflex angle = 360° − the angle measured inside it.
- To draw a reflex angle such as 300° or 270°, draw 360° minus that size (60° or 90°) and mark the outside.
Practice: 15 questions
5 easy, 5 medium, 5 hard. Each one is marked straight away, with a full worked solution.
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