Measuring & drawing angles
🎯 In this topic you will
- Measure angles in degrees.
- Draw angles accurately.
🧠 Key Words
- protractor
Show Definitions
- protractor: A mathematical instrument used to measure and draw angles in degrees.
Measuring and Drawing Angles
When you measure and draw straight lines, you need to use a ruler. When you measure and draw angles, you need to use a protractor.
Understanding a Protractor
A protractor is a flat circular object that has 0° to 360° marked around its edge. It is usually made of plastic so you can see through it. When you measure clockwise from 0° you use the numbers on the outside circle. When you measure anticlockwise from 0° you use the numbers on the inside circle.
❓ EXERCISES
1. Measure the size of each of these acute angles.
👀 Show answer
$a \approx 20^\circ$
$b \approx 55^\circ$
$c \approx 75^\circ$
2. Measure the size of each of these obtuse angles.
👀 Show answer
$d \approx 110^\circ$
$e \approx 130^\circ$
$f \approx 165^\circ$
3. Measure the size of each of these reflex angles.
👀 Show answer
$g \approx 220^\circ$
$h \approx 260^\circ$
$i \approx 300^\circ$
🧠 Think like a Mathematician
Work on your own.
The diagrams show two angles $x$ and $y$.
Task: Find as many different ways as you can to use your protractor to measure the angles.
Method:
- Place the centre of the protractor at the vertex of the angle.
- Align one arm of the angle with $0^\circ$ on the protractor scale.
- Read the number where the other arm crosses the scale.
- Try measuring the angle in different ways, such as using the inside scale, the outside scale, or measuring the smaller angle and calculating the larger one.
Follow-up Questions:
Show Answers
- 1: You can measure the angle using either the inside or outside scale depending on the direction the angle opens.
- 2: Measure the smaller angle first and subtract it from $360^\circ$ to find the reflex angle.
- 3: Aligning the baseline with $0^\circ$ ensures the measurement starts from the correct reference point, giving an accurate angle.
💡 Quick Math Tip
Straight Line Shortcut: If two angles lie on a straight line, you only need to measure one of them because the two angles always add up to $180^\circ$. Subtract the measured angle from $180^\circ$ to find the other.
❓ EXERCISES
4.
a. Draw angles of the following sizes.
i. $30^\circ$ ii. $145^\circ$ iii. $245^\circ$ iv. $350^\circ$
b. In your book, write down three different angles of your choice between $0^\circ$ and $360^\circ$. On a piece of paper accurately draw these angles, but do not write on them the sizes of the angles.
c. Swap your piece of paper with a partner. Measure the angles that they have drawn. Check your answers with their answers. Did you measure each other’s angles correctly? Discuss any mistakes that were made.
👀 Show answer
a. Accurate drawings should show angles of $30^\circ$, $145^\circ$, $245^\circ$ and $350^\circ$.
b. Answers will vary. Any three different angles between $0^\circ$ and $360^\circ$ are acceptable if they are drawn accurately.
c. Answers will vary. Learners should measure the drawn angles correctly and compare with the original values.
5. The diagram shows angles $x$ and $y$.
a. Is Sofia correct? Explain your answer.
b. Write down the sizes of angles $x$ and $y$. Explain the methods you used to find them.
👀 Show answer
a. Yes. Sofia is correct because angles $x$ and $y$ are on a straight line, so they add up to $180^\circ$. If you know one angle, you can find the other by subtraction.
b. Measuring the diagram gives approximately $x = 60^\circ$ and $y = 120^\circ$. One method is to measure $x$ directly with a protractor, then calculate $y = 180^\circ - 60^\circ = 120^\circ$. You could also measure $y$ first and then find $x$ the same way.
6.
a. Measure the angles $v$ and $w$ in this diagram.
b. Explain the calculation you can do to check that your answers to part $a$ are correct.
👀 Show answer
a. Measuring the diagram gives approximately $w = 125^\circ$ and $v = 235^\circ$.
b. Check by adding the two angles around the point: $v + w = 235^\circ + 125^\circ = 360^\circ$.
7. An architect is designing a building. The diagram shows a wheelchair ramp for the building. For a wheelchair ramp to be allowed, the angle of the ramp, $r^\circ$, must be no more than $20^\circ$.
a. Is this wheelchair ramp allowed? Explain your answer.
b. The best wheelchair ramps have an angle of between $7^\circ$ and $15^\circ$. Draw an example of one of these ramps. Make sure you write the angle you have used on your diagram.
👀 Show answer
a. Measuring the diagram gives approximately $r = 25^\circ$, so the ramp is not allowed because $25^\circ > 20^\circ$.
b. Answers will vary. Any accurately drawn ramp with an angle between $7^\circ$ and $15^\circ$ is acceptable, for example $10^\circ$.
8. The diagram shows angles $x$, $y$ and $z$ on a straight line.
a. Measure and write down the sizes of angles $x$, $y$ and $z$.
b. Show how to check your answers to part $a$ are correct.
c. The diagram shows a triangle. Measure and write down the sizes of angles $x$, $y$ and $z$.
d. What do you notice about your answers to parts $a$ and $c$? Discuss and compare what you noticed with other learners in your class.
👀 Show answer
a. Measuring the straight-line diagram gives approximately $x = 45^\circ$, $y = 75^\circ$ and $z = 60^\circ$.
b. Check by adding the three angles on the straight line: $x + y + z = 45^\circ + 75^\circ + 60^\circ = 180^\circ$.
c. Measuring the triangle gives approximately $x = 45^\circ$, $y = 75^\circ$ and $z = 60^\circ$.
d. The answers are the same in both diagrams. In each case, the three angles add up to $180^\circ$.