1. Describing Angles
Lesson 7.2.1

2. Describing Angles 7.2.1 California Standards: What it means for you:
Lesson
7.2.1
Describing Angles
California Standards:
Measurement and Geometry 2.1
Identify angles as vertical, adjacent, complementary, or supplementary and provide descriptions of these terms.
Measurement and Geometry 2.2
Use the properties of complementary and supplementary angles and the sum of the angles of a triangle to solve problems involving an unknown angle.
What it means for you:
You’ll learn about angles – some names for different types, how to measure them, and how to draw them.
Key words:
angle
acute
right angle
obtuse
protractor
ray
vertex

3. Lesson
7.2.1
Describing Angles
Whenever two straight lines meet at a point, there is an angle between them.
This Lesson is about describing angles by naming the different types and by measuring them.

4. Describing Angles 7.2.1 Angles Are Measured in Degrees
Lesson
7.2.1
Describing Angles
Angles Are Measured in Degrees
You can measure how large the angle between two straight lines is.
Line 2
Line 1
The angle is measured at the point where the lines meet.
The units used to measure an angle are called degrees.
The symbol ° means degrees.

5. Lesson
7.2.1
Describing Angles
A right angle measures exactly 90° and looks like the corner of a square or rectangle.
An angle measuring between 0° and 90° is called an acute angle.

6. Lesson
7.2.1
Describing Angles
An angle measuring between 90° and 180° is called an obtuse angle.
So this obtuse angle is approximately:
90°
+ 45°
≈ 135°
~45°
90°
Because you know exactly what a 90° right angle looks like, you can use it to estimate the measure of other angles.

7. Describing Angles 7.2.1 Solution
Lesson
7.2.1
Describing Angles
Example 1
Estimate the sizes of angles A and B shown on the right.
Are these angles acute, obtuse, or right angles?
Solution
The small squares on the picture show where there are right angles, which you know are 90°.
You can use this to estimate angles A and B.
It looks like angle A is about one-third as big as the right angle shown on the left.
So 90 ÷ 3 = 30° might be a good estimate for angle A.
Solution continues…
Solution follows…

8. Describing Angles 7.2.1 Solution (continued)
Lesson
7.2.1
Describing Angles
Example 1
Estimate the sizes of angles A and B shown on the right.
Are these angles acute, obtuse, or right angles?
Solution (continued)
Angle B looks like it is about half the other right angle shown.
So a good estimate of angle B might be 90 ÷ 2 = 45°.
Both angles are less than 90°, so they are both acute.

9. Describing Angles 7.2.1 Guided Practice
Lesson
7.2.1
Describing Angles
Guided Practice
Use the diagram to answer Exercises 1–4:
1. Which of the angles shown are acute and which are obtuse?
2. Estimate the size of angle 1.
3. Estimate the size of angle 3.
4. Estimate the size of angle 4.
Angles 1–4 are less than a right angle, so they are acute. Angle 5 is more than a right angle but less than one half-turn, so it is obtuse.
Angle 1 is a little more than half a right angle, approximately 60°
Angle 3 is less than half a right angle, approximately 20°
Angle 4 is about half a right angle, approximately 40°
Solution follows…

10. Describing Angles 7.2.1 Guided Practice
Lesson
7.2.1
Describing Angles
Guided Practice
Use the diagram to answer Exercises 5–6:
5. Esteban estimates that the size of angle 2 is about 45°. Is this a reasonable estimate? Give a reason for your answer.
No. Angle 2 is more than half a right angle, so the estimate is too small.
6. Abigail estimates that the size of angle 5 is about 175°. Is this a reasonable estimate? Give a reason for your answer.
No. 175° is almost 2 right angles together; angle 5 is clearly smaller than this.
Solution follows…

11. Describing Angles 7.2.1 A Protractor Can Be Used to Measure Angles
Lesson
7.2.1
Describing Angles
A Protractor Can Be Used to Measure Angles
A protractor is a tool you can use to measure the size of an angle.
The protractor has 180 small marks around its edge, each representing an angle of 1°.

12. Lesson
7.2.1
Describing Angles
Example 2
Use a protractor to measure the size of this angle.
Solution
Place the bottom line on the protractor directly over one of the rays of the angle.
The point marking the center should be above the vertex of the angle.
You read the angle at the point where the second ray crosses the curved edge of the protractor.
This angle measures 40°.
Solution follows…

13. Describing Angles 7.2.1 Guided Practice
Lesson
7.2.1
Describing Angles
Guided Practice
Use the diagram to answer Exercises 7–11. Use a protractor to measure the angles.
7. Angle 1
8. Angle 2
9. Angle 3
10. Angle 4
11. Angle 5
17°
100°
24°
65°
35°
Solution follows…

14. Describing Angles 7.2.1 Guided Practice
Lesson
7.2.1
Describing Angles
Guided Practice
Use the diagram to answer Exercises 12–13.
12. Michael used a protractor to measure angle 6. His answer was 139°. What did he do wrong? What is the real size of angle 6?
13. Work out the measure of angle 7 from the values of angles 1–6, then check your answer with a protractor.
He read from the wrong set of numbers on the protractor. Real size: 41°
The sum of angles 1–6 is:
17° + 100° + 24° + 65° + 35° + 41° = 282°
There are 360° in a full turn, so angle 7 is: 360° – 282° = 78°
Solution follows…

15. Describing Angles 7.2.1 A Protractor Can Also Be Used to Draw Angles
Lesson
7.2.1
Describing Angles
A Protractor Can Also Be Used to Draw Angles
So protractors can be used to measure the size of angles. But that’s not all.
You can also use a protractor for drawing angles of a given size.

16. Describing Angles 7.2.1 Draw an angle that measures 54°. Solution
Lesson
7.2.1
Describing Angles
Example 3
Draw an angle that measures 54°.
Solution
Start by drawing a straight line. This will be one of the rays that form the angle.
54°
Line it up with the bottom line on the protractor, with one end of the ray at the center mark of the protractor.
Follow the scale around the protractor until you get to 54°. Make a mark at the 54° point.
Finally, remove the protractor and join the mark up with your original ray.
Solution follows…

17. Describing Angles 7.2.1 Guided Practice
Lesson
7.2.1
Describing Angles
Guided Practice
Use a protractor to draw the following angles.
14. 80° ° °
° ° °
Solution follows…

18. Describing Angles 7.2.1 Independent Practice
Lesson
7.2.1
Describing Angles
Independent Practice
Exercises 1–6 are about hands on a clock. For each time, say whether the angle made by the clock hands will be acute, obtuse, or right.
1. 3: : :15
4. 9: : :55
right
acute
acute
obtuse
acute
obtuse
7. When cutting paper with a pair of scissors, is the angle between the blades of the scissors usually acute, obtuse, or right? Explain why.
Acute. Scissors can open 90° or more, but they don’t have to open very wide to cut paper.
Solution follows…

19. Describing Angles 7.2.1 Independent Practice
Lesson
7.2.1
Describing Angles
Independent Practice
8. Without measuring, match each of these angles with its measure. Choose from 15°, 80°, 90°, 105°, or 168°.
80°
105°
90°
168°
15°
Solution follows…

20. Describing Angles 7.2.1 Independent Practice
Lesson
7.2.1
Describing Angles
Independent Practice
In Exercises 9–10, use the protractor to determine the measure of each angle. Say whether each angle is acute, right, or obtuse.
10. 9.
90° – right
118° – obtuse
Solution follows…

21. Describing Angles 7.2.1 Independent Practice
Lesson
7.2.1
Describing Angles
Independent Practice
In Exercises 11–13, use the protractor to determine the measure of each angle. Say whether each angle is acute, right, or obtuse.
11.
55° – acute
12.
13.
178° – obtuse
32° – acute
Solution follows…

22. Describing Angles 7.2.1 Round Up
Lesson
7.2.1
Describing Angles
Round Up
Angles are useful for describing the different shapes you’ll learn about later in this Chapter.
You should be able to picture how big an angle is when you are told its size in degrees.