Lines and Angles | IT

Question 16

The circle has been divided into 1, 2, 3, 4, 5, 6, 8, 9 10 and 12 parts below. What are the degree measures of the resulting angles? Write the degree measures down near the indicated angles.

Solution

A full circle always measures 360 degrees.

Step 1 — One equal part

The circle is divided into 1 part. This means we consider the whole circle. We divide the total angle by 1. Look at the first circle. It shows the full rotation.

$\text{Angle} = \frac{360 \text{ degrees}}{1}$

$= 360 \text{ degrees}$

$\boxed{360 \text{ degrees}}$

Diagram 1

Step 2 — Two equal parts

The circle is divided into 2 equal parts. We divide the total angle by 2. Look at the second circle. It shows one of these parts.

$\text{Angle} = \frac{360 \text{ degrees}}{2}$

$= 180 \text{ degrees}$

$\boxed{180 \text{ degrees}}$

Diagram 2

Step 3 — Three equal parts

The circle is divided into 3 equal parts. We divide the total angle by 3. Look at the third circle. It shows one of these parts.

$\text{Angle} = \frac{360 \text{ degrees}}{3}$

$= 120 \text{ degrees}$

$\boxed{120 \text{ degrees}}$

Diagram 3

Step 4 — Four equal parts

The circle is divided into 4 equal parts. We divide the total angle by 4. Look at the fourth circle. It shows one of these parts.

$\text{Angle} = \frac{360 \text{ degrees}}{4}$

$= 90 \text{ degrees}$

$\boxed{90 \text{ degrees}}$

Diagram 4

Step 5 — Five equal parts

The circle is divided into 5 equal parts. We divide the total angle by 5. Look at the fifth circle. It shows one of these parts.

$\text{Angle} = \frac{360 \text{ degrees}}{5}$

$= 72 \text{ degrees}$

$\boxed{72 \text{ degrees}}$

Diagram 5

Step 6 — Six equal parts

The circle is divided into 6 equal parts. We divide the total angle by 6. Look at the sixth circle. It shows one of these parts.

$\text{Angle} = \frac{360 \text{ degrees}}{6}$

$= 60 \text{ degrees}$

$\boxed{60 \text{ degrees}}$

Diagram 6

Step 7 — Eight equal parts

The circle is divided into 8 equal parts. We divide the total angle by 8. Look at the seventh circle. It shows one of these parts.

$\text{Angle} = \frac{360 \text{ degrees}}{8}$

$= 45 \text{ degrees}$

$\boxed{45 \text{ degrees}}$

Diagram 7

Step 8 — Nine equal parts

The circle is divided into 9 equal parts. We divide the total angle by 9. Look at the eighth circle. It shows one of these parts.

$\text{Angle} = \frac{360 \text{ degrees}}{9}$

$= 40 \text{ degrees}$

$\boxed{40 \text{ degrees}}$

Diagram 8

Step 9 — Ten equal parts

The circle is divided into 10 equal parts. We divide the total angle by 10. Look at the ninth circle. It shows one of these parts.

$\text{Angle} = \frac{360 \text{ degrees}}{10}$

$= 36 \text{ degrees}$

$\boxed{36 \text{ degrees}}$

Diagram 9

Step 10 — Twelve equal parts

The circle is divided into 12 equal parts. We divide the total angle by 12. Look at the tenth circle. It shows one of these parts.

$\text{Angle} = \frac{360 \text{ degrees}}{12}$

$= 30 \text{ degrees}$

$\boxed{30 \text{ degrees}}$

Diagram 10

Answer

1 part: 360 degrees 2 parts: 180 degrees 3 parts: 120 degrees 4 parts: 90 degrees 5 parts: 72 degrees 6 parts: 60 degrees 8 parts: 45 degrees 9 parts: 40 degrees 10 parts: 36 degrees 12 parts: 30 degrees

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Q10

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Q11

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Q12

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Q13

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Q14

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Q15

What is the measure of a straight angle in degrees? A straight angle is half of a full turn. As a full-turn is 360°, a half turn is 180°. What is the measure of a right angle in degrees? Two right angles together form a straight angle. As a straight angle measures 180°, a right angle measures 90°.

Q16

The circle has been divided into 1, 2, 3, 4, 5, 6, 8, 9 10 and 12 parts below. What are the degree measures of the resulting angles? Write the degree measures down near the indicated angles.

Q17

There are two sets of numbers on the protractor: one increasing from right to left and the other increasing from left to right. Why does it include two sets of numbers?

Q18

What is the degree measure of $\angle$ AOB?

Q19

In Fig. 2.19, we have $\angle AOB = \angle BOC = \angle COD = \angle DOE = \angle EOF = \angle FOG = \angle GOH = \angle HOI = \_\_\_\_\_\_$ . Why?

Q20

Measure all three angles of the triangle shown in Fig. 2.21 (a), and write the measures down near the respective angles. Now add up the three measures. What do you get? Do the same for the triangles in Fig. 2.21 (b) and (c). Try it for other triangles as well, and then make a conjecture for what happens in general! We will come back to why this happens in a later year.

Q21

Let's Explore

In this figure, $\angle\text{TER} = 80^\circ$ . What is the measure of $\angle\text{BET}$ ? What is the measure of $\angle\text{SET}$ ?

Hint: Observe that $\angle\text{REB}$ is a straight angle. Hence, the degree measure of $\angle\text{REB} = 180^\circ$ of which $80^\circ$ is covered by $\angle\text{TER}$ . A similar argument can be applied to find the measure of $\angle\text{SET}$ .