Lines and Angles | IT

Question 9

Suppose we have a transparent circle which can be moved and placed on figures. Let us place the circular paper on the angle made by the first crane. The circle is placed in such a way that its centre is on the vertex of the angle. Let us mark the points A and B on the edge circle at the points where the arms of the angle pass through the circle.

Q. Can we use this to find out if this angle is greater than, or equal to or smaller than the angle made by the second crane?

Solution

We can compare angles by looking at the length of the curved line (arc) they cut on a circle.

Step 1 — Measuring the first angle

Let us take the first crane's angle. Place the transparent circle's center on the angle's vertex. Look at the diagram. Point O is the vertex. The arms of the angle pass through the circle. They mark points A and B on the circle's edge. Let us measure the curved distance from A to B. This curved distance is called an arc.

Diagram 1

Step 2 — Measuring the second angle

Now, let us take the second crane's angle. Place the same transparent circle's center on its vertex. The arms of this second angle will pass through the circle. They will mark two new points on the circle's edge. Let us call these new points C and D. Let us measure the curved distance from C to D. This is the arc for the second angle.

Step 3 — Comparing the angles

Now we compare the two arc lengths we measured. If the arc from A to B is longer than the arc from C to D, then the first angle is bigger. If the arc from C to D is longer, then the second angle is bigger. If both arcs are the same length, then both angles are equal.

Answer

Yes, we can use this method to compare the angles. The angle that cuts a longer arc on the circle is the bigger angle.

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Is it always easy to compare two angles?

Here are some angles. Label each of the angles. How will you compare them? Draw a few more angles; label them and compare.

Q. Can we use this to find out if this angle is greater than, or equal to or smaller than the angle made by the second crane?

Q10

Let us place the circle on the angle made by the second crane so that the vertex coincides with the centre of the circle and one of the arms passes through OA.

Q. Can you now tell which angle is bigger?

Q11

Which crane was making the bigger angle?

If you can make a circular piece of transparent paper, try this method to compare the angles in Fig. 2.10 with each other.

Q12

Is it possible to draw $\overrightarrow{OC}$ such that the two angles are equal to each other in size?

Q13

If a straight angle is formed by half of a full turn, how much of a full turn will form a right angle?

Q14

Angles are classified in three groups as shown below. Right angles are shown in the second group. What could be the common feature of the other two groups?

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 $\angle\text{SET}$ .