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 and 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?

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

Understand the Question

The size of an angle represents the amount of turn or opening between its two arms.
By placing the center of the same circle at the vertex of an angle, the arms intersect the circle to form an arc along its edge.
A larger angle intercepts a longer arc, while a smaller angle intercepts a shorter arc, allowing us to compare any two angles.

Step 1 · Measure the Arc for the First Angle

Place the center $O$ of the transparent circle at the vertex of the first crane's angle. Diagram 1

The arms of the angle pass through the circle at points $A$ and $B$ . Measure the curved distance (arc length) from $A$ to $B$ .

Step 2 · Measure the Arc for the Second Angle

Place the same transparent circle with its center at the vertex of the second crane's angle.

Mark the points $C$ and $D$ where the arms cut the circle and measure the arc length from $C$ to $D$ .

Step 3 · Compare the Two Arc Lengths

Compare the measured arc lengths:

If $\text{arc } AB > \text{arc } CD$ , the first angle is greater.
If $\text{arc } AB < \text{arc } CD$ , the first angle is smaller.
If $\text{arc } AB = \text{arc } CD$ , both angles are equal.

Answer

Yes, we can use this method to compare the angles. The angle that subtends a longer arc is greater.

Common Mistakes

Using Circles of Different Radii: The same circle (or circles of identical size) must be used for both angles; otherwise, comparing arc lengths directly will yield incorrect results.
Incorrect Center Alignment: The center of the circle must be placed precisely on the vertex of each angle to ensure accurate arc measurements.

More questions in IT

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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 $360^\circ$ , a half turn is $180^\circ$ . What is the measure of a right angle in degrees? Two right angles together form a straight angle. As a straight angle measures $180^\circ$ , a right angle measures $90^\circ$ .

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 $\angle \text{AOB}$ ?

Q19

In Fig. 2.19, we have $\angle AOB = \angle BOC = \angle COD = \angle DOE = \angle EOF = \angle FOG = \angle GOH = \angle HOI = \text{______}$ . 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 $\angle \text{SET}$ .