Question 8
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.
We can compare angles accurately by using a simple tracing method.
Step 1 — Ease of comparison
No, it is not always easy to compare angles. Sometimes, angles can look very similar. Our eyes might trick us. So, we need a proper way to compare them.
Step 2 — Labeling the angles
Let us label the angles in the given picture. From left to right, we can call them Angle 1, Angle 2, Angle 3, and Angle 4. Angle 1 has its vertex at the bottom left. Angle 2 has its vertex at the top left. Angle 3 has its vertex at the top right. Angle 4 has its vertex at the bottom right.
Step 3 — Comparing the given angles
To compare Angle 1 and Angle 2, we use tracing paper. First, trace Angle 1 onto a piece of tracing paper. Now, place this tracing over Angle 2. Make sure the vertex of the traced Angle 1 sits exactly on the vertex of Angle 2. Also, make one arm of the traced Angle 1 lie exactly on one arm of Angle 2. Then, look at the other arm of the traced Angle 1. If this arm falls inside Angle 2, then Angle 1 is smaller than Angle 2. If this arm falls outside Angle 2, then Angle 1 is larger than Angle 2. If both arms match perfectly, then Angle 1 and Angle 2 are equal. We can use this same method to compare any pair of angles from the picture.
Step 4 — Drawing and comparing new angles
Let us draw two new angles. We can call them Angle A and Angle B. Draw Angle A using a ruler. Draw Angle B, making it look a bit different from Angle A. To compare Angle A and Angle B, we use the same tracing method. Trace Angle A onto tracing paper. Place the tracing over Angle B. Align their vertices and one arm. Observe the position of the other arm to see which angle is larger or smaller.
Answer
(i) No, it is not always easy to compare two angles. (ii) To compare the angles in the image, I will place one angle over the other by tracing or using transparent paper to directly compare which angle is larger or smaller. If I place one arm of each angle one over another, it can be observed easily (by looking at the other arm) which one is lesser or greater. (iii) I will draw a few more angles and label them. I will compare them using the same tracing method as described above.
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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.
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?
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?
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.
Is it possible to draw such that the two angles are equal to each other in size?
If a straight angle is formed by half of a full turn, how much of a full turn will form a right angle?
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?
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°.
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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?
What is the degree measure of AOB?
In Fig. 2.19, we have . Why?
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.
Let's Explore
In this figure, . What is the measure of ? What is the measure of ?
Hint: Observe that is a straight angle. Hence, the degree measure of of which is covered by . A similar argument can be applied to find the measure of .