**Math Labs with Activity – Paper Cutting and Folding Method**

**OBJECTIVE**

To verify by the method of paper cutting and folding that

- the angle in a semicircle is a right angle
- the angle in a minor segment is an obtuse angle
- the angle in a major segment is an acute angle

**Materials Required**

- Three sheets of white paper
- A tracing paper
- A geometry box

**Procedure**

To verify that the angle in a semicircle is a right angle.

**Step 1:** Take a sheet of white paper and mark a point O . on this paper. With O as centre draw a circle of any radius.

**Step 2:** Draw a diameter AB of this circle. Draw any chord AP. Join PB. We thus get ΔAPB such that ∠APB is the angle in a semicircle [see Figure 20.1(a)].

**Step 3:** Make two exact replicas of A APB using a tracing paper. Make cut-outs of these two replicas of ΔAPB and label them as ΔA_{1}P_{1}B_{1} and ΔA_{2}P_{2}B_{2}.

**Step 4:** Place the two cut-outs adjacent to each other as shown in Figure 20.1(b). What do you observe?

**Observations**

We observe that the two line segments A_{1}P_{1} and P_{2}B_{2} lie on a straight line.

Therefore, ∠A_{1}P_{1}B_{1} and ∠A_{2}P_{2}B_{2} are supplementary angles, i.e., ∠A_{1}P_{1}B_{1 }+ ∠A_{2}P_{2}B_{2} =180°.

Since each of the angles ∠A_{1}P_{1}B_{1} and ∠A_{2}P_{2}B_{2} is equal to ∠APB, therefore 2∠APB=180°, i.e., ∠APB = 90°.

**Procedure**

To verfiy that the angle in a minor segment is an obtuse angle.

**Step 1:** Take a white sheet of paper and mark a point O on it. With O as centre draw a circle of any radius.

**Step 2:** Draw a chord MN (such that MN is not a diameter) in this circle.

**Step 3:** Draw a ΔMQN so that point Q lies on the minor arc of the circle. Thus, ∠MQN is the angle in a minor segment [see Figure 20.2(a)].

**Step 4:** Make an exact replica of ΔMQN. Place it over a right-angled ΔXYZ (right angled at point Y) in such a way that side QN of ΔMQN falls over the side YZ of ΔXYZ as shown in Figure 20.2(b). What do you observe?

**Observations**

∠MQN is greater than ∠XYZ, i.e., ∠MQN > 90°, i.e., ∠MQN is an obtuse angle.

Procedure To verify that the angle in a major segment is an acute angle.

**Step 1:** Take a white sheet of paper and mark a point O on it. With O as centre draw a circle of any radius.

**Step 2:** Draw a chord GH (such that GH is not a diameter) in this circle.

**Step 3:** Draw a ΔGSH so that point S lies on the major arc of the circle. Thus, ∠GSH is the angle in a major segment [see Figure 20.3(a)].

**Step 4:** Make an exact replica of ΔGSH. Place it over the right-angled ΔXYZ in such a way that side SG of ΔGSH falls over the side YZ of ∠XYZ as shown in Figure 20.3(b). What do you observe?

**Observations**

∠GSH is less than ∠XYZ, i.e., ∠GSH <90°, i.e., ∠GSH is an acute angle.

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