**CBSE Class 9 Maths Lab Manual – Angles in the Same Segment**

**Objective**

To show that the angles subtended by the chord of a circle in the same segment are equal, experimentally.

**Prerequisite Knowledge**

- Concept of a circle.
- Concept of angle subtended by an arc/chord.

**Materials Required**

Glazed papers, sketch pens, a pair of scissors, gluestick, geometry box, white sheet.

**Procedure**

- Take red colour glazed paper and draw a circle of any radius say 2.5 cm on the white side of the paper.
- Cut this circle with centre O and radius 2.5 cm.
- Mark any point P on the circumference of the circle.
- Mark two other points A and B on the circumference of the circle.
- Fold and press the circle along AB to get a crease which is a chord of the circle.
- Join PA and PB to get ∠APB as shown in fig. (i).

- Take another coloured glazed paper say, yellow, cut circle of same radius.
- Draw a chord CD on second circle with centre O’, such that AB = CD.
- Mark a point R on the circumference of the second circle. Join RC and RD fig (ii).

- Cut out ∆RCD and paste it on the first circle such that R lies on P and RD lies on PB and RC lies on PA as shown in fig. (iii).

∠APB superimposes ∠DRC

**Observation**

Since the chord AB = chord CD and ∠APB superimposes ∠DRC, their arcs are same. When two angles superimpose each other it means their two arms lie on one another. This verifies that angles in the same segments are equal.

**Result**

We verified that two angles subtended by the chord of a circle in the same segment are equal.

**Learning Outcome**

We learnt that two or more angles subtended by same chord in the same segment of a circle are equal.

**Activity Time**

- This theorem can be verified by taking any point on the minor segment of circle.
- Prove this theorem for different chords, such as diameter, chord in minor segment, chord in major segment.

**Viva Voce**

**Question 1.**

Define chord of a circle.

**Answer:**

A line segment AB joining two points A and B of the circle is called a chord of the circle.

**Question 2.**

How many longest chords are there in a circle ?

**Answer:**

There are infinity many longest chords in a circle passing through the centre.

**Question 3.**

Define minor segment of a circle.

**Answer:**

A chord divides a circle into two parts the smaller part is called as minor segment.

**Question 4.**

Define major segment of a circle.

**Answer:**

The larger part of the circle is called as major segment.

**Question 5.**

What is the measure of an angle formed in a semicircle ?

**Answer:**

90°

**Question 6.**

If the angle subtended by an arc at the centre is 30°, what is the value of angle subtended by it on the remaining part of the circle.

**Answer:**

15°

**Question 7.**

If the angle subtended by an arc at the centre in major segment is reflex angle, then what is the angle in the minor segment ?

**Answer:**

Obtuse angle.

**Question 8.**

The angle subtented by an arc at the circle in minor segment is obtuse angle. What is the values of angle subtended by it in the major segment ?

**Answer:**

Acute angle.

**Multiple Choice Questions**

**Question 1.**

Find ∠BDC, if ∠ABC = 70°, ∠ACB = 30°:

(i) 80°

(ii) 100°

(iii) 70°

(iv) none of these

**Question 2.**

If ∠BDC = 30°, find ∠BAC:

(i) 30°

(ii) 60°

(iii) 15°

(iv) none of these

**Question 3.**

In a circle with centre O, AB is a chord and P, Q, R are three points on the same side of the chord AB of circle. If ∠APB = 40°, what is value of ∠AQB, ∠ARB?

(i) 20°, 40°

(ii) 80°, 80°

(iii) 40°, 40°

(iv) none of these

**Question 4.**

Calculate the measure of ∠PQB, where O is the centre of the circle:

(i) 50°

(ii) 60°

(iii) 90°

(iv) none of these

**Question 5.**

If ∠PQS = 40°, ∠SPR = 65°, find ∠RSP.

(i) 70°

(ii) 75°

(iii) 105°

(iv) none of these

**Question 6.**

In the fig, ∠ABC = 69°, ∠ACB = 31°. Find ∠BDC:

(i) 80°

(ii) 100°

(iii) 70°

(iv) none of these

**Question 7.**

In the fig, ∠BEC = 130°, ∠ECD = 20°. Find ∠BAC:

(i) 50°

(ii) 110°

(iii) 70°

(iv) none of these

**Question 8.**

In a circle, ABCD is a cyclic quadrilateral in which ∠DBC = 80° and ∠BAC = 40°, find ∠DCB:

(i) 60°

(ii) 70°

(iii) 80°

(iv) none of these

**Question 9.**

In a circle, AB is a diameter, Q, P are any points on the circle. What is value of ∠AQB and ∠APB ?

(i) 80°, 90°

(ii) 90°, 90°

(iii) 50°, 50°

(iv) none of these

**Answers**

- (i)
- (i)
- (iii)
- (i)
- (ii)
- (i)
- (ii)
- (i)
- (ii)

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