Lesson 2, Topic 5
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# Angles in the Same Segment of a Circle

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Theorem: Angles in the same segment are equal.

Given: Consider the circle as shown above, in which chord ππ subtends angles ππ΄π and ππΆπ at any two points π΄ and πΆ on the circumference of a circle.

To prove: β ππ΄π = β ππΆπ

### Proof:

In the last lesson, we learnt that the angle subtended by the chord at the centre is double the angle subtended by it at any point on the circle.

Thus, β πππ = 2β ππ΄πβ¦β¦.(π)

Similarly, β πππ = 2β ππΆπβ¦β¦.(ππ)

Equate (π) and (ππ)

2β ππ΄π = 2β ππΆπ

β΄ β ππ΄π = β ππΆπ

Therefore angles in the same segments of a circle are equal.

These angles can be more than two. Therefore, we can say that all angles in the major segment are equal.

For the major segment in the circle below all the angles are equal.

β΄ β a = β b = β c = β d

Similarly, all angles in a minor segment are all equal.

For the minor segment in the circle below all the angles are equal.

β΄ β x = β y = β z

### Example

In the figure below β π΅π΄π· = 36Β° and β πΆπ΅π΄ = 37Β°, find β π΅πΆπ· and β πΆπ·π΄.

Solution:

Let’s add the given angles to the diagram.

If we draw lines to join BD and AC, we get two chords that separate the circle into segments.

If we consider the major segment of chord BD

β π΅πΆπ· = β π΅π΄π· = 36Β° (angles in the same segment)

If we consider the major segment of chord AC

β πΆπ·π΄ = β πΆπ΅π΄ = 37Β° (angles in the same segment)

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