Class 9 Maths Circles Angle Subtended by Arc of Circle

Angle Subtended by Arc of Circle

If two chords of a circle are equal, then their corresponding arcs are congruent and conversely, if two arcs are congruent, then their corresponding chords are equal. If Arc AB = Arc CD, then Chord AB = Chord CD.

Congruent arcs (or equal arcs) of a circle subtend equal angles at the centre. In the figure, Arc AB = Arc CD & thus, it is observed that ∠1 =∠2..

 

Theorem 8: The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

In the figure shown that ∠2 =Twice of ∠1.

Refer ExamFear video lessons for Proof of this theorem

 

Theorem 9: Angles in the same segment of a circle are equal

In the figure, ∠1 = ∠3.

 

Refer ExamFear video lessons for Proof of this theorem

 

Converse Theorem 10: If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle (i.e. they are  concyclic).

In the figure, if ∠1 = ∠3, then point A, B, C & D lies on a circle.

Refer ExamFear video lessons for Proof of this theorem

 

 

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