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Arcs of a Circle
Lesson Demo

Theorem: The angle subtended by an arc at the centre is double the angle subtended by the arc at any point on the circle.

Given:

Arc AB.

Point C on the circle is outside AB.

To prove: ∠ AOB = 2 x ∠ ACB

Construction: Draw a line CO extended till point D.

Proof: In Δ OAC in each of these figures,

∠ AOD = ∠ OAC + ∠ OCA (Exterior angle of a triangle is equal to sum of two opposite interior angles)

OA = OC (Radii of same circle)

Thus, ∠ OAC = ∠ OCA (Angles opposite equal sides of a triangle are equal)

∠ AOD = ∠ OAC + ∠ OCA

⇒∠ AOD = 2 x ∠ OCA

Similarly, in Δ OBC,

∠ BOD = 2 x ∠ OCB

∠ AOD = 2 x ∠ OCA

and ∠ BOD = 2 x ∠ OCB

⇒∠ AOD + ∠ BOD = 2∠ OCA + 2∠ OCB

∠ AOD + ∠ BOD = 2 x (∠ OCA + ∠ OCB)

Or ∠ AOB = 2 x ∠ ACB

Hence, the theorem is proved.

Theorem: Angles subtended by an arc at all points within the same segment of the circle are equal.

Given:

An arc AB.

Points C and D are on the circle in the same segment.

To prove: ∠ ACB = ∠ ADB

Proof: By the theorem that the angle subtended by an arc at the centre is double the angle subtended by the arc at any point on the circle:

∠ AOB = 2 x ∠ ACB

Also, ∠ AOB = 2 x ∠ ADB