Arc  and Angles

Arc and angles Subtended by them
Some results on angles subtended by arcs of a circle either at the center of the circle or at a point on its circumference.

1) The angle subtended by an arc of a circle at the center is double the angle subtended by it at any point on the remaining part of the circle.

∠POQ = 2 ∠PRQ or ∠ PRQ = ½ ∠ POQ
2) Angles in the same segment of a circle are equal.

∠PRQ = ∠PSQ ( both the angles are subtended in the same arc PQ )
3) The angle in a semicircle is a right angle.

∠PRQ = 90⁰ (PQ is a diameter ) Some solved examples on arc and angles

1) In the given figure, A,B,C and D are four points on a circle. AC and BD intersect at a point E such that ∠BEC = 130⁰ and ∠ECD = 20⁰. Find ∠BAC.


Solution : ∠BEC = 130⁰ and ∠ECD = 20⁰
∠BEC + ∠CED = 180⁰ ( Linear pair angles)
130 + ∠CED = 180

∴ ∠CED = 50⁰

∠EDC = 180 – (50 +20) [ In ΔCDE, sum of all the angles in a Δ = 180⁰ )

∠EDC = 110⁰

But ∠EDC = ∠CDB ( angles in the same segment are equal )

∴ ∠CDB = 110⁰
2) Two circles are drawn with sides AB and AC of a triangle ABC as diameters. The circles intersect at a point D. Prove that D lies on BC.

Solution : Join AD.

As AC is a diameter of circle C1
∴ ∠ADC = 90⁰ [ Angle in a semicircle is a right angle (90) ]

As AB is a diameter of circle C2

∴ ∠ADB = 90⁰ [ Angle in a semicircle is a right angle (90) ]

∠ ADC + ∠ADB = 90 + 90 = 180⁰

⇒ BDC is a straight line ⇒ D lies on BC.

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Circles

• Circles
• Parts of Circle
• Arc and Chords
• Equal Chords of a Circle
• Arc and Angles
• Cyclic Quadrilaterals
• Tangent to Circle

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