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 = 90

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 = 130

∠BEC + ∠CED = 180

130 + ∠CED = 180

∴ ∠CED = 50

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

∠EDC = 110

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

∴ ∠CDB = 110

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.

As AC is a diameter of circle C1

∴ ∠ADC = 90

As AB is a diameter of circle C2

∴ ∠ADB = 90

∠ ADC + ∠ADB = 90 + 90 = 180

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

• Circles

• Parts of Circle

• Arc and Chords

• Equal Chords of a Circle

• Arc and Angles

• Cyclic Quadrilaterals

• Tangent to Circle