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.