Math's: Angles in a Circle: Their Properties, Concyclic Points and Cyclic Quadrilateral

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Angles in a Circle

Central Angle

The angle made at the center of a circle by the radii at the end points of an arc (or a chord) is called the central angle or angle subtended by an arc (or chord) at the center. is the central angle made by .

Central Angle

Central Angle

This central angle helps in measuring the length of the arc.

Inscribed Angle

The angle subtended by an arc (or chord) on any point on the remaining part of the circle is called an inscribed angle. is the angle inscribed by at point of the remaining part of the circle or by the chord at the point .

Inscribed Angle

Inscribed Angle

Some Important Properties of Angles in a Circle

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

  • Thus, in the figure (i) given below .

  • Consider a semicircle as in fig (ii). The central angle , then the inscribed angle . Hence, we conclude that the angle inscribed in a semicircle is a right angle.

Properties of Angles

Properties of Angles

Angles subtended in the same segment of a circle are equal or conversely, if a line segment joining two points subtends equal angles at two other points on the same side of the line containing the segment, the four points lie on a circle. In the below circle with center O the angles and are in the same segment formed by the arc . Therefore,

Properties of Angles

Properties of Angles

Concyclic Points

Points which lie on a circle are known as concyclic points.

Given one or two points there are infinitely many circles passing through them.

Concyclic Points

Concyclic Points

Three non-collinear points are always concyclic and there is only one circle passing through all of them.

Concyclic Points

Concyclic Points

  • Three collinear points are not concyclic (or noncyclic). That is, it is not possible to draw a circle passing through three collinear points.

  • Four non-collinear points may or may not be concyclic.

Concyclic Points

Concyclic Points

Cyclic Quadrilateral

Cyclic Quadrilateral

Cyclic Quadrilateral

  • A cyclic quadrilateral is a quadrilateral whose all four vertices are concyclic i.e. all the four vertices lie on a circle. is a concyclic quadrilateral.

  • Sum of the opposite angles of a cyclic quadrilateral is . Thus, in the adjoining concyclic quadrilateral , and , .

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