### Class 9 Chapter 10 Circles Theorem 10.12 : If the sum of a pair of opposite angles of a quadrilateral is 180º, the quadrilateral is cyclic.

### Class 9 Chapter 10 Circles Theorems

- Theorem 10.1 : Equal chords of a circle subtend equal angles at the centre.
- Theorem 10.2 : If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal
- Theorem 10.3 : The perpendicular from the centre of a circle to a chord bisects the chord.
- Theorem 10.4 : The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
- Theorem 10.5 : There is one and only one circle passing through three given non-collinear points.
- Theorem 10.6 : Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centres).
- Theorem 10.7 : Chords equidistant from the centre of a circle are equal in length.
- Theorem 10.8 : The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
- Theorem 10.9 : Angles in the same segment of a circle are equal.
- Theorem 10.10 : If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle (i.e. they are concyclic).
- Theorem 10.11 : The sum of either pair of opposite angles of a cyclic quadrilateral is 180º.
- Theorem 10.12 : If the sum of a pair of opposite angles of a quadrilateral is 180º, the quadrilateral is cyclic.