## Class 9 Maths All Important Theorems

### Class 9 Maths Chapter 6 Theorems Lines and Angles

- Theorem 6.1 If two lines intersect each other, then the vertically opposite angles are equal.
- Theorem 6.2 Class 9 If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.
- Theorem 6.3 Class 9 If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.
- Theorem 6.4 Class 9 If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary.
- Theorem 6.5 Class 9 If a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel.
- Theorem 6.6 Class 9 Lines which are parallel to the same line are parallel to each other.
- Theorem 6.7 Class 9 The sum of the angles of a triangle is 180º.
- Theorem 6.8 Class 9 If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.

### Class 9 Maths Chapter 7 Theorems Triangles

- Theorem 7.1 Class 9 (ASA congruence rule) Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of other triangle.
- Theorem 7.2 Class 9 Angles opposite to equal sides of an isosceles triangle are equal.
- Theorem 7.3 Class 9 The sides opposite to equal angles of a triangle are equal.
- Theorem 7.4 Class 9 (SSS congruence rule) If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.
- Theorem 7.5 Class 9 (RHS congruence rule) If in two right triangles the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent.
- Theorem 7.6 Class 9 If two sides of a triangle are unequal, the angle opposite to the longer side is larger (or greater).
- Theorem 7.7 Class 9 In any triangle, the side opposite to the larger (greater) angle is longer.
- Theorem 7.8 Class 9 The sum of any two sides of a triangle is greater than the third side.

### Class 9 Chapter 8 Quadrilaterals Important Theorems

- Theorem 8.1 A Diagonals of a parallelogram divides it into two congruent triangles.
- Theorem 8.2 In a parallelogram, opposite sides are equal.
- Theorem 8.3 If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.
- Theorem 8.4 In a parallelogram, opposite angles are equal.
- Theorem 8.5 If in a Quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.
- Theorem 8.6 The diagonals of a parallelogram bisect each other.
- Theorem 8.7 If the diagonals of a Quadrilateral bisect each other, then it is a parallelogram.
- Theorem 8.8 A Quadrilateral is a parallelogram if a pair of opposite sides are equal and parallel.
- Theorem 8.9 The line segment joining the mid-points of two sides of a triangle is parallel to the third side.
- Theorem 8.10 The line drawn through the mid-point of one side of a triangle, parallel to another side bisects the third side.

### Class 9 Chapter 9 Areas of Parallelograms and Triangles Important Theorems

- Theorem 9.1 : Parallelograms on the same base and between the same parallels are equal in area.
- Theorem 9.2 : Two triangles on the same base (or equal bases) and between the same parallels are equal in area.
- Theorem 9.3 : Two triangles having the same base (or equal bases) and equal areas lie between the same parallels.

### Class 9 Chapter 10 Circles Important 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.

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