Class 9 Maths

Congruency In Triangles

Theorems

  1. Two figures are congruent, if they are of the same shape and of the same size.
  2. Two circles of the same radii are congruent.
  3. Two squares of the same sides are congruent.
  4. If two triangles ABC and PQR are congruent under the correspondence A – P, B-Q and C-R, then symbolically, it is expressed as Δ ABC Δ PQR.

SAS Congruence Rule

If two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle, then the two triangles are congruent. (Axiom: This result cannot be proved with the help of previously known results.)

ASA Congruence Rule

If two angles and the included side of one triangle are equal to two angles and the included side of the other triangle, then the two triangles are congruent (ASA Congruence Rule).

Construction: Two triangles are given as follows in which:
`∠ABC=∠DEF` and `∠ACB=∠DEF`
Sides `AB=DE`

To prove: `ΔABC≅ ΔDEF`

Proof: `∠ABC=∠DEF` (given)
`AB=DE`
`AC=DF`
(Sides opposite to corresponding angles are in the same ratio as ratio of angles)
Hence, `ΔABC≅ ΔDEF` (SAS rule)

Triangle Triangle

AAS Congruence Rule

If two angles and one side of one triangle are equal to two angles and the corresponding side of the other triangle, then the two triangles are congruent.
This theorem can be proved in similar way as the previous one.

  • Angles opposite to equal sides of a triangle are equal.
  • Sides opposite to equal angles of a triangle are equal.
  • Each angle of an equilateral triangle is of 60°.

SSS Congruence Rule

If three sides of one triangle are equal to three sides of the other triangle, then the two triangles are congruent.

RHS Congruence Rule

If in two right triangles, hypotenuse and one side of a triangle are equal to the hypotenuse and one side of other triangle, then the two triangles are congruent (RHS Congruence Rule).

  • In a triangle, angle opposite to the longer side is larger (greater).
  • In a triangle, side opposite to the larger (greater) angle is longer.
  • Sum of any two sides of a triangle is greater than the third side.

Theorem: Angles opposite to equal sides of an isosceles triangle are equal.

Theorem: The sides opposite to equal angles of a triangle are equal.

Theorem: If two sides of a triangle are unequal, the angle opposite to the longer side is larger (or greater).

Theorem: In any triangle, the side opposite to the larger (greater) angle is longer.

Theorem: The sum of any two sides of a triangle is greater than the third side.


Number System

Numbers are in the form of p/q or can be written in the form of p/q are called rational numbers.

Polynomials

Algebraic expression with many terms having variables and coefficients is called polynomial.

Coordinate Geometry

The distance of a point from the y - axis is called its x-coordinate, or abscissa, and the distance of the point from the x-axis is called its y-coordinate, or ordinate.

Linear Equations in Two Variables

A linear equation in two variables has infinite number of solutions.

Euclid's Geometry

The basic concept about a line is that it should be straight and that it should extend indefinitely in both the directions.

Lines & Angles

An angle is formed by two rays originating from the same end point. The rays making an angle are called the arms of the angle and the end-points are called the vertex of the angle.

Triangles

Two figures are congruent, if they are of the same shape and of the same size. Two circles of the same radii are congruent. Two squares of the same sides are congruent.

Quadrilaterals

Sum of the angles of a quadrilateral is 360°. A diagonal of a parallelogram divides it into two congruent triangles.

Parallelograms

In a parallelogram: opposite sides are equal, opposite angles are equal, and diagonals bisect each other

Circles

The collection of all the points in a plane, which are at a fixed distance from a fixed point in the plane, is called a circle.

Heron's Formula

Heron's formula is generally used for calculating area of scalene triangle.

Surface Area

How to calculate surface area of cylinder, cone and sphere?

Volume

How to calculate volume of cylinder, cone and sphere?

Statistics

Learn to make bar chart and histogram from given data.

Probability

Probability tells about likelihood of an event to occur.