## त्रिभुज(About area of triangle)

A plane closed figure bounded by three line segments is called a triangle. The triangle is denoted by . A triangle has six parts – three sides and three angles.

## Area of triangle if height and base are known

**Base-base, height=height**

Heron’s Formula(About area of triangle)

Heron’s Formula

Formula is Where is the semiperimeter of the triangle and , And The lengths of the sides of the triangle are

Here S=(a+b+c)/2 and a, b and c are the three sides of the triangle.

## Types of Triangles

Triangles are divided on two bases:

(i) on the basis of angles

(ii) at the base of the sides

## (i) Types of Triangles on the Basis of Angles

### Right-angled triangle :

A triangle whose measure of one angle is 90° is called a right angled triangle. The side opposite to the right angle is called the hypotenuse of the triangle, the remaining two sides are called the base and the perpendicular respectively.

**(hypotenuse)^2 = (base)^2 + (perpendicular)^2**

### Acute-angled triangle :

A triangle whose measure of each angle is greater than 0° and less than 90° is called an acute triangle.

### Obtuse-angled triangle :

A triangle whose measure of one angle is greater than 90° but less than 180° is called an obtuse triangle.

## (ii) Types of Triangle on the Basis of Sides

### Scalene triangle :

A triangle whose all the sides are of different lengths is called an equilateral triangle.

### Isosceles triangle

A triangle whose two sides are equal is called an isosceles triangle.

### Equilateral triangle

A triangle whose three sides are equal is called an equilateral triangle.

Area of Equilateral triangle = ,**where s=length of the side**

## Some Useful Results on Triangles

• The sum of the three angles of a triangle is 180°.

• The measure of the exterior angle formed by extending one side of a triangle is equal to the sum of the measures of two opposite interior angles.

• The exterior angle of a triangle is greater than any opposite interior angle.

• The angle between the interior of one base angle of a triangle and the exterior angle of another angle is half of the angle.

• The sum of any two sides of a triangle is greater than its third side.

•If the bisector of the vertex angle of a triangle bisects the base, then the triangle is an isosceles triangle.

• The perpendiculars from the vertices of an isosceles triangle on opposite sides are equal.

•The medians of an equilateral triangle are equal.

## Congruent Triangle :

If all the angles and sides of one triangle are equal to the corresponding angles and corresponding sides of another triangle, then both the triangles are said to be congruent.

• Every triangle is congruent to itself.

## Triangle Congruency Axiom

**Assumption 1: **If two sides of one triangle and the angle between them are equal to the corresponding sides and corresponding angle of the other triangle. (SAS axiom)

**Assumption 2: **If two angles of one triangle and the common side of these angles are equal to the corresponding angles and common side of the other triangle. (SAA axiom)

**Assumption 3:** If all the three sides of one triangle are equal to the three corresponding sides of another triangle.(SSS axiom)

**Assumption 4: **If the hypotenuse and one side of one triangle are equal to the hypotenuse and the corresponding side of the other triangle. (RHS axiom)

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