A triangle is a polygon with three sides, three angles, and three vertices, and the sum of all three angles of any triangle is 180°.

**Properties of a triangle:**

- A triangle has three sides, three angles, and three vertices.
- The sum of all internal angles of a triangle is always equal to 180°. This is called the angle sum property of a triangle.
- The sum of the length of any two sides of a triangle is greater than the length of the third side.
- The side opposite the largest angle of a triangle is the largest side.
- Any exterior angle of the triangle is equal to the sum of its interior opposite angles. This is called the exterior angle property of a triangle.

**Based on the angle measurement, there are three types of triangles:**

**1. Acute angle triangle:**

An acute angle triangle is a triangle in which* *all three angles are less than 90*°*.

**2. Right angle triangle:**

A triangle that has one angle that measures exactly 90*°* is a right-angle triangle.

- The other two angles of a right-angle triangle are acute angles.
- The side opposite the right angle is the largest side of the triangle and is called the hypotenuse.
**Pythagoras theorem:**In a right-angled triangle, the sum of the squares of the perpendicular sides is equal to the square of the hypotenuse.

For example, considering the right-angled triangle ACB, we can say:

(AC)2 + (CB)2 = (AB)2

Vice versa, we can say that if a triangle satisfies the Pythagoras condition, then it is a right-angled triangle.

**3. Obtuse angle triangle:** - A triangle that has one angle that measures more than 90
*°*is an obtuse angle triangle.

**Based on the length of the sides, triangles are classified into three types:**

**1. Scalene triangle:**

A scalene triangle is a triangle in which all three sides are of different lengths.

- Since all the three sides are of different lengths, the three angles will also be different.

**2. Isosceles triangle:**

A triangle that has two sides of the same length and the third side of a different length is an isosceles triangle.

- The angles opposite to the equal sides measure the same.

**3.Equilateral triangle:**

A triangle which has all the three sides of the same length is an equilateral triangle.

- Since all the three sides are of the same length, all the three angles will also be equal.
- Each interior angle of an equilateral triangle is equal to 60°.

**Median:**

The line joining the midpoint of a side with the opposite vertex is called a median.

**Altitude:**

The perpendicular drawn from a vertex to the opposite side is called the altitude.

**Perpendicular bisector:**

A line that bisects and also makes a right angle with the same side of the triangle is called a perpendicular bisector.

**Angle bisector:**

The line that divides the angle at one of the vertices into two parts is called an angular bisector.

**Note:**

- All points on the angular bisector are equidistant from both arms of the angle.
- All points on the perpendicular bisector of a line are equidistant from both ends of the line.
- In an equilateral triangle, the perpendicular bisector, median, angle bisector, and altitude (drawn from a vertex to a side) coincide.

## **Geometric centres**

**Orthocentre:**

The point of intersection of the three altitudes is the orthocentre.

**Centroid:**

The point of intersection of the three medians is the centroid.

**Circumcentre:**

The three perpendicular bisectors of a triangle meet at a point called the circumcentre. A circle drawn from this point with the circumradius would pass through all the vertices of the triangle.

**Incentre:**

The three angle bisectors of a triangle meet at a point called the incentre of a triangle. The incentre is equidistant from the three sides, and a circle drawn from this point with the inradius would touch all the sides of the triangle.

**Apollonius theorem**

In a triangle ABC, if AD is the median to side BC then by Apollonius theorem,

**Midpoint theorem**

The line joining the midpoint of any two sides in a triangle is parallel to the third side and is half the length of the third side. If X is the midpoint of CA and Y is the midpoint of CB, then XY will be parallel to AB and XY = ½ * AB.

**Basic proportionality theorem**

If a line is drawn parallel to one side of a triangle and it intersects the other two sides at two distinct points, then it divides the two sides in the ratio of respective sides.

If in a triangle ABC, D and E are the points lying on AB and BC, respectively, and DE is parallel to AC, then AD/DB = EC/BE.

**Interior angular bisector theorem**

In a triangle, the angular bisector of an angle divides the side opposite to the angle, in the ratio of the remaining two sides. In a triangle ABC, if AD is the angle bisector of angle A then AD divides the side BC in the same ratio as the other two sides of the triangle, i.e., BD/ CD= AB/AC.

**Exterior angular bisector theorem**

The angular bisector of the exterior angle of a triangle divides the opposite side externally in the ratio of the sides containing the angle. In a triangle ABC, if CE is the angular bisector of the exterior angle BCD, then AE/BE = AC/BC.

**Similar triangles**

If two triangles are similar, then their corresponding angles are equal, and the corresponding sides will be in proportion.

For any two similar triangles:

- Ratio of sides = Ratio of medians = Ratio of heights = Ratio of circumradii = Ratio of angular bisectors
- Ratio of areas = Ratio of the square of the sides

Tests of similarity: (AA/SSS / SAS)

**Congruent triangles**

If two triangles are congruent, then their corresponding angles and corresponding sides are equal.

Tests of congruence: (SSS/SAS/AAS/ASA)

**Area of a triangle, A**

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