A polygon which has three sides is called a triangle. It is formed by joining three non-collinear points by a straight line. The lines AB, BC and CA are called the sides of the triangle. The angles formed by these three lines are called the angles of the triangle and they are represented by A, B and C.
Basic properties of triangles
1. The sum of the angles of a triangle is 180o. In the figure, ∠A + ∠B + ∠C = 180o.
2. The sum of any two sides of a triangle is greater than the third side of the triangle. In the figure above, AB + BC > CA or BC + CA > AB or CA + AB > BC.
3. An angle formed by one side and the extended portion of an adjacent side at the 3. common vertex is called the exterior angle of the triangle. The exterior angle of a triangle is equal to the sum of the interior opposite angles.
Classification of triangles on the basis of sides
Scalene Triangle: A triangle in which all the three sides are unequal is a scalene triangle.
Isosceles triangle: A triangle in which two sides are equal is an isosceles triangle. The angles opposite to the equal sides are also equal.
In the above figure, sides AB and AC of the triangle ABC are equal. Therefore ∠B = ∠C.
Equilateral triangle: A triangle in which all the three sides are equal is an equilateral triangle. All the three angles of an equilateral triangle are equal to 60o.
Classification of triangles on the basis of angles
Acute angled triangle: A triangle in which all the three angles are acute is an acute-angled triangle.
Obtuse angled triangle: A triangle in which one angle is obtuse is an obtuse angled triangle. The side opposite to the obtuse angle is the greatest side in the triangle.
Right Angled Triangle: A triangle in which one angle is right angle is a right-angled triangle. The side opposite to the right angle is called the hypotenuse and is the greatest side in the triangle. In the right triangle ABC below, B is the right angle and AC is the hypotenuse. Here, AC2 = AB2 + BC2. This is called the Pythagoras theorem.
Special Right-Angled Triangles
A 30-60-90 triangle is a triangle with angles 30o, 60o, and 90o. The lengths of the sides of a 30-60-90 triangle will always be in the ratio 1:√3:2. Therefore, if any one side of a triangle is known, then the other two sides can be found out using the above ratio. For example, if the length of the side opposite the 60o angle is 3 units, then by using the ratio 1:√3:2, the length of the side opposite the 30o angle is √3 units and the length of the hypotenuse i.e. side opposite the 90o angle is 2√3 units.
A 45-45-90 triangle is a triangle with angles 45o, 45o, and 90o. The lengths of the sides of a 45-45-90 triangle will always be in the ratio 1:1:√2. This is an isosceles right triangle.