# Properties of Triangle

In this post we will try to understand some important properties of triangle which will help us develop deep understanding of the concepts and also guide us to solve problems related to triangles.

## Properties of Triangle

A triangle has 3 sides, 3 interior angles and 3 exterior angles

### Angle sum property of a triangle

The sum of interior angles of a triangle is always equal to 180o.
Let us prove if this statement is correct or not

Statement – Sum of interior angle of a triangle is always equal to 180o
Given – A triangle PQR
To construct – Construct a line XY parallel to side QR of the triangle. Name the angles formed as shown in the figure.

Proof – Since, XY is a straight line

Therefore, ∠4 + ∠1 + ∠5 = 180o                ………………………………………………..(i)

Since, XY || QR, where PQ and PR are transversals

Therefore, ∠4 = ∠2 (a pair of alternate interior angles)            ……………….. (ii)

Also, ∠5 = ∠3          (a pair of alternate interior angles)             ….………….. (iii)

From eq. (i), (ii) and (iii)

∠1 + ∠2 + ∠3 = 180o        (replacing ∠4 and ∠5 with ∠2 and ∠3 respectively)

Or Sum of all interior angles of a triangle = 180o                          Proved

### Exterior angle property of a triangle theorem

If any side of a triangle is extended, then the exterior angle so formed is the sum of the two opposite interior angles of the triangle.

Let us prove this statement step by step.

Given: A triangle ABC
To Construct: Extend the side BC to D and make an exterior angle do as shown in the figure

Proof: From the above figure,

∠ACB + ∠ACD = 180o         (both the angles form a linear pair)

bo + do = 180o                                        ………………………………………………………………..(i)

By Angle sum property i.e., sum of interior angles of a triangle = 180o

∠BAC + ∠ACB + ∠ABC = 180o

ao + bo + co = 180o                                     …………………………………………………………..(ii)

From eq. (i) and (ii)

ao + bo + co  =  bo + do

ao + co  =  do

Or

∠ACD = ∠BAC + ∠ACB                                                                        Proved.

### The exterior angles of a triangle always add up to 360 degrees

Given: A triangle ABC with its exterior angles
To prove: ∠4+∠5+∠6 = 360°

Proof:

From the above figure,

∠1 + ∠4 = 180°      (both the angles form a linear pair)                      ………………(i)

∠2 + ∠5 = 180°     (both the angles form a linear pair)                      ………………(ii)

∠3 + ∠6 = 180°     (both the angles form a linear pair)                     ………………(iii)

Adding eq.(i), (ii) and (iii), we get

∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 180° + 180° + 180°             …………………………………(iv)

By angle sum property,

∠1 + ∠2 + ∠3 = 180°                                           …………….……………………………………(v)

From eq.(iv) and (v), we can replace ∠1 + ∠2 + ∠3 with 180°

180 + ∠4 + ∠5 + ∠6 = 180° + 180° + 180° ∠4+∠5+∠6 = 360°                                                                                Proved

### Property of Length of Triangle

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Similarly, the difference between the lengths of any two sides of a triangle is less than the length of the third side.

For this triangle, Sum of any two sides i.e., 5 + 6 = 11 cm which is greater than third side i.e., 8 cm
Similarly, 8 – 6 = 2 cm which is less than the third side i.e., 5 cm

### Property of angle of Triangle

The shortest side is always opposite the smallest interior angle. Similarly, the longest side is always opposite the largest interior angle

Here, the side opposite to the smallest angle (23o) is smallest (5 cm) and side opposite to the largest angle (109o) is largest (8 cm)

### Perimeter of Triangle

The total length of the outer boundary is called the perimeter of the triangle

i.e. Perimeter of triangle = sum of all sides

For the triangle with length of sides as a, b and c as shown in the figure

Perimeter (P) = a + b + c

Note – the unit of perimeter is same as the unit of the sides of the triangle

### Area of Triangle

The region occupied by the triangle in 2-dimensional space is called the area of triangle.
Area is measured in square units (sq.cm, sq.m, sq.km etc.)

#### Area of triangle if height and base is given

If “b” is the base and “h” is the height of the triangle, then

Area = half of product of height and base

Area = ½ x b x h

#### Area of triangle using Heron’s formula

If all the sides of the triangle are of different length, then area is calculated using heron’s formula

Area = √s (s – a) (s – b) (s – c)

Where, s = semi-perimeter = (a + b + c)/2