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
The sum of consecutive interior and exterior angle is supplementary
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
