Triangle Inequality Property

Triangle inequality property : The sum of any two sides of a triangle is greater than the third side.


In triangle ΔABC,
b + c > a

c + a > b

a + b > c

This important property of a triangle is known as Triangle inequality.

Note : In a triangle, the angle opposite the longest side is the largest .

If all the above triangle inequality property satisfied then the triangle is possible.

Examples :

Q.1 State if these numbers could possibly be the lengths of the sides of a triangle.

1) 2,3,4

Solution :
We have,
2 +3 > 4 ; 3 + 4 > 2 and 4 + 2 > 3

Thus, the sum of any two sides of a triangle is greater than the third side.

So, 2,3, 4 are the sides of triangle.

2) 7,3,1

Solution :
We have,
7 +3 > 1 ; 3 + 1< 7 and 1 + 7 > 3

As 3 + 1 < 7

So, the 7,3,1 are not the sides of the triangle.

Q.2 State which angle of the triangle is largest and which angle is smallest.

1) In ΔPQR, PQ = 4 cm; QR = 7 cm and PR = 5 cm.

Solution :
In a triangle, the angle opposite the greatest side is the largest.

Here, QR is the longest side so the angle opposite to it is ∠P.

PQ is the shortest side so the angle opposite to it is ∠R.

Greatest angle = ∠P

Smallest angle = ∠R

2) In ΔABC, AB = 5 cm ; BC = 3 cm and AC = 4 cm.

Solution :
In a triangle, the angle opposite the greatest side is the largest.

Here, AB is the longest side so the angle opposite to it is ∠C.

BC is the shortest side so the angle opposite to it is ∠A.

Greatest angle = ∠C

Smallest angle = ∠A

Q.3 In ΔABC, ∠A = 100
0 ; ∠B = 30 0 and ∠C = 50 0 . Name the shortest and the largest sides of the triangle.

Solution :
As ∠A = 100
0 is the greatest angle so side opposite to angle A is the longest
Longest side = BC

And ∠B = 50
0 is the smallest angle so side opposite to it is the shortest side

Shortest side = AC

Triangles

Introduction to Triangles
Types of Triangles on the basis of Sides
Types of Triangles on the basis of Angles
Angle Sum Property of Triangles
Exterior and Interior angles of Triangle
Triangle Inequality Property
Congruent Triangles
Postulates of Congruent Triangle
Inequality in Triangle

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