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 = 1000; ∠B = 300 and ∠C = 500. Name the shortest and the largest sides of the triangle.

Solution :
As ∠A = 1000 is the greatest angle so side opposite to angle A is the longest
Longest side = BC

And ∠B = 500 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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