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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