# Triangle Inequality Property

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**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

• 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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