In this section, we are going to see "The Triangle Inequality Theorem" which can be used to know whether a triangle actually exists.

**Triangle Inequality Theorem : **

The sum of the lengths of any two sides of a triangle is greater than the third side.

**Example : **

A triangle with the side lengths 5 cm, 6 cm and 4 cm actually exists.

Because, sum of the lengths of any two sides is greater than the third side.

That is, 5 cm + 6 cm > 4 cm.

6 cm + 4 cm > 5 cm

5 cm + 4 cm > 6 cm

The diagram given below illustrates the theorem :

**Problem 1 : **

State if the three numbers given below can be the measures of the sides of a triangle.

8, 12 and 9

**Solution :**

According to the theorem explained above, if the sum of the lengths of any two sides is greater than the third side, then the given sides will form a triangle.

Let us apply the theorem for the given numbers.

8 + 12 > 9

12 + 9 > 8

8 + 9 > 12

Because the given numbers meet the condition said in the theorem, the numbers 8, 12 and 9 can be the measures of the sides of a triangle.

**Problem 2 : **

State if the three numbers given below can be the measures of the sides of a triangle.

10, 7 and 13

**Solution :**

According to the theorem explained above, if the sum of the lengths of any two sides is greater than the third side, then the given sides will form a triangle.

Let us apply the theorem for the given numbers.

10 + 7 > 13

7 + 13 > 10

10 + 13 > 7

Because the given numbers meet the condition said in the theorem, the numbers 10, 7 and 13 can be the measures of the sides of a triangle.

**Problem 3 : **

State if the three numbers given below can be the measures of the sides of a triangle.

6, 12 and 3

**Solution :**

According to the theorem explained above, if the sum of the lengths of any two sides is greater than the third side, then the given sides will form a triangle.

Let us apply the theorem for the given numbers.

6 + 12 > 3

12 + 3 > 6

6 + 3 < 12 (Does not satisfy the theorem)

Because the given numbers do not meet the condition said in the theorem, the numbers 6, 12 and 3 can not be the measures of the sides of a triangle.

**Problem 4 : **

Two sides of a triangle have the measures 6 and 7. Find the range of possible measures for the third side.

**Solution :**

Let "x" be the length of the third side of the triangle.

Sum of the lengths of the given two sides :

6 + 7 = 13

Because the sum of the lengths of the two sides 6 and 7 is 13, the maximum length of the third side must be less than 13.

That is

x < 13 -----(1)

Let us find the minimum value of "x".

According to the theorem, we must have

x + 6 > 7

x + 7 > 6

To satisfy both the inequalities above, the value of "x" must be greater than 1.

That is

x > 1 (or) 1 < x -----(2)

From (1) and (2), the range of "x" is

1 < x < 13

**Problem 5 : **

Find the range of possible measures of x in the following given sides of a triangle :

10, 7, x

**Solution :**

Sum of the lengths of the given two sides :

10 + 7 = 13

Because the sum of the lengths of the two sides 10 and 7 is 17, the value of "x" must be less than 17.

That is

x < 17 -----(2)

Let us find the minimum value of "x".

According to the theorem, we must have

x + 10 > 7

x + 7 > 10

To satisfy both the inequalities above, the value of "x" must be greater than 3.

That is

x > 3 (or) 3 < x -----(2)

From (1) and (2), the range of "x" is

3 < x < 17

**Shortcut : **

For better understanding, problem 4 and 5 have been explained in detail.

But there is a shortcut to find the range of possible measures for the third side.

**Problem 4 : **

Lengths of the given two sides are 6 and 7.

Difference of the lengths = 7 - 6 = 1

Sum of the lengths = 7 + 6 = 13

Hence, the range of possible measures for the third side is

1 < x < 13

**Problem 5 : **

Lengths of the given two sides are 10 and 7.

Difference of the lengths = 10 - 7 = 3

Sum of the lengths = 10 + 7 = 17

Hence, the range of possible measures for the third side is

3 < x < 17

After having gone through the stuff given above, we hope that the students would have understood "Triangle inequality theorem".

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