Triangle Inequality theorem states that the sum of any two sides of a triangle should always be greater than the measure of the third side. With this law we can say that, a triangle is not formed if the sum of two sides is less than or equal to the third side, thus proving that any given measure for the sides does not form a triangle. For the given 3 side lengths, a triangle is only formed if the sum of any two sides is greater than the third side length.
Example 1: Is a triangle formed with the side lengths 12, 4 and 7?
In order to check if the given side lengths form a triangle or not, we should check if the sides satisfy the
Triangle Inequality Theorem.
This implies: 12 + 4 = 16 > 7 Yes!
12 + 7 = 19 > 4 Yes!
4 + 7 = 11 > 12 No!
Since the theorem is not satisfied when two sides are added in the third case, therefore a triangle cannot be formed with the given side lengths.
Example 2: Is a triangle formed with the side lengths 5, 6 and 10?
Using the Triangle Inequality Theorem, we get: 5 + 6 = 11 > 10 Yes!
5 + 10 = 15 > 6 Yes!
6 + 10 = 16 > 5 Yes!
Since the theorem is satisfied all in all the three cases, sum of any two sides is always greater than the third side, and hence a triangle can be formed with the given side lengths.