Ash and his brother Jerry propose to go for a local Football Match. Ash sets out to the ground straight, while Jerry wants to pick up his friend Monika to the match. The topographic positions of Ash's House, Monika's house and the Foot Ball ground are shown below.

Triangle Inequality

If the question is asked who is expected to reach the ground first, Ash or his brother Jerry, then we are intuitive enough to say that Ash will reach the ground first. We expect that the the sum of the distances Jerry needs to cover is longer than the straight length Ash is moving on. The triangle inequality theorem states exactly this fact.
It states that, the sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Triangle Inequality Theorem
AB + BC > AC
BC + AC > AB
AC + AB > BC

Let us now prove this theorem using a construction.

Given: Triangle ABC
Prove: AB + BC > AC
BC + AC > AB
AC + AB > BC
Triangle Inequality Theorem Proof
Statement
Reasons
1. Extend CB to D such that BD = AB
1. Ruler Postulate
2.< DAB ≅ < BDA 2. Isosceles triangle theorem
3. m < DAB = m < BDA 3. Congruent Angles
4. m < DAB + m < BAC > m < BDA 4. Quantity added to one side of the inequality
5. m < DAC > m < BDA 5. Angle Addition Postulate
6. DC > AC 6. Side opposite to the greater angle.
7. DB + BC > AC 7. Segment addition Postulate
8. AB + BC > AC 8. Substitution DB = AB

In a similar manner, the other two inequalities can also be proved.

Hence, the Triangle inequality theorem is proved.

Solved Examples

Question 1: Ruth proposes to draw a triangle, with side lengths 4 in, 6 in and 10 in. Will she able to construct the triangle ? Explain.
Solution:
 
If we take the sum of the two shorter side in Ruth's choice, we get 6 + 4 = 10 in.
   This is equal to the length of the third side.  By triangle inequality theorem the sum of any two sides of a triangle must be greater than the length of the third side. Hence Ruth will not be able to draw this triangle.
 

Question 2: The sides of Δ ABC are marked as follows:

   Triangle Inequality Theorem Example

   Solve a range of values for x, using triangle inequality theorem.
Solution:
 
Applying triangle inequality theorem we get three inequalities
   AB + BC > AC  ⇒    (x + 2) + (3x -2  > x + 3
   BC + AC > AB  ⇒    (3x - 2) + (x + 3) > x + 2
   AC + AB > BC  ⇒    (x + 3) + (x + 2) > 3x - 2
   Solving the inequalities separately we get,
   x + 2 + 3x - 2 > x + 3
                   4x > x + 3      ⇒   3x > 3      ⇒    x > 1      ................(1)
   3x -2 + x + 3 > x +2       
            4x + 1 > x + 2       ⇒  3x > 1       ⇒   x > $\frac{1}{3}$      ................(2)
   x + 3 + x + 2 > 3x - 2
             2x + 5 > 3x - 2     ⇒    7 > x       ⇒   x < 7      .................(3)
Combining the three inequalities, x should satisfy the inequality 1 < x < 7.