Triangles Inequality Theorem

Table of Contents

• Triangle Inequality Theorem
• Application of Triangle Inequality Theorem
• What’s Next?

In the previous segment, we solved an example based on the Angle sum property of a triangle. In this segment, we will use the Triangle inequality theorem to prove an interesting property of a triangle.

What is the Triangle inequality theorem?

The triangle inequality theorem states that ‘the sum of the lengths of any two sides of a

triangle is greater than the third side’.

This means for any triangle ABC,

• AB + BC > CA
• BC + CA > AB
• AB + CA > BC

Application of triangle inequality theorem

Let us now use the triangle inequality theorem to prove that the sum of lengths of the sides of a triangle is greater than twice the length of the segment joining a vertex to the opposite side.

That is, in △ABC, to prove that AB + BC + CA > 2AD.

Figure for Proof

	Statement	Reason
1	In △ABD, AB + BD > AD AD + BD > AB AB + AD > BD	Triangle inequality theorem