**
Triangle Inequality **is
the relation between the sides and angles of triangles which helps us
understand the properties and solutions related to triangles. Triangles are
the most fundamental geometric shape as we can’t make any closed shape with
two or one side. Triangles consist of three sides, three angles, and three
vertices.

The construction possibility of a triangle based on its side is given by the
theorem named** “Triangle
Inequality Theorem.”** The

**states the inequality relation between the triangle’s three sides. In this article, we will explore the Triangle Inequality Theorem and some of its applications as well as the other various inequalities related to the sides and angles of triangles.**

**Triangle Inequality Theorem**
In this article, we’ll delve into the concept of **triangle
inequality, triangle inequality theorem, its significance, and its practical
applications.**

## Table of Content

## What is Triangle Inequality?

**
Triangle Inequality **is
a fundamental geometric principle that plays a vital role in various
mathematical and real-world applications. It lays the foundation for
understanding relationships between the sides of a triangle, contributing to
fields such as geometry, physics, and computer science.

## Triangle Inequality Theorem

**
Triangle Inequality Theorem** states
that “the sum of the length of any two sides of a triangle must be greater
than the length of the third side.” If the sides of a triangle are a, b, and
c then the

**Theorem can be represented mathematically as:**

**Triangle Inequality**- a + b > c,
- b + c > a,
- c + a > b

##
**
****
Triangle Inequality Proof**

**Triangle Inequality Proof**

In this section, we will learn the proof of the triangle inequality theorem. To prove the theorem, assume there is a triangle ABC in which side AB is produced to D and CD is joined.

Notice that the side BA of Δ ABC has been produced to a point D such that AD = AC. Now, since ∠BCD > ∠BDC.

By the properties mentioned above, we can conclude that BD > BC.

We know that, BD = BA + AD

So, BA + AD > BC

= BA + AC > BC

So, this proved sum of two sides triangle is always greater than the other side.

Let’s see an example based on Triangle Inequality Theorem to understand its concept more clearly.

**
Example: D is a point on side BC of triangle ABC such that AD = DC. Show
that AB > BD. **

**
Solution**:

In triangle DAC,

AD = AC,∠ADC = ∠ACD (Angles opposite to equal sides)

∠ ADC is an exterior angle for ΔABD.

∠ ADC > ∠ ABD

⇒ ∠ ACB > ∠ ABC

⇒ AB > AC (Side opposite to larger angle in Δ ABC)

⇒ AB > AD (AD = AC)