Theorems of Inequality

Instructor: Usha Bhakuni

Usha has taught high school level Math and has master's degree in Finance

In this lesson, we will learn about the geometrical inequality relationships that exist between the sides and angles of a triangle, with the help of illustrative images.

Triangles and Inequalities

Triangles are one of the most common geometrical shapes found around us. They consist of three sides containing three angles within. So, let's explore some geometrical theorems related to inequalities in triangles.

Exterior Angle Inequality Theorem

An exterior angle of a triangle is the angle formed between any side of the triangle and an external extension of the adjacent side. There are six external angles formed in a triangle, two at each of the vertexes.

Exterior Angle

The exterior angle inequality theorem states that the measure of any exterior angle of a triangle is greater than both of the non-adjacent interior angles. This rule is satisfied by all the six external angles of a triangle.

Exterior Angle Inequality theorem

In the image above, we can see that angle ACD is an external angle. So,

Exterior Angle Inequality


Exterior Angle Inequality

Triangle Inequality Theorem

A triangle cannot be formed by just any set of three random lines. All triangles must observe the triangle inequality theorem. It states that the sum of lengths two sides of the triangle will always be greater than the length of the third side. This rule is satisfied by all the three sides of a triangle. It means if we have the lengths of two sides, we can safely bet that the length of the third side will be smaller than the sum of the two sides.

triangle inequality theorem

In the image, there are three inequalities,

triangle inequality theorem

triangle inequality theorem

triangle inequality theorem

Let's try to draw an imaginary triangle with sides 3cm, 4cm, and 10cm. We can see that 4 + 10 > 3, 3 + 10 > 4, but 4 + 3 < 10. Therefore, the sides with these dimensions do not follow the triangle inequality theorem. Hence, a triangle like this cannot exist.

Angle-Side Relationship

In a triangle, the angle opposite a longer side will be greater than the angle opposite a shorter side. Let's look at the angle-side relationship in a triangle that has three unequal sides, with AC being the smallest and BC being the longest.

angle side relationship

In this image, there are three inequalities that exist-

angle-side relationship

angle-side relationship

angle-side relationship

To unlock this lesson you must be a Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use

Become a member and start learning now.
Become a Member  Back
What teachers are saying about
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account