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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


and


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


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