*Usha Bhakuni*Show bio

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

Lesson Transcript

Instructor:
*Usha Bhakuni*
Show bio

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

In trigonometry, inequality theorems describe the relationships between the sides and angles of a triangle. Discover the theorems of inequality, including the theorems of exterior angle inequality, triangle inequality, and angle-side relationship.
Updated: 12/14/2021

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.

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 vertices.

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.

In the image appearing on your screen, we can see that âˆ ACD is an external angle. So the measure of âˆ ACD > âˆ CAB and the measure of âˆ ACD > âˆ CBA.

A triangle can't 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 of 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.

In the image on your screen, there are three inequalities:

- AB + BC > AC
- AB + AC > BC
- BC + AC > AB

Let's try to draw an imaginary triangle with sides of 3 cm, 4 cm, and 10 cm. We can see that 4 + 10 > 3, 3 + 10 > 4, but 4 + 3 < 10. Therefore, the sides with these dimensions don't follow the triangle inequality theorem. This is why we can confidently say that a triangle like this can't exist.

In a triangle, the largest side in a triangle will be opposite to the largest angle and the shortest side will be opposite to the smallest angle, known as the **angle-side relationship**. 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.

In the image on your screen, there are three inequalities that exist:

- âˆ A > âˆ B
- âˆ A > âˆ C
- âˆ C > âˆ B

The converse of this relationship is also true, i.e., the side opposite a greater interior angle of a triangle will be greater than the side opposite a smaller angle. See the triangle appearing on your screen with âˆ A > âˆ B > âˆ C.

The following inequalities will exist here:

- âˆ A > âˆ B â†’ BC > AC
- âˆ B > âˆ C â†’ AC > AB
- âˆ A > âˆ C â†’ BC > AB

In short, the largest side in a triangle will be opposite to the largest angle and vice-versa.

All right, let's now take a moment to review what we learned in this lesson about theories of inequality. As we learned, there are three important theorems of inequality that dictate geometric rules about triangles specifically.

First, the exterior angle in a triangle is greater than either of the non-adjacent interior angles, according to the **exterior angle inequality theorem**. Remember that the **exterior angle of a triangle** is the angle formed between any side of the triangle and an external extension of the adjacent side. Second, the **triangle inequality theorem** states that the sum of the lengths of any two sides of a triangle will be greater than the length of the third side. And third, in the **angle-side relationship**, the largest side in a triangle will be opposite to the largest angle and the shortest side will be opposite to the smallest angle. All the three sides and angles of every triangle must satisfy these rules.

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