# Applying Inequalities to Triangles

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• 0:00 Inequalities and Triangles
• 1:25 Problem 1
• 2:35 Problem 2
• 3:50 Problem 3
• 5:10 Lesson Summary
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Lesson Transcript
Instructor: Elizabeth Foster

Elizabeth has been involved with tutoring since high school and has a B.A. in Classics.

In this lesson, you'll apply what you know about inequalities to solving problems about triangles. Learn how to compare side lengths and angle measures without knowing the precise numbers.

## Inequalities and Triangles

Here's a math problem you might have encountered in real life (see video).

John is walking to the gym. John can choose one of two walking routes, either the orange path or the purple path. John is late for his personal trainer, so he wants to get to the gym as fast as possible. Which route should he take? Purple, right? But how did you know, since there aren't any numbers on the picture? How do you know it's shorter?

Probably because you used an inequality. You don't know the exact measure of the two paths, but you can see that the orange path and the purple path together make a triangle. The orange path represents the two shorter legs of the triangle, and the purple path represents the hypotenuse. You know that the hypotenuse is always shorter than the sum of the legs, so the purple distance will always be shorter, regardless of what the distances are specifically. If John wanted a workout, he should have taken the orange path, but to get to his trainer as fast as possible, the purple is the way to go.

This lesson is about problems like that: inequalities with triangles. Instead of figuring out precise side lengths or angle measures, you have to look at two angles or two sides and decide which is bigger and which is smaller, or whether they're both equal. Let's get into it!

## Problem 1

Fill in the appropriate inequality symbol in the red box to describe the relationship between angle B and angle C (see video).

To solve this one, you have to know about the relationships between the sides and the angles of a triangle. In any triangle, the length of each side is proportional to the size of the angle opposite it. So the biggest angle is opposite the biggest side, and the smallest angle is opposite the smallest side.

That's how you can solve this one. Even if you don't know the precise measures of every angle, it doesn't matter because you know the lengths of the sides opposite them. So you know that angle A must be the biggest because it's opposite the longest side. Angle B is in the middle, and angle C is the smallest because angle B is opposite the middle side, and angle C is opposite the smallest side.

So you would fill in the greater than symbol in the box, because angle B is greater than angle C. It doesn't matter what the angles are exactly, only which one is bigger.

## Problem 2

Time for something a little tougher? Here's your second problem (see video). Again, your job is to fill in the box to show the correct relationship between angle A and angle B.

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