Triangle Inequalities

What kinds of things do you know are true about triangles? If you have seen the movie The Wizard of Oz you have certainly heard the Scarecrow recite the Pythagorean theorem! Is that the Pythagorean theorem true for all triangles? It turns out the answer is no, you need it to be a right triangle. But there are some special triangle inequalities that are true for all triangles, with or without a right angle!

Definition of a Triangle Inequality

Let's start by looking at what a triangle inequality actually is.

In this article you will see a couple of different kinds of triangle inequalities, along with examples of each one.

Triangle Inequality Theorem

One of the most important inequality theorems about triangles is that if you add up the length of any two sides, it will be larger than the length of the remaining side.

Let's take a look at a triangle for reference.

Fig. 1. A general triangle.

Remember that the notation for a side of the triangle refers to the corners, so the side connecting points \(A\) and \(B\) would be written \(AB\). The length of \(AB\) is written \(|AB|\). So you can rephrase the triangle inequality theorem as

\[ |AC| + |BC| > |AB|.\]

This is true for any pair of sides, so the triangle inequality theorem also tells you that

\[ |AB| + |AC| > |BC|\]

and

\[ |AB| + |BC| > |AC|.\]

That looks good, but why do you know it is true?

Triangle Inequality Proof

You certainly don't want to have to show all three of those inequalities are true, so the idea is to pick one statement at random and prove that. Then the other two are done in the same way. So let's try and prove that

\[ |AC| + |BC| > |AB|.\]

A geometric proof is the easiest way. Draw two arcs, one from corner \(A\) and one from corner \(B\). The radius of the arc centered at \(A\) is \(|AC|\), and the radius of the arc centered at \(B\) is \(|BC|\).

Fig. 2. Triangle with arcs drawn from corners A and B.

By the way the arcs are drawn, you know that the distance from \(A\) to \(E\) is the same as \(|AC|\), and the distance from \(B\) to \(D\) is the same as \(|BC|\). Looking at the picture you know that

\[ |AD| + |DE| + |EB| = |AB|,\]

so

\[ \begin{align} |AB| &< |AD| + 2|DE| + |EB| \\ &= \left( |AD| + |DE| \right) + \left(|DE| + |EB| \right) \\ &= |AC| + |BC|, \end{align} \]

which is exactly what you were trying to show! Notice we used a neat trick of adding in \(|DE| \) to make the whole thing bigger. This only works because \(|DE| \) is a positive number.

Inequality Theorem About Triangle Sides and Angles

You can also talk about the relationships between the sides and the angles of a triangle. Remember that the measure of an angle is how many degrees (or radians) are in it. The notation might be

• \(m (\angle A)\)

• \(\text{meas }\angle A\), or

• \(\measuredangle A\)

depending on the book you are looking at. Just to be consistent this article uses \( m ( \angle A)\).

Let's go back to the picture of the triangle.

Fig. 3. A general triangle.

The Angle Side theorem says that if one side is longer than another, then their angle opposite the longest side is bigger than the angle opposite the shorter side. In other words:

• if \(|AC| > |AB|\) then \( m(\angle B) > m(\angle C)\);

• if \(|BC| > |AC|\) then \(m( \angle A) > m(\angle B)\);

and so forth for the other comparisons.

Exterior Angle Inequality Theorem

One last triangle inequality theorem, this time involving exterior angles. Let's start with a picture.

Fig. 4. Triangle with interior and exterior angles labeled.

In the picture above, angles \(b\), \(f\), and \(c\) are the interior angles of the triangle, while angles \(a\), \(g\), and \(d\) are the exterior angles.

You also need to know that the two interior angles of a triangle that are not next to a given exterior angle are called the remote interior angles. So for exterior angle \(g\), the remote interior angles are angle \(b\) and angle \(c\).

The exterior angle triangle inequality says that the measure of an exterior angle is bigger than the measure of either of its two remote interior angles. In other words,

\[m(\angle a) > m(\angle c) \text{ and } m (\angle a) > m(\angle f).\]

You can say the same thing for the other two exterior angles of this triangle.

Applications of Triangle Inequalities

Let's take a look at a couple of the ways that triangle inequalities can be used.

Let's look at another example.