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!

Triangle Inequalities Triangle Inequalities

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Contents
Table of contents

    Definition of a Triangle Inequality

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

    A triangle inequality is an inequality that is true about any kind of triangle.

    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.

    Triangle Inequalities triangle with corners A, B, and C, but not necessarily a right triangle StudySmarterFig. 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 \(|BC|\).

    Triangle Inequalities triangle with corners A B and C with arcs drawn from corners A and B intersecting at corner C StudySmarterFig. 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.

    Triangle Inequalities triangle with corners A, B, and C, but not necessarily a right triangle StudySmarterFig. 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.

    Triangle Inequalities triangle with interior and exterior angles labeled StudySmarter 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.

    A triangle has the side lengths \(5\) and \(9\), find the possible lengths of the third side of the triangle.

    Answer:

    It always helps to draw a picture first! Below is a picture of a triangle with the three corners and two sides labeled.

    Triangle Inequalities triangle with corners A, B and C, side AC length 9 and side AB length 5 StudySmarterFig. 5. Triangle with two sides labeled.

    The goal is to find possible lengths of side \(BC\). Since you don't have any angles given, the idea is to use the inequality with the length of the sides.

    Let's state the inequalities you can use:

    \[ \begin{align} &|AB| + |BC| >|AC|\\ &|AC| + |AB| > |BC|\\ &|BC| + |AC| > |AB| .\end{align}\]

    Now you can plug in \( |AB| = 5\) and \(|AC| = 9\) into each of the inequalities to get

    \[ \begin{align} &5 + |BC| >9\\ &9 +5 > |BC|\\ &|BC| +9 > 5 .\end{align}\]

    If you do a little algebra to simplify these you can see that

    \[ \begin{align} & |BC| >4\\ &14 > |BC|\\ &|BC| > -4 .\end{align}\]

    The last one, \( |BC| > -4 \) doesn't actually do you much good since you already know that lengths are bigger than zero. But the first two are useful, telling you that \(|BC| > 4\) and \(|BC| < 14\). Putting this into a single inequality,

    \[ 4 < |BC| < 14.\]

    Let's look at another example.

    Suppose angle \(x\) is such that it is the largest angle in the triangle. Can you tell which side is the longest?

    Triangle Inequalities triangle with sides a b and c and an angle labeled x opposite side c StudySmarterFig. 6. Triangle with sides labeled and angle \(x\).

    Answer:

    You can use the triangle sides and angles theorem for this! You know that the longest side is opposite the largest angle. Since angle \(x\) is the largest angle, then side \(c\) is the longest side.

    If the question had asked for which side was the shortest you would have been in trouble because there wouldn't be enough information to answer it.

    Triangle Inequalities - Key takeaways

    • For any triangle, if you add up the length of any two sides, it will be larger than the length of the remaining side. This is the triangle inequality theorem.
    • For any triangle, if one side is longer than another, then their angle opposite the longest side is bigger than the angle opposite the shorter side. This is the angle side triangle theorem.
    • 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.
    Frequently Asked Questions about Triangle Inequalities

    How do you proof triangle inequality?

    Using properties of angles and similar triangles.

    How to calculate triangle inequality?

    This is a tough one to answer, since you don't really calculate triangle inequalities, you use them.

    What is triangle inequality?

    There are lots of them, including the Angle-Side-Angle inequality, the exterior angle inequality, and the one about the length of the sides of a triangle.

    What is the triangle inequality theorem? 

    It says that the sum of any two sides of a triangle is longer than the remaining side.

    What is the formula for calculating triangle inequality?

    The triangle inequality says that the sum of any two sides of a triangle is longer than the remaining side.

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