Geometry (A): READ: Triangle Inequality Reading

Inequalities in Triangles

Learning Objectives

Determine relationships among the angles and sides of a triangle.
Apply the Triangle Inequality Theorem to solve problems.

Introduction

In this lesson we will examine the various relationships among the measure of the angles and the lengths of the sides of triangles. We will do so by stating and proving a few key theorems that will enable us to determine the types of relationships that hold true.

Look at the following two triangles

We see that the first triangle is isosceles while in the the second triangle, $\overline{DE}$ is longer than $\overline{AB}$ . How are the measures of the angles at $C$ and $F$ related to the lengths of $\overline{AB}$ and $\overline{DE}?$ It appears (and, in fact it is the case) that the measure of the angle at vertex $F$ is larger than angle $C$ .

In this section we will formally prove theorems that reveal when such relationships hold. We will start with the following theorem.

Relationship Between the Sides and Angles of a Triangle

Theorem: If two sides of a triangle are of unequal length, then the angles opposite these sides are also unequal. The larger side will have a larger angle opposite it.

Larger angle has longer opposite side: If one angle of a triangle has greater measure than a second angle, then the side opposite the first angle is longer than the side opposite the second angle.

Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Proof. Consider $\triangle ABC$ . We must show the following:

$AB + BC > AC$
$AC + BC > AB$
$AB + AC > BC$

Suppose that $\overline{AC}$ is the longest side. Then statements 2 and 3 above are true.

In order to prove 1. $AB+BC>AC$ , construct the perpendicular from point $B$ to $X$ on the opposite side as follows:

Now we have two right triangles and can draw the following conclusions:

Since the perpendicular segment is the shortest path from a point to a line (or segment), we have $\overline{AX}$ is the shortest segment from $A$ to $\overline{XB}$ . Also, $\overline{CX}$ is the shortest segment from $C$ to $\overline{XB}$ . Therefore $AB > AX$ and $BC > CX$ and by addition we have

$AB + BC > AX + XC = AC.$

So, $AB + BC > AC$ . $\blacklozenge$

Example 1

Can you have a triangle with sides having lengths $4, 5, 10$ ?

Without a drawing we can still answer this question—it is an impossible situation, we cannot have such a triangle. By the Triangle Inequality Theorem, we must have that the sum of lengths of any two sides of the triangle must be greater than the length of the third side. In this case, we note that $4 + 5 = 9 < 10$ .

Example 2

Find the angle of smallest measure in the following triangle.

$\angle{B}$ has the smallest measure. Since the triangle is a right triangle, we can find $x = 6$ using the Pythagorean Theorem (which we will prove later).

By the fact that the longest side is opposite the largest angle in a triangle, we can conclude that $m\angle{B}< m\angle{A}< \angle{C}$ .

Lesson Summary

In this lesson we:

Stated and proved theorems that helped us determine relationships among the angles and sides of a triangle.
Introduced the method of indirect proof.
Applied the Triangle Inequality Theorem to solve problems.

Points to Consider

Knowing these theorems and the relationships between the angles and sides of triangles will be applied when we use trigonometry. Since the size of the angle affects the length of the opposite side, we can show that there are specific angles associated with certain relationships (ratios) between the sides in a right triangle, and vice versa.

The following questions are for your own review. The answers are listed below to help you check your work and understanding.

Review Questions

Name the largest and smallest angles in the following triangles:
Name the longest side and the shortest side of the triangles.
Is it possible to have triangles with the following lengths? Give a reason for your answer.
1. $6, 13, 6$
2. $8, 9, 10$
3. $7, 18 ,11$
4. $3, 4, 5$
Two sides of a triangle have lengths $18$ and $24$ . What can you conclude about the length of the third side?
The base of an isosceles triangle has length $30$ . What can you say about the length of each leg?

In exercises 6 and 7, find the numbered angle that has the largest measure of the triangle.

In exercises 8-9, find the longest segment in the diagram.

Review Answers

1. $\angle{R}$ is largest and $\angle{S}$ is smallest.
2. $\angle{C}$ is largest and $\angle{A}$ is smallest.
1. $\overline{AC}$ is longest and $\overline{AB}$ is the shortest.
2. $\overline{BC}$ is longest and $\overline{AB}$ is the shortest.
1. No, $6 + 6 = 12 < 13$ .
2. Yes
3. No, $7 + 11 = 18$ .
4. Yes
The third side must have length $x$ such that $6 < x < 42$ .
The legs each must have length greater than $15$ .
$\angle{2}$
$\angle{1}$
$\overline{DF}$
$\overline{XY}$

Last modified: Friday, November 12, 2010, 11:56 AM