Geometry (A): READ: Triangle Sums Reading

Triangle Sums

Learning Objectives

Identify interior and exterior angles in a triangle.
Understand and apply the Triangle Sum Theorem.
Utilize the complementary relationship of acute angles in a right triangle.
Identify the relationship of the exterior angles in a triangle.

Introduction

In the first chapter of this course, you developed an understanding of basic geometric principles. The rest of this course explores specific ideas, techniques, and rules that will help you be a successful problem solver. If you ever want to review the basic problem solving in geometry return to Chapter 1. This chapter explores triangles in more depth. In this lesson, you’ll explore some of their basic components.

Interior and Exterior Angles

Any closed structure has an inside and an outside. In geometry we use the words interior and exterior for the inside and outside of a figure. An interior designer is someone who furnishes or arranges objects inside a house or office. An external skeleton (or exo-skeleton) is on the outside of the body. So the prefix “ex” means outside and exterior refers to the outside of a figure.

The terms interior and exterior help when you need to identify the different angles in triangles. The three angles inside the triangles are called interior angles. On the outside, exterior angles are the angles formed by extending the sides of the triangle. The exterior angle is the angle formed by one side of the triangle and the extension of the other.

Note: In triangles and other polygons there are TWO sets of exterior angles, one “going” clockwise, and the other “going” counterclockwise. The following diagram should help.

But, if you look at one vertex of the triangle, you will see that the interior angle and an exterior angle form a linear pair. Based on the Linear Pair Postulate, we can conclude that interior and exterior angles at the same vertex will always be supplementary. This tells us that the two exterior angles at the same vertex are congruent.

Example 1

What is $m \angle{RQS}$ in the triangle below?

The question asks for $m \angle{RQS}$ . The exterior angle at vertex $\angle{RQS}$ measures $115^\circ.$ Since interior and exterior angles sum to $180^\circ,$ you can set up an equation.

$\text{interior angle} + \text{exterior angle} & = 180^\circ \\ m \angle{RQS} + 115 &=180 \\ m \angle{RQS} + 115 - 115 &=180-115\\ m\angle{RQS} &= 65$

Thus, $m\angle{RQS}=65^{\circ}$ .

Triangle Sum Theorem

Probably the single most valuable piece of information regarding triangles is the Triangle Sum Theorem.

Triangle Sum Theorem

The sum of the measures of the interior angles in a triangle is $180^\circ$

Regardless of whether the triangle is right, obtuse, acute, scalene, isosceles, or equilateral, the interior angles will always add up to $180^\circ.$ Examine each of the triangles shown below.

Notice that each of the triangles has an angle that sums to $180^\circ.$

$100^\circ + 40^\circ + 40^\circ & = 180^\circ \\ 90^\circ + 30^\circ +60^\circ &= 180^\circ \\ 45^\circ + 75^\circ + 60^\circ &= 180^\circ$

You can also use the triangle sum theorem to find a missing angle in a triangle. Set the sum of the angles equal to $180^\circ$ and solve for the missing value.

Example 2

What is $m\angle{T}$ in the triangle below?

Set up an equation where the three angle measures sum to $180^\circ.$ Then, solve for $m \angle{T}$ .

$82 + 43 + m \angle{T} & = 180\\ 125 + m \angle{T} & = 180\\ 125 -125 + m \angle{T} & = 180-125\\ m \angle{T} & = 55$

$m \angle{T} = 55^\circ$

Now that you have seen an example of the triangle sum theorem at work, you may wonder, why it is true. The answer is actually surprising: The measures of the angles in a triangle add to $180^\circ$ because of the Parallel line Postulate. Here is a proof of the triangle sum theorem.

Given: $\triangle {ABC}$ as in the diagram below,
Prove: that the measures of the three angles add to $180^\circ,$ or in symbols, that $m \angle{1} + m \angle{2} + m \angle{3} = 180^\circ$ .

Statement	Reason
1. Given $\triangle {ABC}$ in the diagram	1. Given

2. Through point $B$ , draw the line parallel to $\overline{AC}$ . We will call it $\overleftrightarrow{BD}.$	2. Parallel Postulate

3. $\angle{4} \cong \angle{1}\; (m \angle{4} = m \angle{1})$	3. Alternate interior Angles Theorem
4. $\angle{5} \cong \angle{3}\; (m \angle{5} = m \angle{3})$	4. Alternate interior Angles Theorem
5. $m \angle{4} + m \angle{2} = m \angle{DBC}$	5. Angle Addition postulate
6. $m \angle{DBC} + m \angle{5} = 180^\circ$	6. Linear Pair Postulate
7. $m \angle{4} + m \angle{2} + m \angle{5} = 180^\circ$	7. Substitution (also known as “transitive property of equality”)
8. $m \angle{1} + m \angle{2} + m \angle{3} = 180^\circ$	8. Substitution (Combining steps 3, 4, and 7).

And that proves that the sum of the measures of the angles in ANY triangle is $180^\circ.$

Acute Angles in a Right Triangle

Expanding on the triangle sum theorem, you can find more specific relationships. Think about the implications of the triangle sum theorem on right triangles. In any right triangle, by definition, one of the angles is a right angle—it will always measure $90^\circ.$ This means that the sum of the other two angles will always be $90^\circ,$ resulting in a total sum of $180^\circ.$

Therefore the two acute angles in a right triangle will always be complementary and as one of the angles gets larger, the other will get smaller so that their sum is $90^\circ$ .

Recall that a right angle is shown in diagrams by using a small square marking in the angle, as shown below.

So, when you know that a triangle is right, and you have the measure of one acute angle, you can easily find the other.

Example 3

What is the measure of the missing angle in the triangle below?

Since the triangle above is a right triangle, the two acute angles must be complementary. Their sum will be $90^\circ.$ We will represent the missing angle with the variable $g$ and write an equation.

$38^\circ + g = 90^\circ$

Now we can use inverse operations to isolate the variable, and then we will have the measure of the missing angle.

$38+g&=90\\ 38 + g - 38 &= 90-38^\circ\\ g&= 52$

The measure of the missing angle is $52^\circ.$

Exterior Angles in a Triangle

One of the most important lessons you have learned thus far was the triangle sum theorem, stating that the sum of the measure of the interior angles in any triangle will be equal to $180^\circ.$ You know, however, that there are two types of angles formed by triangles: interior and exterior. It may be that there is a similar theorem that identifies the sum of the exterior angles in a triangle.

Recall that the exterior and interior angles around a single vertex sum to $180^\circ,$ as shown below.

Imagine an equilateral triangle and the exterior angles it forms. Since each interior angle measures $60^\circ,$ each exterior angle will measure $120^\circ.$

What is the sum of these three angles? Add them to find out.

$120^\circ + 120^\circ + 120^\circ = 360^\circ$

The sum of these three angles is $360^\circ.$ In fact, the sum of the exterior angles in any triangle will always be equal to $360^\circ.$ You can use this information just as you did the triangle sum theorem to find missing angles and measurements.

Example 4

What is the value of $p$ in the triangle below?

You can set up an equation relating the three exterior angles to $360^\circ.$ Remember that $p$ does not represent an exterior angle, so do not use that variable. Solve for the value of the exterior angle. Let's call the measure of the exterior angle $e$ .

$130^\circ + 110^\circ + e &= 360^\circ\\ 240^\circ + e &= 360^\circ\\ 240^\circ + e - 240^\circ &= 360^\circ - 240^\circ\\ e &=120^\circ$

The missing exterior angle measures $120^\circ.$ You can use this information to find the value of $p$ , because the interior and exterior angles form a linear pair and therefore they must sum to $180^\circ.$

$120^\circ + p &= 180^\circ\\ 120^\circ + p - 120^\circ & = 180^\circ -120^\circ\\ p &= 60^\circ$

Exterior Angles in a Triangle Theorem

In a triangle, the measure of an exterior angle is equal to the sum of the remote interior angles.

We won’t prove this theorem with a two-column proof (that will be an exercise), but we will use the example above to illustrate it. Look at the diagram from the previous example for a moment. If we look at the exterior angle at $D$ , then the interior angles at $A$ and $B$ are called “remote interior angles.”

Notice that the exterior angle at point $D$ measured $120^\circ.$ At the same time, the interior angle at point $A$ measured $70^\circ$ and the interior angle at $B$ measured $50^\circ.$ The sum of interior angles $m\angle{A}+m\angle{B}=70^\circ + 50^\circ = 120^\circ$ . Notice the measures of the remote interior angles sum to the measure of the exterior angle at $D$ . This relationship is always true, and it is a result of the linear pair postulate and the triangle sum theorem. Your job will be to show how this works.

Lesson Summary

In this lesson, we explored triangle sums. Specifically, we have learned:

How to identify interior and exterior angles in a triangle.
How to understand and apply the Triangle Sum Theorem
How to utilize the complementary relationship of acute angles in a right triangle.
How to identify the relationship of the exterior angles in a triangle.

These skills will help you understand triangles and their unique qualities. Always look for triangles in diagrams, maps, and other mathematical representations.

Points to Consider

Now that you understand the internal qualities of triangles, it is time to explore the basic concepts of triangle congruence.

Last modified: Monday, June 28, 2010, 2:19 PM