3.4: Triangle Sum and Exterior Angle Theorems
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.
Interior and Exterior Angles
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.
The angles on the inside of a triangle are called __________________ angles.
The angles on the outside of a triangle are called _________________ angles.
You can see that the words “interior” and “exterior” share the same 6 letters at the end “-terior.” However, they have different prefixes:
“in-” means “inside” and “ex-” means “outside.”
Can you think of some other pairs of words that share an ending but have opposite prefixes “in” and “ex”?
internal and external
include and exclude
... ?
Note: In triangles and other polygons there are TWO sets of exterior angles, one “going” clockwise, and the other “going” counterclockwise.
Clockwise goes around a circle in the same direction as a clock tells time:
Counterclockwise goes around a circle in the opposite direction as a clock tells time:
The following diagram helps you see the difference between counterclockwise exterior angles and clockwise exterior angles on the same triangle:
Exterior angles can be measured in two directions, ______________ and ________________ - __________________.
Linear Pair Postulate
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.
The two exterior angles at the same vertex are _____________________.
Example 1
What is \begin{align*}m \angle RQS\end{align*}
\begin{align*}\text{interior angle} + \text{exterior angle} & \ = \ \ 180^\circ\\
m \angle RQS + 115 & \ = \ \ 180\\
- 115 & \ \ \ - 115\\
m \angle RQS & \ = \ \ 65\\
\text{Thus,} \ m \angle RQS & \ = \ \ 65^\circ\end{align*}
Reading Check:
1. What is \begin{align*}m \angle ABC\end{align*} in the triangle below?
2. What other angle in the triangle above would be congruent to \begin{align*}\angle ABC\end{align*}? Why?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
Triangle Sum Theorem
The sum of the measures of the interior angles in a triangle is \begin{align*}180^\circ\end{align*}.
- All three angles in a triangle add up to ___________________.
Regardless of whether the triangle is right, obtuse, acute, scalene, isosceles, or equilateral, the interior angles will always add up to \begin{align*}180^\circ\end{align*}. Examine each of the triangles shown below.
Notice that each of the triangles has an angle that sums to \begin{align*}180^\circ\end{align*}.
\begin{align*}100^\circ + 40^\circ + 40^\circ &= 180^\circ\\ 90^\circ + 30^\circ + 60^\circ &= 180^\circ\\ 45^\circ + 75^\circ + 60^\circ &= 180^\circ\end{align*}
Reading Check:
1. True or false: The interior angles of only right triangles sum to \begin{align*}180^\circ\end{align*}.
\begin{align*}{\;}\end{align*}
2. True or false: The interior angles of scalene triangles are equiangular and sum to \begin{align*}180^\circ\end{align*}.
\begin{align*}{\;}\end{align*}
3. Draw a triangle below and label each of the three interior angles. Show your work to make sure your angles add up to \begin{align*}180^\circ\end{align*}.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}\underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} + \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} + \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} = 180^\circ\end{align*}
You can also use the Triangle Sum Theorem to find a missing angle in a triangle. Set the sum of the angles equal to \begin{align*}180^\circ\end{align*} and solve for the missing value.
Example 2
What is \begin{align*}m \angle T\end{align*} in the triangle below?
Set up an equation where the three angle measures sum to \begin{align*}180^\circ\end{align*}. Then, solve for \begin{align*}m \angle T\end{align*}.
\begin{align*}82^\circ + 43^\circ + m \angle T & \ = \ \ 180^\circ\\ 125 + m \angle T & \ = \ \ 180\\ - 125 & \ \ \ - 125\\ m \angle T & \ = \ \ 55\\ \text{Thus,} \ m \angle T & \ = \ \ 55^\circ\end{align*}
Reading Check:
What is \begin{align*}m \angle Y\end{align*} in the triangle below?
Acute Angles in a Right Triangle
Expanding on the Triangle Sum Theorem, you can find more specific relationships. For example, in any right triangle, by definition, one of the angles will measure \begin{align*}90^\circ\end{align*}. This means that the sum of the other two angles will always be \begin{align*}90^\circ\end{align*}, resulting in a total sum of \begin{align*}180^\circ\end{align*} (since \begin{align*}90 + 90 = 180\end{align*}).
Therefore, the two acute angles in a right triangle will always be complementary.
- In a right triangle, the two acute angles will add up to _______________.
Example 3
What is the measure of the missing angle \begin{align*}g\end{align*} in the triangle below?
Since the triangle above is a right triangle, the two acute angles must be complementary, which means their sum will be \begin{align*}90^\circ\end{align*}. We will represent the missing angle with the variable \begin{align*}g\end{align*} and write an equation.
The two acute angles are __________________ and __________________ so:
\begin{align*}38^\circ + g = 90^\circ\end{align*}
Now we can use inverse operations to isolate the variable, and then we will have the measure of the missing angle.
\begin{align*}& \ \quad 38 + g \ = \ 90\\ & -38 \qquad \ \ -38\\ & \qquad \quad \ \ g \ = \ 52\end{align*}
The measure of the missing angle \begin{align*}g\end{align*} is \begin{align*}52^\circ\end{align*}
Reading Check:
1. True or false:
In a right triangle, the right angle is \begin{align*}90^\circ\end{align*} and the other two angles are equiangular.
2. Why are the two acute angles in a right triangle complementary but not supplementary? Explain.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
3. What is the measure of the missing angle \begin{align*}a\end{align*} in the triangle below?
Exterior Angles in a Triangle
Recall that the exterior and interior angles around a single vertex create a linear pair, so they add up to \begin{align*}180^\circ\end{align*} as shown below.
\begin{align*}120^\circ + 60^\circ = 180^\circ\end{align*}
A linear pair describes two angles that add up to ________________.
Imagine an equilateral triangle and the exterior angles it forms, like in the diagram below.
Since each interior angle measures \begin{align*}60^\circ\end{align*} in an equilateral triangle, each exterior angle will measure \begin{align*}120^\circ\end{align*}.
- Every interior angle in an equilateral (or equiangular) triangle equals ______________.
- Every exterior angle in an equilateral (or equiangular) triangle equals ______________.
What is the sum of the three exterior angles? Add them to find out:
\begin{align*}120^\circ + 120^\circ + 120^\circ = 360^\circ\end{align*}
The sum of these three exterior angles is \begin{align*}360^\circ\end{align*}.
The sum of the exterior angles in any triangle will always be equal to \begin{align*}360^\circ\end{align*}.
- For any triangle, all exterior angles will add up to ________________.
You can use this information just as you did the Triangle Sum Theorem to find missing angles and measurements.
As a review,
The Triangle Sum Theorem says that all _________________ angles in a triangle add up to \begin{align*}180^\circ\end{align*}.
We just discovered that all __________________________ angles in a triangle add up to \begin{align*}360^\circ\end{align*}.
Example 4
What is the value of the angle \begin{align*}P\end{align*} in the triangle below?
You can set up an equation relating the three exterior angles to \begin{align*}360^\circ\end{align*}. Notice that \begin{align*}p\end{align*} is an interior angle in the triangle, not an exterior angle, so be careful with how you set up this equation! Solve for the value of the exterior angle. Let's call the measure of the exterior angle \begin{align*}e\end{align*}.
Label \begin{align*}e\end{align*} in the diagram above.
Now, set up an equation with the sum of the exterior angles equal to \begin{align*}360^\circ\end{align*}:
\begin{align*}& 130^\circ + 110^\circ + e \ = \ 360^\circ\\ & \quad \ \ 240^\circ + e \quad \ \ = \ 360^\circ\\ & \ - 240^\circ \qquad \qquad - 240^\circ\\ & \qquad \qquad \qquad e \ = \ 120^\circ\end{align*}
The missing exterior angle measures \begin{align*}120^\circ\end{align*}. However, this is not your final answer! You have one more step to find the value of \begin{align*}p\end{align*}.
You can use \begin{align*}e\end{align*} (that you found above) to find \begin{align*}p\end{align*} because the interior and exterior angles (angle \begin{align*}e\end{align*} and angle \begin{align*}p\end{align*}) form a linear pair.
Linear pairs add up to _______________ so:
\begin{align*}& \quad \ 120^\circ + p \ = \ 180^\circ\\ & - 120^\circ \qquad \ \ - 120^\circ\\ & \qquad \qquad \ p \ = \ \ 60^\circ\end{align*}
Your final answer for the measure of angle \begin{align*}p\end{align*} is \begin{align*}60^\circ\end{align*}!
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.
Look at the diagram from the previous example for a moment. If we focus on the exterior angle at angle \begin{align*}D\end{align*}, then the interior angles at angles \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are called remote interior angles. Every exterior angle has 2 remote interior angles that correspond to it:
The \begin{align*}120^\circ\end{align*} exterior angle at \begin{align*}\angle D\end{align*} above has 2 remote ____________________________ angles, one at angle __________ and the other at angle __________.
Notice that the exterior angle at point \begin{align*}D\end{align*} measured \begin{align*}120^\circ\end{align*}. At the same time, the interior angle at point \begin{align*}A\end{align*} measured \begin{align*}70^\circ\end{align*} and the interior angle at point \begin{align*}B\end{align*} measured \begin{align*}50^\circ\end{align*}.
The sum of interior angles \begin{align*}m \angle A + m \angle B = 70^\circ + 50^\circ = 120^\circ\end{align*}
Notice the measures of the remote interior angles sum to the measure of the exterior angle at point \begin{align*}D\end{align*}.
This relationship is always true, and it is a result of the Linear Pair Postulate and the Triangle Sum Theorem.
- The measure of an exterior angle is equal to the ______________ of the measures of its ______________________ interior angles.
Reading Check:
1. To what angle measure do all three interior angles in a triangle sum?
2. To what angle measure do all three exterior angles in a triangle sum?
3. Name the two remote interior angles to the exterior angle at point \begin{align*}H\end{align*} in the diagram below.
4. Find the measure of the exterior angle at point \begin{align*}H\end{align*} in the diagram above.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
5. When used as an adjective, the word “remote” means “removed in space, time, or relation.” Based on this definition, why do you think that word is used to identify a particular set of interior angles in a triangle?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
6. Can you think of another way to find the measure of the exterior angle at point \begin{align*}H\end{align*}? Describe your step-by-step method:
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
Graphic Organizer for Lesson 2
Angles | Draw a picture of the angles | What do these angles add up to? |
---|---|---|
All three interior angles in a triangle | ||
All three clockwise exterior angles of a triangle | ||
All three counter clockwise exterior angles of a triangle | ||
The two acute angles in a right triangle | ||
One interior angle in a triangle and the exterior angle at the same vertex | ||
Two remote interior angles in a triangle |
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