6.6: Understanding the Angle Measures of Triangles
Have you ever seen a house being built? Take a look at this dilemma.
The roof construction of this home is using a triangle for stability. It also allows for water runoff during rain or when snow melts. Using triangles makes a lot of sense.
If two of these triangles have the same measure? Do you know what kind of triangle it is? If the sum of two of the angles is equal to
Understanding angle measures will help you to figure out this dilemma. You will see it again at the end of the Concept.
Guidance
If you look at all of the angles in a triangle, you will notice something consistent about each one of them. If we add up the number of degrees in each angle of a triangle, you will see that the sum of the angle measures is equal to
That is a great question. That short answer is, yes, it is always true. But let’s look at an example to understand this a little further.
We can start by looking at an equilateral triangle. The three angles of an equilateral triangle are all equal. Each of these angles is 60 degrees. Here is an equilateral triangle.
Now let’s look at what happens when we cut out the
Now let’s look at how we can use this information to find missing angle measures.
What is the missing angle measure of this triangle?
Now we have a triangle here.
First, we can use our known information to figure out what kind of triangle it is.
Let’s begin by looking at the angles in this triangle. There are two small angles. These are 25 degree angles and they are acute. We can see by looking at this third unknown angle, that is an obtuse angle. This is an obtuse triangle.
Next, we can write an expression to help us to figure out the missing measure.
Now, we can write this expression into an equation with a sum of 180 degrees. Then we can solve it for the value of
The measure of the missing angle is
Let's look at another one.
What is the measure of the two missing angles if this is an isosceles triangle?
Here we have two missing angles. We know from the problem that this is an isosceles triangle. That means that side lengths are the same and we can see that the two base angles are also congruent. Our given angle is
We can expand this expression to an equation that is equal to 180 degrees.
Next, we combine the like terms before solving this.
Each of the base angles is equal to
You can also identify missing angle measures when lines intersect. Take a look.
Find the value of the missing angles
Now to work through this problem, you will need to apply all of the things that you have learned to problem solve the measures of the missing angles.
Let’s start by looking at angle
You can see that angle
There are two ways to find the measure of angle
The second way is to use vertical angles. You can see that the angle
Find each missing angle measure.
Example A
Solution:
Example B
Solution:
Example C
Solution:
Now let's go back to the dilemma from the beginning of the Concept.
The sum of two of the angles of the triangle is equal to
We know that the sum of the three angles of a triangle is equal to
Therefore, the measure of the missing angle is
Vocabulary
 Acute Triangle

A triangle where all three angles are less than
90∘ .
 Right Triangle

A triangle with one
90∘ angle and two acute angles.
 Obtuse Triangle

a triangle with one angle that is greater than
90∘ .
 Equilateral Triangle
 all three side lengths and all three angles are congruent.
 Isosceles Triangle
 two side lengths are the same.
 Scalene Triangle
 all three side lengths are different
 Congruent
 means exactly the same, having the same measure.
Guided Practice
Here is one for you to try on your own.
Think about architecture. Why do you think triangles are used in designs? Take a look at this bridge. See if you can understand why triangles are often a fundamental part of architectural design.
Solution
This is a truss bridge.
Looking at this bridge, you can see how the basic shape of a triangle is fundamental to the design of the bridge. The triangle helps to keep the bridge stable because of the strength of its foundation. The triangle is a shape that because of its base is very stable and won’t give to pressure. It is a balanced figure.
Video Review
Practice
Directions: Using what you have learned about the interior angles of a triangle, determine the missing angle in each triangle.

45∘,45∘,? 
60∘,60∘,? 
90∘,50∘,? 
100∘,40∘,? 
110∘,30∘,? 
50∘,10∘,? 
145∘,15∘,? 
55∘,45∘,?  \begin{align*}70^{\circ}, 35^{\circ}, ?\end{align*}
 \begin{align*}50^{\circ}, 50^{\circ}, ?\end{align*}
 \begin{align*}63^{\circ}, 42^{\circ}, ?\end{align*}
 \begin{align*}18^{\circ}, 75^{\circ}, ?\end{align*}
Directions: Identify three triangles in the room around you.
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Term  Definition 

Acute Triangle  An acute triangle has three angles that each measure less than 90 degrees. 
Congruent  Congruent figures are identical in size, shape and measure. 
Equilateral Triangle  An equilateral triangle is a triangle in which all three sides are the same length. 
Isosceles Triangle  An isosceles triangle is a triangle in which exactly two sides are the same length. 
Obtuse Triangle  An obtuse triangle is a triangle with one angle that is greater than 90 degrees. 
Right Triangle  A right triangle is a triangle with one 90 degree angle. 
Scalene Triangle  A scalene triangle is a triangle in which all three sides are different lengths. 
Triangle Sum Theorem  The Triangle Sum Theorem states that the three interior angles of any triangle add up to 180 degrees. 
Image Attributions
Here you'll understand the angle measures of triangles and that the sum of the interior angles of a triangle is equal to @$\begin{align*}180^{\circ}\end{align*}@$.