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Triangle Sum Theorem

Interior angles add to 180 degrees

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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 run-off 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 120\begin{align*}120^{\circ}\end{align*}, what is the measure of the third angle?

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 180\begin{align*}180^{\circ}\end{align*}.

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 60\begin{align*}60^{\circ}\end{align*} angles. Three 60\begin{align*}60^{\circ}\end{align*} angles are equal to 180\begin{align*}180^{\circ}\end{align*} and there are 180\begin{align*}180^{\circ}\end{align*} in a straight line. The sum of the angles of a triangle is 180\begin{align*}180^{\circ}\end{align*}, and this will happen no matter what the angle measures are. The angles of a triangle will always form a straight line and be equal to 180\begin{align*}180^{\circ}\end{align*}.

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.

25+25+x

Now, we can write this expression into an equation with a sum of 180 degrees. Then we can solve it for the value of x\begin{align*}x\end{align*}.

25+25+x50+xxx=180=180=18050=130

The measure of the missing angle is 130\begin{align*}130^{\circ}\end{align*}.

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 50\begin{align*}50^{\circ}\end{align*}, so we can write a variable expression to help us figure out the measure of the missing base angles.

x+x+50

We can expand this expression to an equation that is equal to 180 degrees.

x+x+50=180

Next, we combine the like terms before solving this.

2x+502xx=180=130=65

Each of the base angles is equal to 65\begin{align*}65^{\circ}\end{align*}.

You can also identify missing angle measures when lines intersect. Take a look.

Find the value of the missing angles x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*}

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 x\begin{align*}x\end{align*}.

You can see that angle x\begin{align*}x\end{align*} is an acute angle. It is also an adjacent angle with the 140 degree angle already labeled. We know that the sum of adjacent angles is 180\begin{align*}180^{\circ}\end{align*}. Now we can write an equation and solve for the missing angle measure.

140+xx=180=40

There are two ways to find the measure of angle y\begin{align*}y\end{align*}. One is to use the sum of the angle measures given that we know the measure of x\begin{align*}x\end{align*}. Let’s do that one first.

y+40+85y+125y=180=180=55

The second way is to use vertical angles. You can see that the angle 125\begin{align*}125^{\circ}\end{align*} is labeled. This means that the angle vertical to this labeled angle is also 125\begin{align*}125^{\circ}\end{align*}. The angle y\begin{align*}y\end{align*} forms a straight line with that angle and therefore is 55\begin{align*}55^{\circ}\end{align*}.

Find each missing angle measure.

Example A

55,35,?\begin{align*}55^{\circ}, 35^{\circ}, ?\end{align*}

Solution:  90\begin{align*}90^{\circ}\end{align*}

Example B

105,25,?\begin{align*}105^{\circ}, 25^{\circ}, ?\end{align*}

Solution:  60\begin{align*}60^{\circ}\end{align*}

Example C

42,15,?\begin{align*}42^{\circ}, 15^{\circ}, ?\end{align*}

Solution:  123\begin{align*}123^{\circ}\end{align*}

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 120\begin{align*}120^{\circ}\end{align*}.

We know that the sum of the three angles of a triangle is equal to 180\begin{align*}180^{\circ}\end{align*}.

Therefore, the measure of the missing angle is 60\begin{align*}60^{\circ}\end{align*}.

Vocabulary

Acute Triangle
A triangle where all three angles are less than 90\begin{align*}90^{\circ}\end{align*}.
Right Triangle
A triangle with one 90\begin{align*}90^{\circ}\end{align*} angle and two acute angles.
Obtuse Triangle
a triangle with one angle that is greater than 90\begin{align*}90^{\circ}\end{align*}.
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.

Practice

Directions: Using what you have learned about the interior angles of a triangle, determine the missing angle in each triangle.

1. 45,45,?\begin{align*}45^{\circ}, 45^{\circ}, ?\end{align*}
2. 60,60,?\begin{align*}60^{\circ}, 60^{\circ}, ?\end{align*}
3. 90,50,?\begin{align*}90^{\circ}, 50^{\circ}, ?\end{align*}
4. 100,40,?\begin{align*}100^{\circ}, 40^{\circ}, ?\end{align*}
5. 110,30,?\begin{align*}110^{\circ}, 30^{\circ}, ?\end{align*}
6. 50,10,?\begin{align*}50^{\circ}, 10^{\circ}, ?\end{align*}
7. 145,15,?\begin{align*}145^{\circ}, 15^{\circ}, ?\end{align*}
8. 55,45,?\begin{align*}55^{\circ}, 45^{\circ}, ?\end{align*}
9. 70,35,?\begin{align*}70^{\circ}, 35^{\circ}, ?\end{align*}
10. 50,50,?\begin{align*}50^{\circ}, 50^{\circ}, ?\end{align*}
11. 63,42,?\begin{align*}63^{\circ}, 42^{\circ}, ?\end{align*}
12. 18,75,?\begin{align*}18^{\circ}, 75^{\circ}, ?\end{align*}

Directions: Identify three triangles in the room around you.

1. .
2. .
3. .

Vocabulary Language: English

Acute Triangle

Acute Triangle

An acute triangle has three angles that each measure less than 90 degrees.
Congruent

Congruent

Congruent figures are identical in size, shape and measure.
Equilateral Triangle

Equilateral Triangle

An equilateral triangle is a triangle in which all three sides are the same length.
Isosceles Triangle

Isosceles Triangle

An isosceles triangle is a triangle in which exactly two sides are the same length.
Obtuse Triangle

Obtuse Triangle

An obtuse triangle is a triangle with one angle that is greater than 90 degrees.
Right Triangle

Right Triangle

A right triangle is a triangle with one 90 degree angle.
Scalene Triangle

Scalene Triangle

A scalene triangle is a triangle in which all three sides are different lengths.
Triangle Sum Theorem

Triangle Sum Theorem

The Triangle Sum Theorem states that the three interior angles of any triangle add up to 180 degrees.