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Angle Measures in Given Triangles

Use equations to find missing angle measures given the sum of 180 degrees.

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Angle Measures in Given Triangles
Source: https://pixabay.com/en/wooden-bridge-nature-wood-768663/
License: CC BY-NC 3.0

Georgianna is hiking with her family and a tour guide. The tour guide draws a triangle to represent the area where the family is safe to travel. Georgiana notices that two of the angles for the triangle are labeled. One angle is 40 degrees and the other angle is 58 degrees. The third angle is missing. What is the value of the third angle?

In this concept, you will learn how to find missing angles in a triangle and how to draw specific types of triangles.

Finding Angle Measures

There are some problem solving aspects of working with triangles. Some of this consists of figuring out missing angles, and some of it concerns drawing specified triangles. Angle measures are important in both of these topics.

The sum of the angles of a triangle is \begin{align*}180^\circ\end{align*}180. This information can help you find a missing angle.

Let's look at an example.

This is a right triangle. One of the angles is equal to 90 degrees. To figure out the measure of the missing angle, use a variable to represent the unknown quantity. Here is the equation.

\begin{align*}55 + 90 + x & = 180 \\ 145 + x & = 180 \\ 180 - 145 & = x \\ x & = 35^\circ .\end{align*}55+90+x145+x180145x=180=180=x=35.

The answer is \begin{align*}35^\circ\end{align*}35.

You can use the angles to draw a specific type of triangle using a ruler and a protractor. You can use the protractor to figure out the measure of an angle and draw in the rest of the triangle with the ruler.

To draw an obtuse triangle, begin by drawing the obtuse angle. Use your protractor to measure an angle that is greater than 90 degrees.

Here is a protractor where an angle that is 105 degrees has been drawn. Next, draw in the rest of the triangle using the ruler.

Here is an obtuse triangle.

Next you can draw an acute triangle. An acute triangle has all three angles that are smaller than 90 degrees. You will need to be careful as you draw and measure this triangle. Let’s begin with one angle.

Draw a  \begin{align*}30^\circ\end{align*}30 angle then draw in the other two angles by making sure that they are less than 90 degrees.

 

Here is an acute triangle.

Examples

Example 1

Earlier, you were given a problem about Georgianna and the triangle.

The triangle has two given angles. If the two angles are 40 degrees and 58 degrees, what is the measure of the missing angle?

First, write the equation.

 \begin{align*}40 + 58 + x = 180\end{align*}40+58+x=180

Then, write the answer.

 \begin{align*}x = 82\end{align*}x=82

The answer is \begin{align*}82^o\end{align*}82o.

Example 2

Look at the following angle sums. Figure out the measure of the missing angle.

\begin{align*}25 + 45 + x = 180^\circ\end{align*}25+45+x=180

First, add up the measures of the two given angles.

\begin{align*}25 + 45 = 70\end{align*}25+45=70

Then, subtract that measure from 180.

\begin{align*}180 - 70 = 130\end{align*}18070=130

The answer is 130 degrees. 

Example 3

Find the missing angle.

First, set up the equation.

 \begin{align*}90 + 20 + x = 180\end{align*}90+20+x=180

Then, solve for the missing angle.

 \begin{align*}x = 70\end{align*}x=70

The answer is \begin{align*}70^o\end{align*}70o.

Example 4

Find the missing angle.

First, set up the equation.

 \begin{align*}71 + 38 + x = 180\end{align*}71+38+x=180

Then, solve for the missing angle.

 \begin{align*}x = 71\end{align*}x=71

The answer is\begin{align*}71^o\end{align*}71o

Example 5

Find the missing angle.

A triangle with the following angles.

\begin{align*}90 + 45 + x = 180^\circ\end{align*}90+45+x=180

First, solve for the missing value.

 \begin{align*}x = 45\end{align*}x=45

Then, state the value.

 \begin{align*}45^o\end{align*}45o

The answer is \begin{align*}45^o\end{align*}45o.

Review

Each question combines information about the angles and side lengths. Answer each question carefully.

  1. True or false. If a triangle is equiangular, it can also be equilateral.
  2. True or false. A scalene triangle can not be an equilateral triangle.
  3. True or false. The word “equiangular” applies to side lengths.
  4. True or false. An isosceles triangle can be an obtuse or acute triangle.
  5. A ___________________ is a tool used to measure angles.
  6. A __________ angle is equal to 90 degrees.
  7. A __________angle is equal to 180 degrees.
  8. An ________ angle is less than 90 degrees.
  9. An _________ angle is greater than 90 but less than 180 degrees.
  10. The prefix “tri” means ______________.
  11. How many angles are there in a triangle?

Look at each and determine the missing angle measure.


  1. \begin{align*}20 + 70 + x = 180^\circ\end{align*}20+70+x=180

  2. \begin{align*}60 + 60 + x = 180^\circ\end{align*}60+60+x=180

  3. \begin{align*}90 + 15 + x = 180^\circ\end{align*}90+15+x=180

  4. \begin{align*}110 + 45 + x = 180^\circ\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 9.9. 

Resources

Vocabulary

Acute Triangle

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

Equilateral Triangle

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

Interior angles

Interior angles are the angles inside a figure.

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.

Protractor

A protractor is a tool used to measure an angle in terms of 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.

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