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Lesson 5Using Equations to Solve for Unknown Angles

Let’s figure out missing angles using equations.

Learning Targets:

  • I can write an equation to represent a relationship between angle measures and solve the equation to find unknown angle measures.

5.1 Is This Enough?

Tyler thinks that this figure has enough information to figure out the values of a and b .

A straight angle is made of angles a, b, and 90 degrees.

Do you agree? Explain your reasoning.

5.2 What Does It Look Like?

Elena and Diego each wrote equations to represent these diagrams. For each diagram, decide which equation you agree with, and solve it. You can assume that angles that look like right angles are indeed right angles.

  1. Elena: x=35
    Diego: x+35=180
Angles x, w, and 35 degrees
  1. Elena: 35+w+41=180
    Diego: 35+w=180
Angles 35, 41, and w degrees.
  1. Elena:  w + 35 = 90
    Diego: 2w+35=90
Angles w, w, and 35 degrees.
  1. Elena: 2w + 35 = 90
    Diego: w+35=90
Angles w, w, and 35 degrees.
  1. Elena: w + 148 = 180
    Diego: x+90=148
Angles x, w, and 128 degrees

5.3 Calculate the Measure

Find the unknown angle measures. Show your thinking. Organize it so it can be followed by others.


  1. A straight angle is made up of 124 and w degrees.

  1. A straight is formed out of angles that are 52, 23, and x degrees.
  1. Lines \ell and m are perpendicular.
    Two angles with missing values and an angle of 23 degrees.

  1. There is an angle that is 120 degrees. It's opposite angle is made up of two angles labeled as m and one angle of 66 degrees.

Are you ready for more?

The diagram contains three squares. Three additional segments have been drawn that connect corners of the squares. We want to find the exact value of  a+b+c .

  1. Use a protractor to measure the three angles. Use your measurements to conjecture about the value of  a+b+c .
  2. Find the exact value of  a+b+c by reasoning about the diagram.
Angles A, B, and C

Lesson 5 Summary

To find an unknown angle measure, sometimes it is helpful to write and solve an equation that represents the situation. For example, suppose we want to know the value of x in this diagram.

An angle is 144 degree and it forms a straight angle with the missing angle next to it. The undetermined angle creates a right angle with x angles labeled as x degrees.

Using what we know about vertical angles, we can write the equation 3x + 90 = 144 to represent this situation. Then we can solve the equation.

\begin{align} 3x + 90 &= 144 \\ 3x + 90 - 90 &= 144 - 90 \\ 3x &= 54 \\ 3x \boldcdot \frac13 &= 54 \boldcdot \frac13 \\ x &= 18 \end{align}

Lesson 5 Practice Problems

  1. Segments AB , DC , and EC intersect at point C . Angle DCE measures 148^\circ . Find the value of x .

    Angles ACD and BCE are the same. Together with Angle DCE, they form a straight angle. Angle DCE is 148 degrees.

  2. Line \ell is perpendicular to line m . Find the value of x and w .

    A 138 degree and angle w form a straigh angle. Angles W and X create a right angle with the 19 degree angle next to it.

  3. If you knew that two angles were complementary and were given the measure of one of those angles, would you be able to find the measure of the other angle? Explain your reasoning.

  4. For each inequality, decide whether the solution is represented by x < 4.5 or x > 4.5 .

    1. \text-24>\text-6(x-0.5)
    2. \text-8x + 6 > \text-30
    3. \text-2(x + 3.2) < \text-15.4

  5. A runner ran \frac23 of a 5 kilometer race in 21 minutes. They ran the entire race at a constant speed.

    1. How long did it take to run the entire race?
    2. How many minutes did it take to run 1 kilometer?

  6. Jada, Elena, and Lin walked a total of 37 miles last week. Jada walked 4 more miles than Elena, and Lin walked 2 more miles than Jada. The diagram represents this situation:

    Three tape diagrams are labeled “Elena”, “Jada,” and “Lin.” Elena’s tape diagram is of 1 part labeled m. Jada’s tape diagram is partitioned into 2 parts labeled m and 4. Lin’s tape diagram is partitioned into 3 parts labeled m, 4, and 2. A brace is drawn that contains of all 3 diagrams and is labeled 37.

    Find the number of miles that they each walked. Explain or show your reasoning.

  7. Select all the expressions that are equivalent to \text-36x+54y-90 .

    1. \text-9(4x-6y-10)
    2. \text-18(2x-3y+5)
    3. \text-6(6x+9y-15)
    4. 18(\text-2x+3y-5)
    5. \text-2(18x-27y+45)
    6. 2(\text-18x+54y-90)