Lesson 13
Using Equations to Solve for Unknown Angles
13.1: Is This Enough? (5 minutes)
Warmup
In this activity, students consider whether there is enough information given to solve for the unknown angle measures. In previous lessons, students were given the measures of some angles in a figure and asked to solve for another. In this warmup, the figure contains two unknowns and students are asked to critique Tyler’s thinking (MP3).
The discussion addresses the case in which angles \(a\) and \(b\) are equal to each other, in preparation for future activities in this lesson that have multiple unknown angles with the same measure. Monitor for students who agree and disagree with Tyler’s thinking, and ask them to share during the discussion.
Launch
Arrange students in groups of 2. Give students 1 minute of quiet think time followed by time to discuss their reasoning with their partner. Follow with a wholeclass discussion.
Student Facing
Tyler thinks that this figure has enough information to figure out the values of \(a\) and \(b\).
Do you agree? Explain your reasoning.
Student Response
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Anticipated Misconceptions
Some students may want to use tools from their geometry toolkits to measure the angles. Explain that the question is asking if they can solve the problem by only looking at the figure, not by measuring it.
Activity Synthesis
Poll the class on whether or not they agree with Tyler. Invite students to share their reasoning until they reach an agreement that Tyler is incorrect.
Ask students to come up with an equation to represent the angle measures in the figure. (\(a+90+b=180\) or equivalent) Record their answers for all to see.
Display this image. Invite students to share how this figure is the same as the figure from the task and how it is different.
If students do not mention any of these points, make sure to point them out:
 Some things that are the same are the fact that there are still two angles with unknown measures and the measures of the three angles sum to 180 degrees. The two unknown angles are still complementary.
 The main difference is that the two unknown angles have the same measure.
 This figure can be represented with the equation \(a +90 + a = 180\) or equivalent.
 Because both unknown angles have the same measure, we have enough information to know the value of \(a\).
 \(a=45\)
13.2: What Does It Look Like? (15 minutes)
Activity
The purpose of this activity is for students to practice solving equations that represent relationships between angles, in preparation for the next activity where students will write such equations themselves.
The last three figures include right angles, but they are not marked (except that the task statement says to assume angles that look like right angles are right angles). This may come up in discussion after students have had time to work.
Launch
Tell students that each diagram has two possible equations, and their job is to choose the equation that best represents a relationship between angles in the diagram. Then, solve their chosen equation.
Keep students in the same groups. Give 5 minutes of quiet work time followed by time to discuss reasoning with a partner. Follow with a wholeclass discussion.
Supports accessibility for: Memory; Conceptual processing
Student Facing
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.

Elena: \(x = 35\)
Diego: \(x+35=180\)

Elena: \(35+w+41=180\)
Diego: \(w+35=180\)

Elena: \(w + 35 = 90\)
Diego: \(2w+35=90\)

Elena: \(2w + 35 = 90\)
Diego: \(w+35=90\)

Elena: \(w + 148 = 180\)
Diego: \(x+90=148\)
Student Response
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Anticipated Misconceptions
If students struggle working with equations, encourage them to start with the diagram and label any angles they can figure out with their degree measure. The thinking necessary to figure out the measures of other angles may help them recognize a corresponding equation. Prompt students to recall what it looks like when an angle measures 90 degrees and what it looks like when an angle measures 180 degrees.
Activity Synthesis
Select students to share equations they agreed with and angle measures they found for each problem. As students share their explanations consider asking these questions:
 “Where do you see the relationship expressed in the equation in the given figure? (and vice versa)”
 “Did you and your partner agree on the equations and angle measures?”
For the last question, have students who used different equations to figure out the unknown angle measures share their explanations. Ask students:
 “What angle relationship did you need to recognize to use Elena’s equation?” (That the angle with a measure of \(w\) degrees and the angle measuring 148 degrees were supplementary.)
 “What angle relationship did you need to recognize to use Diego’s equation?” (That the angle measuring 148 degrees formed a vertical angle with the right angle and the angle measuring \(x\) degrees.)
 “Does either method get us the same answer for both unknown angle measures?” (Yes.)
Explain to students that there might be multiple ways to get an answer because of the many angle relationships found in some figures. Encourage them to look for different methods in the next activity.
Design Principle(s): Optimize output (for explanation)
13.3: Calculate the Measure (10 minutes)
Activity
This activity is a culmination of all the work students have done with angles in this unit. With less support than in previous activities, students come up with equations that represent the relationships between angles in a figure. Then, students solve their equation to find each unknown angle measure.
Launch
Encourage students to write an equation for each problem. Give students 2–3 minutes of quiet work time followed by a wholeclass discussion.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Find the unknown angle measures. Show your thinking. Organize it so it can be followed by others.
Student Response
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Student Facing
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 \(a+b+c\).
 Use a protractor to measure the three angles. Use your measurements to conjecture about the value of \(a+b+c\).
 Find the exact value of \(a+b+c\) by reasoning about the diagram.
Student Response
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Anticipated Misconceptions
If students struggle to see the angle relationships in the figures, prompt them to look for any angles that are vertical, complementary, or supplementary to get them started.
Activity Synthesis
The goal of this discussion is for students to see different equations that can be used to represent and solve for the same unknown angle measures.
Select students to share their answers to each problem. Consider asking some of the following questions:
 “Did anyone use a different equation for this same problem? If so, did you get the same answer?”
 “Were any of the questions harder than others? Why?”
 “Were there any questions you used a strategy that was new to you?”
Design Principle(s): Optimize output (for explanation); Support sensemaking
Lesson Synthesis
Lesson Synthesis
 How can equations help us solve for an unknown angle measure? (They allow us to represent relationships among angles. Then we can solve the equation to find the unknown angle measures.)
 Is there only one way to solve for an unknown angle measure? (No, there are usually a few different equations that can be used, based on the relationships present in the figure.)
13.4: Cooldown  In Words (5 minutes)
CoolDown
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Student Lesson Summary
Student Facing
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.
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.
\(\begin{align} 3x + 90 &= 144 \\ 3x + 90  90 &= 144  90 \\ 3x &= 54 \\ 3x \boldcdot \frac13 &= 54 \boldcdot \frac13 \\ x &= 18 \end{align}\)