Fishtank Learning

8th Grade Math
Unit 3: Transformations and Angle Relationships
Lesson 19

Objective

Solve for missing angle measures in parallel line diagrams using equations.

Warm Up

A ray extends from line $$p$$ creating two angles whose measurements are shown below.

Ray from line p forms two angles with measures 5x and 3x + 20.

What are the measures of the two angles?

Anchor Problems - Problem 1

Lines $${L_1 }$$ and $${L_2}$$ are parallel and cut by transversal $$p$$.

$Parallel lines $L_1$ and $L_2$ cut by transversal $p$; angles marked with expressions $4x + 11$ and $8x - 25$.$

What are the measures of the angles shown by the algebraic expressions?

Purpose

The purpose of this problem is for students to combine their understanding of corresponding angles in a parallel line diagram with their skills and knowledge of equations to determine missing angle measures.

Facilitation Guidance

Launch the problem by engaging students in a notice and wonder of the diagram, as well as estimating the measures of the angles marked by expressions. Students may wonder how they can find numerical values for the angle measures if they are only given expressions with unknown values.
- What do you notice about the diagram and the information you are given?
- What do you wonder?
- What is a reasonable estimate for the measures of the two marked angles?
- What is a measure that is too big? What is a measure that is too small?
Raise the question (or re-raise if it was brought up in the launch) of how you can get to a numerical measurement for the angles when the only information shared is in the form of an algebraic expression. Rather than giving students the answer to this, first collect student ideas and guide them towards the conclusion with questioning if needed:
- Start with what you know. What do you know about this line diagram?
- Make a connection to other problems you’ve looked at. How is this similar to other problems you’ve looked at? What did you do in those problems and is there anything that could be applied here?
- Think about your end goal. What are you trying to find out? What is preventing you from finding that information? How can you target that part of the problem?
- Think about the vocabulary and definitions you’ve learned in this topic. How might those apply here?
The goal of this conversation is for students to understand that even if they do not know angle measurements, they do know angle relationships, and they can use that information to represent the situation in a way that further information can be revealed.
Give students time to work on the problem with a partner. As you circulate, check that students accurately represent the angle relationship by setting the two expressions equal to one another. Together, students can apply their equation solving skills to find the value for $$x$$, checking each other’s work as they go.
Debrief the problem by having students share the angle measures they found.
- If students share the value for $$x$$ as their final response, invite other students to offer feedback and see if the students can make an adjustment to their answers.
- Check the angle measure against estimations made in the launch to check for reasonability.

Anchor Problems - Problem 2

Find the measures of the angles marked with expressions in the parallel line diagram below.

Parallel lines with intersecting transversal, angles marked with expressions.

Purpose

This problem extends on Anchor Problem 1 in that it involves different angle relationships (supplementary, vertical, and alternate exterior angles) as well as two different variables. Students think carefully about which angle relationships they are using and how they represent them algebraically.

Facilitation Guidance

Launch the problem by having students look closely at the diagram and observe how it is similar and different from the diagram in Anchor Problem 1.
- How is this diagram similar to the one in Anchor Problem 1? What does that suggest you’ll be able to do?
- How is this diagram different from the one in Anchor Problem 1? How do you think that will impact how you approach this problem?
Put students into new pairs and give them time to work on the problem. Before they jump in, encourage them to make estimations with their partner. As you circulate, look for these potential first steps which may need redirecting:
- Students who use the vertical angle relationship to write an equation. While this is a true relationship and an equation of equality can be created, it does not do much to help solve the problem because two variables are involved. Ask students if this equation helps them determine any new information, and if not, what they could try differently.
- Similar to above, students who use a corresponding angle relationship to write an equation that results in two different variables.
- Students who identify supplementary angles but set them equal to each other rather than adding up to $$180^{\circ}$$. Ask students to describe the relationship to you in words, or ask them what estimations they made for the angles, and if those fit into the equation they created.
Take note of the different approaches and angle relationships you see students using to spotlight in the debrief, as there are various angle relationships students may have chosen to use.
Debrief the problem as a class. Invite student pairs to share their process and equations they used. Invite any specific partner groups that you observed while circulating to share as well. Additional questions to ask include:
- Which angle relationships were helpful in finding the angle measures?
- Did you need to find the values of both $$x$$ and $$y$$? Why or why not?
- If you know one angle measure out of 8 angles in a parallel line diagram like this, can you determine all of the angle measures? Explain.
- What is the relationship between the angle marked $$17x - 31$$ and the angle marked $$10x - 5$$? What does this look like as an equation?

Problem Set

View Problem Set

Target Task

Lines $$b$$ and $$c$$ are parallel cut by transversal line $$a$$. Determine all of the angle measures.

Parallel lines b and c cut by transversal a; angles labeled with expressions involving x.