Lesson 14

This task is designed to prompt students to recall their prior work with supplementary angles. While they have seen this material in grade 7, this is the first time it has come up explicitly in grade 8. As students work on the task, listen to their conversations specifically for the use of vocabulary such as supplementary and vertical angles. If no students use this language, make those terms explicit in the discussion.

Some students may wish to use protractors, either to double check work or to investigate the different angle measures. This is an appropriate use of technology (MP5), but ask these students what other methods they could use instead.

Provide access to geometry toolkits. Before students start working, make sure they are familiar with the convention for naming an angle using three points, where the middle letter denotes the angle’s vertex.

Display the image for all to see. Invite students to share their responses, adding onto the image as needed to help make clear student thinking. If not mentioned by students, make sure to highlight the term supplementary angles to describe, for example, angles \(FGJ\) and \(JGH\), and vertical angles to describe, for example, angles \(JGF\) and \(HGI\).

In this task, students explore the relationship between angles formed when two parallel lines are cut by a transversal line. Students investigate whether knowing the measure of one angle is sufficient to figure out all the angle measures in this situation. They also consider whether the relationships they found hold true for any two lines cut by a transversal. 

The last two questions in this activity are optional, to be completed if time allows. Make sure to leave enough time for the next activity, “Alternate Interior Angles are Congruent.”

As students work with their partners, they begin to fill in supplementary angles and vertical angles. To find the measures of corresponding and alternate interior, students may use tracing paper and some of the strategies found earlier in the unit. For example, they may use tracing paper to translate vertex \(B\) to vertex \(E\). They might try to translate \(B\) to \(E\) in the third picture and observe that the angles at those two vertices are not congruent. Similarly, to find measures of vertical angles students may use a \(180^\circ\) rotation like they did earlier in this unit when showing that vertical angles are congruent. Monitor for students who use these different strategies and select them to share during the discussion.

For students who finish early, ask them to think of different methods they could use to determine the angles: For example, all of the congruent angles can be shown to be congruent with transformations.

A transversal (or transversal line) for a pair of parallel lines is a line that meets each of the parallel lines at exactly one point. Introduce this idea and provide a picture such as this picture where line \(k\) is a transversal for parallel lines \(\ell\) and \(m\):

Arrange students in groups of 2. Provide access to geometry toolkits. Give students 1 minute of quiet think time to plan on how to find the angle measures in the picture then time to share their plan with their partner. Give partners time for the rest of the task, followed by a whole-class discussion. Instruct students to stop after the third question if you've decided to skip the last two questions.

In the first image, students may fill in congruent angle measurements based on the argument that they look the same size. Ask students how they can be certain that the angles don't differ in measure by 1 degree. Encourage them a way to explain how we can know for sure that the angles are exactly the same measure.

The purpose of this discussion is to make sure students noticed relationships between the angles formed when two parallel lines are cut by a transversal and to introduce the term alternate interior angles to students. Display the images from the Task Statement for all to see one at a time and invite groups to share their responses. Encourage students to use precise vocabulary, such as supplementary and vertical angles, when describing how they figured out the different angle measurements. After students point out the matching angles at the two vertices, define the term alternate interior angles and ask a few students to identify some pairs of angles from the activity.

Consider asking some of the following questions:

• "What were some tools you used to find or confirm angle measures?" (Tracing paper, protractor, transformations)
• "What were some angle relationships you used to find missing measures?" (Vertical angles, supplementary angles)
• "What do you notice about the angles at vertex \(B\) compared to the angles at vertex \(E\)?" (They have the same angle measures for angles in the same position relative to the transversal.)
• "Which angle relationships were true for all three pictures? Which were true for only one or two of the pictures?" (Congruent vertical and supplementary angles around a vertex were always true. Congruent angles in corresponding positions at the two vertices were only true in the first two pictures, which had parallel lines.)

The goal of this task is to experiment with rigid motions to help visualize why alternate interior angles (made by a transversal connecting two parallel lines) are congruent. This result will be used in a future lesson to establish that the sum of the angles in a triangle is 180 degrees. The second question is optional if time allows. This provides a deeper understanding of the relationship between the angles made by a pair of (not necessarily parallel) lines cut by a transversal.

Expect informal arguments as students are only beginning to develop a formal understanding of parallel lines and rigid motions. They have, however, studied the idea of 180 degree rotations in a previous lesson where they used this technique to show that a pair of vertical angles made by intersecting lines are congruent. Consider recalling this technique especially to students who get stuck and suggesting the use of tracing paper.

Given the diagram, the tracing paper, and what they have learned in this unit, students should be looking for ways to demonstrate that alternate interior angles are congruent using transformations. Make note of the different strategies (including different transformations) students use to show that the angles are congruent and invite them to share their strategies during the discussion. Approaches might include:

• A 180 degree rotation about \(M\)
• First translating \(P\) to \(Q\) and then applying a 180-degree rotation with center \(Q\)

Provide access to geometry toolkits. Tell students that in this activity, we will try to figure out why we saw all the matching angles we did in the last activity.

Select students to share their explanations. Pay close attention to which transformations students use in the first question and make sure to highlight different possibilities if they arise. Ask students to describe and demonstrate the transformations they used to show that alternate interior angles are congruent.

Highlight the fact that students are using many of the transformations from earlier sections of this unit. The argument here is especially close to the one used to show that vertical angles made by intersecting lines are congruent. In both cases a 180 degree rotation exchanges pairs of angles. For vertical angles, the center of rotation is the common point of intersecting lines. For alternate interior angles, the center of rotation is the midpoint of the transverse between the two parallel lines. But the structure of these arguments is identical.