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 the diagram.

Monitor for students who use these different strategies to find the angle measures:

• Measure every angle using a protractor 
• Measure some angles and use tracing paper to identify congruent angles
• Use what they know about vertical and supplementary angles to identify some angles

Select these students to share during the whole-class discussion. 

The goal of this discussion is for students to describe the relationships they notice between the  angles formed when two parallel lines are cut by a transversal. 

Display the image from the problem for all to see and invite groups to share what they noticed. Encourage students to use precise vocabulary, such as "supplementary angles" and "vertical angles," when describing how they figured out the different angle measurements (MP6). After students point out the matching angles at the two vertices, define the term alternate interior angles: Alternate interior angles are created when two parallel lines are crossed by another line called a transversal. Alternate interior angles are inside the parallel lines and on opposite sides of the transversal.

Ask a few students to identify the pairs of alternate interior angles from the activity.

The goal of this task is for students to connect their work with rigid transformations with the property that alternate interior angles are congruent. In this activity, students use properties of 180-degree rotations to find corresponding angles in the figure. As students describe their rigid transformations to their partner and listen to their partner’s reasoning, they construct and critique the argument that these angles are congruent (MP3).

Listen for different strategies students use to show that the angles are congruent and select these students to share their strategies during the discussion. Approaches might include:

• A 180-degree rotation about
• First translating to and then applying a 180-degree rotation with center

Invite previously selected students to share their explanations. Ask students to describe and demonstrate the transformations they used to show that alternate interior angles are congruent. If any students connect this diagram to earlier work with 180-degree rotations or their justifications that vertical angles are congruent, invite them to share their observations.

Consider displaying this image for all to see as students share their thinking:

If any students use a translation to take to or vice versa then claim that vertical angles are congruent, encourage more precision in their description by asking what rigid transformation tells them that vertical angles are congruent. 

This activity is optional because it provides additional opportunities for students to find supplementary and vertical angles on a diagram. Students also compare their reasoning about alternate interior angles and transversals with a diagram that does not include parallel lines. This allows students to conclude that when two lines which are not parallel are cut by a third line, there are no alternate interior angles formed. 

The goal of this discussion is for students to articulate that for pairs of alternate interior angles to be formed, a transversal must cut two parallel lines. If the lines are not parallel, then we cannot use rigid transformations to show these pairs of angles are congruent.

Display both of these images for all to see:
 

Here are some questions for discussion:

• "What differences do you see between these diagrams?" (One diagram has a set of parallel lines, the other does not have any. One diagram has the same angle measures around point and , but the other has different angle measures around and ).
• "Which of these diagrams has a transversal and alternate interior angles?" (The diagram on the left with the parallel lines.)
• "If you know the angle measure of , what other angles can you determine on each diagram?" (On the left diagram, you can determine all of the angles around point and then use alternate interior angles to determine the angles around point . On the right diagram, you can only know the angles around point .)
   

In this activity, students solve for an unknown angle measure in a multi-step problem but do not initially have enough information to do so. To bridge the gap, they need to exchange questions and ideas.

The Information Gap structure requires students to make sense of a problems by determining what information is necessary, and then to ask for the information they need to solve the problem. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and to ask increasingly more precise questions until they get the information they need (MP6).

After students have completed their work, share the correct answers, and ask students to discuss the process of solving the problems. Here are some questions for discussion:

• “Did you need all of the information from the data card?” (No, it is possible to calculate the angle measure without using all of the information from the data card.)
• “How was your approach to the first card different from your approach to the second card?”

Highlight for students that writing and solving equations is an efficient strategy to show their reasoning about multi-step angle problems.

Math Community
Invite 2–3 students to share the norm they chose and how it supported the work of the group or a realization they had about a norm that would have worked better in this situation. Provide these sentence frames to help students organize their thoughts in a clear, precise way:

• “I picked the norm ‘.’ It really helped me/my group because .”
• “I picked the norm ‘.’ During the activity, I realized the norm ‘’ would be a better focus because .”