Lesson 10

Students begin connecting their geometric transformation understanding of parallel lines with slope by calculating the slope of translated lines. Identify students with different starting slopes so the class can see the slope remains constant for positive, negative, steep, and shallow lines.

Arrange students in groups of 2. Tell students there are many possible answers for the questions. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.

The purpose of the discussion is to elicit the conjecture that slopes of parallel lines are equal. Select students with positive, negative, steep, and shallow slopes to share. Here are some questions for discussion:

• “What do you notice about the slopes of the translated lines?” (The slopes stayed the same when the lines were translated.)
• “What do you know about the 3 lines you drew?” (They are parallel because translations take lines to parallel lines or the same line.)
• “Make a conjecture about slopes of parallel lines.” (They appear to be equal.)

Tell students that they will prove this conjecture in the next activity.

In this activity, students provide reasons for a proof that parallel lines must have equal slopes. To prepare them for this reasoning, students begin by noticing and wondering about an image of parallel lines on a coordinate plane. Identify students who annotate their image.

Display the graph for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion. Move through this relatively quickly so students have time to engage in the activity.

Things students may notice:

• There are 2 parallel lines.
• There are 4 angles marked congruent.
• The slopes of the lines are positive.
• There are 2 triangles that are shaded differently.

Things students may wonder:

• Why are the triangles shaded differently?
• Are the triangles similar?
• What are the slopes of the lines?

If students are stuck, suggest they mark angle \(\theta\) in the diagram. They can also consider shading the 2 triangles and labeling the lengths of the legs of the triangles with arbitrary letters so it’s easier to talk about them.

Invite students to share their reasoning in response to each question. Highlight any students who annotated their image to make their explanations clearer.

Ask students, “What if the lines are horizontal?” (All horizontal lines are parallel to each other and have a slope of 0, so the idea of parallel lines having equal slopes applies to this situation.) “What if the lines are vertical?” (Slopes of vertical lines are undefined, so the equal slopes criterion does not apply to them.)

Tell students that the converse is also true: if 2 lines have equal slopes, then they are parallel. Ask students to add this theorem to their reference charts as you add it to the class reference chart:

Lines are parallel if and only if they have equal slopes. (Theorem)

Students apply the theorem they proved in the previous activity to problems. First, students write an equation of a parallel line by recognizing the slope must remain constant. Identify students who write the equation in different forms. Next, students prove a quadrilateral is a parallelogram. Most students will use slope in their proof but monitor for students who use any other valid proof.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Distribute graph paper to students. Remind students they can use any form to write equations of lines.

Students can look in their reference chart for ways to prove a quadrilateral is a parallelogram.

If students aren’t sure how to graph lines, remind them they can use slope triangles or they can find intercepts to help graph the line. Ask how many points are needed to draw the graph of a line (just 2).

Invite previously selected students to share their equations. Ask students to justify that all of their answers are correct despite looking different.

Invite previously selected students to share their proofs. Start with the most common proof (probably two pairs of parallel sides), then share alternate approaches (two pairs of congruent sides, diagonals bisect each other, translation).