# Lesson 10

Parallel Lines in the Plane

## 10.1: Translating Lines (5 minutes)

### Warm-up

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.

### Launch

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.

### Student Facing

- Draw any non-vertical line in the plane. Draw 2 possible translations of the line.
- Find the slope of your original line and the slopes of the images.

### Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

### Activity Synthesis

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.

## 10.2: Priya’s Proof (15 minutes)

### 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.

### Launch

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?

*Action and Expression: Internalize Executive Functions.*Provide students with a graphic organizer in which they can list one thing they notice and one thing that they wonder about the graph.

*Supports accessibility for: Language; Organization*

### Student Facing

Priya writes a proof saying:

Consider any 2 parallel lines. Assume they are not horizontal or vertical. Therefore they must pass through the \(x\)-axis as well as the \(y\)-axis. This forms 2 right triangles with a second congruent angle. Call the angle \(\theta\). The tangent of \(\theta\) is equal for both triangles. Therefore the lines have the same slope.

- How does Priya know the right triangles have a second congruent angle?
- Show or explain what it means that the tangent of \(\theta\) is equal for both triangles.
- How does this prove the slopes of parallel lines are equal?

### Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

### Anticipated Misconceptions

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.

### Activity Synthesis

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*)

## 10.3: Prove Your Parallelogram (15 minutes)

### Activity

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).

### Launch

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

*Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.*Use this routine to help students improve their written responses for the last question. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “What is the definition of a parallelogram?”, “How do you know that the slope of segment \(AB\) is equal to the slope of segment \(DC\)?”, “How do you know that the lengths of segments \(AB\) and \(DC\) are equal?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify why the quadrilateral is a parallelogram.

*Design Principle(s): Optimize output (for justification); Cultivate conversation*

*Representation: Internalize Comprehension.*Activate or supply background knowledge about slope-intercept form. Ask students to describe the line \(y=2x+3\).

*Supports accessibility for: Memory; Conceptual processing*

### Student Facing

- Write the equation of a line parallel to \(y = 2x + 3\), passing through \((\text-4, 1)\).
- Graph both lines described in the previous question.
- Draw a parallelogram using the 2 lines you graphed and using \((\text-4, 1)\) as one of the vertices.
- Prove that your figure is a parallelogram.

### Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

### Student Facing

#### Are you ready for more?

Prove algebraically that the translation \((x,y)\rightarrow (x+p,y+q)\) takes the line \(y=mx+b\) to a line with the same slope.

### Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Extension Student Response.

### Anticipated Misconceptions

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).

### Activity Synthesis

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).

## Lesson Synthesis

### Lesson Synthesis

Display this image for all to see.

Ask students to translate segment \(AC\) by the directed line segment \(CD\). Then ask students to translate segment \(AC\) by the directed line segment \(AB\). What do they notice? (Both translations take \(AC\) to \(BD\).) What does this tell you about quadrilateral \(ABDC\)? (\(AC\) is parallel to \(BD\) because translation takes lines to parallel or coinciding lines. \(AB\) is parallel and congruent to \(CD\) because they both take \(AC\) to the same image. So \(ABDC\) is a parallelogram.) Verify your claim using slope. (Each of the slopes of segments \(AC\) and \(BD\) is equal to \(\frac17\). Each of the slopes of segments \(AB\) and \(CD\) is equal to -4.)

## 10.4: Cool-down - Parallel? (5 minutes)

### Cool-Down

Teachers with a valid work email address can click here to register or sign in for free access to Cool-Downs.

## Student Lesson Summary

### Student Facing

The solid line has been translated in 2 different ways:

- by the directed line segment from \((2,2)\) to \((2,4)\) to produce the dashed line above
- by the directed line segment from \((2,2)\) to \((2,\text-2)\) to produce the dotted line below

The 3 lines look parallel to one another, as we would expect. We know that translations of lines result in parallel lines.

What happens to the slopes of these lines? If we draw in the slope triangles that go through the origin, we can see right triangles. Since we know the lines are parallel, the corresponding pairs of angles in the triangles must be congruent by the Alternate Interior Angles Theorem. Triangles with congruent angles are similar, and similar slope triangles result in lines with the same slope. Here we see slopes of \(\text-\frac{5}{10}, \text-\frac36,\) and \(\text-\frac12\), which are all equal.

We can use similar reasoning to show that any 2 parallel lines that aren’t vertical have the same slope, and also that any 2 lines with the same slope are parallel.

What if we wanted to find the equation of a line parallel to these 3 lines that goes through the point \((6,\text-1)\)? We know the line must have the same slope of \(\text-\frac12\). We can use point-slope form and get \(y+1=\text-\frac{1}{2}(x-6)\).