Lesson 19

Evidence, Angles, and Proof

19.1: Math Talk: Supplementary Angles (5 minutes)

Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for determining the angle measures in pairs of intersecting lines or for pairs of angles that make a straight angle. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to explain why vertical angles are congruent. In this activity, students have an opportunity to notice and make use of structure (MP7) when they identify supplementary angles.

Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Facing

Mentally evaluate all of the missing angle measures in each figure.

Figure A

Line EB with ray AD roughly at the midpoint of line FD at point A. Angle DAB is 50 degrees.

Figure B

Lines IG and HF intersect at point E, angle GEF is 30 degrees.

Figure C

Line FD with ray CE roughly at the midpoint of line FD at point C. Angle ECD is 65 degrees.

Figure D

Lines DC and BE intersect at point A, angle CAE is 135 degrees.

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

  • “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
  • “Did anyone have the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
  • “Do you agree or disagree? Why?”
Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . ."  Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. 
Design Principle(s): Optimize output (for explanation)

19.2: That Can’t Be Right, Can It? (15 minutes)

Activity

The purpose of this activity is for students to take an informal conjecture and describe it more precisely by labeling a figure. Students begin by describing three examples of angle bisectors and forming a conjecture. By engaging with this explicit prompt to take a step back and become familiar with a context and the mathematics that might be involved, students are making sense of problems (MP1). When students start out labeling the diagram they may use a variety of methods to make sense and explain. The ways students attempt to formulate the conjecture more precisely will be refined in the discussion.

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

Launch

Display three examples of angle bisectors of linear pairs for all to see:

3 supplementary angles bisected.

Ask students, “What do you notice? What do you wonder?”

Things students may notice:

  • there are solid and dashed lines
  • there is a horizontal line in all three
  • the two solid angles make a linear pair

Things students may wonder:

  • Are the dashed lines angle bisectors?
  • Do the dashed lines make a right angle?

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. If the conjecture that the angle between the angle bisectors is always a right angle does not come up during the conversation, ask students to discuss this idea.

If students have access to GeoGebra Geometry from Math Tools, suggest that it might be a helpful tool in this activity.

Action and Expression: Internalize Executive Functions. Provide students with a table to record what they notice and wonder prior to being expected to share these ideas with others. 
Supports accessibility for: Language; Organization

Student Facing

Here is a figure where ray \(r\) meets line \(\ell\). The dashed rays are angle bisectors.

Ray r meets line l, 2 dashed rays form angle bisectors.
  1. Diego made the conjecture: “The angle formed between the angle bisectors is always a right angle, no matter what the angle between \(r\) and \(\ell\) is.” It is difficult to tell specifically which angles Diego is talking about in his conjecture. Label the diagram and rephrase Diego’s conjecture more precisely using your labels.
  2. Is the conjecture true? Explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students get stuck, ask them to estimate the measure of one angle and then make arguments based on angle measure like in the warm-up. If time allows, invite students to generalize.

Activity Synthesis

The purpose of this discussion is to introduce the concept of marking angles as congruent, labeling points, and labeling angles.

Ask students to share convincing arguments why Diego’s conjecture is true or false. Label an image displayed for all to see with the information they provide. Students need not write a formal proof at this point, but encourage students to rephrase using more precise language.

Build on the ideas students have shared about how they labeled the figure to introduce conventions about marking angles as congruent, labeling points, and labeling variable angle measures. For an example of one way to mark the figure, see the sample student response.

Writing, Speaking: MLR 1 Stronger and Clearer Each Time. Use this with successive pair shares to give students a structured opportunity to revise and refine their response to “For each pair of congruent angles that you find, explain to your partner how you know the angles are congruent.” Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help teams strengthen their ideas and clarify their language (e.g., "Can you explain how…?" "You should expand on...," etc.). Students can borrow ideas and language from each partner to strengthen the final product. 
Design Principle(s): Optimize output (for generalization)

19.3: Convince Me (15 minutes)

Activity

The purpose of this activity is for students to prove that vertical angles are congruent and work towards a more formal, rigorous way of expressing themselves when giving arguments based on rigid transformations. As students work, remind them to label points and make markings on the diagram to help in the process of explaining their ideas.

Monitor for arguments based on:

  • transformations
  • supplementary angles 

Launch

Display two intersecting lines for all to see. Remind students that the pairs of angles opposite the intersection point are called vertical angles. 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 whole-class discussion.

Engagement: Internalize Self Regulation. Demonstrate giving and receiving constructive feedback. Use a structured process and display sentence frames to support productive feedback. For example, “How do you know…?,” “That could/couldn’t be true because…,” and “We can agree that.…”
Supports accessibility for: Social-emotional skills; Organization; Language

Student Facing

Here are 2 intersecting lines that create 2 pairs of vertical angles:

Intersecting lines.

1. What is the relationship between vertical angles? Write down a conjecture. Label the diagram to make it easier to write your conjecture precisely.

2. How do you know your conjecture is true for all possible pairs of vertical angles? Explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

One reason mathematicians like to have rigorous proofs even when conjectures seem to be true is that sometimes conjectures that are made turn out to not be true. Here is one famous example. If we draw \(n\) points on a circle and connect each pair of points how many regions does that divide the circle into? If we draw only 1 point there are no line segments to connect and so just 1 region in the circle. If we draw 2 points they are connected by a line segment which divides the circle into 2 regions.

  1. If we draw 3 points on a circle and connect each pair of points with a line segment how many regions do we get in our circle?
  2. If we draw 4 points on a circle and connect each pair of points with a line segment how many regions do we get in our circle?
  3. If we draw 5 points on a circle and connect each pair of points with a line segment how many regions do we get in our circle?
  4. Make a conjecture about how many regions we get if we draw \(n\) points on a circle and connect each pair of points with a line segment.
  5. Test your conjecture with 6 points on a circle. How many regions do we get?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students are stuck, suggest they label one of the acute angles as \(x^\circ\). Ask what else they can label or figure out based on that information.

Activity Synthesis

The purpose of discussion is to refine students’ arguments into convincing proofs. With input from students, label points on the figure so that everyone can discuss the same objects consistently. Ask previously identified students to share their transformational argument. Then invite previously identified students to share their supplementary angles argument.

Ask if either argument relied on the specifics of the particular angles given. Tell students that a proof has to work for any angle measure, otherwise it's an example.

Students will be writing an explanation that vertical angles are congruent in the cool-down, so be sure students understand at least one of these arguments.

Lesson Synthesis

Lesson Synthesis

Tell students “In everyday life, it is often complicated to understand all of the reasons why statements are true or false. For example, think about the economic system, international relations, or the history of a country. In geometry, the objects of study are not as complex: angles, lines, points, triangles, and so on. In this way, geometry is a great training ground for understanding the reasons why ideas are true and communicating those reasons to others.”

In the previous activity, students came up with different explanations for why vertical angles are congruent. Discuss the difference between arguments based on angle measure and arguments based on transformations. Ask students,

  • “Which argument makes more sense to you, rigid transformations that take one vertical angle onto the other, or using straight angles to look at 180 degree sums?” (Transformations make more sense because it’s possible to see how one angle is taken onto the other. I like algebra better so I prefer the 180 degree sums.)
  • “What is the difference between angle and angle measure?” (It’s like the difference between a segment and its length. Segments and angles are geometric figures, but lengths and angle measures are numbers used to describe how large or small the segments and angles are.)

Ask students to add this theorem to their reference charts as you add it to the class reference chart:

Vertical angles are congruent.

(Theorem)

Two intersecting lines with vertical angles marked.

Provide the tip: Look for vertical angles whenever two lines intersect.

19.4: Cool-down - Plead Your Case (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

In many situations, it is important to understand the reasons why an idea is true. Here are some questions to ask when trying to convince ourselves or others that a statement is true:

  • How do we know this is true?
  • Would these reasons convince someone who didn’t think it was true?
  • Is this true always, or only in certain cases?
  • Can we find any situations where this is false?

In this lesson, we reasoned that pairs of vertical angles are always congruent to each other:

Intersecting lines forming a pair of congruent vertical angles.

We saw this by labeling the diagram and making precise arguments having to do with transformations or angle relationships. For example, label the diagram with points:

Lines CD and AB intersect at E, from the top and clockwise forming angles y degrees, w degrees, z degrees and x degrees. Angles x degrees and w degrees are congruent.

Rotate the figure 180 degrees around point \(E\). Then ray \(EA\) goes to ray \(EB\) and ray \(ED\) goes to ray \(EC\). That means the rotation takes angle \(AED\) onto angle \(BEC\), and so angle \(AED\) is congruent to angle \(BEC\).

Many true statements have multiple explanations. Another line of reasoning uses angle relationships. Notice that angles \(AED\) and \(AEC\) together form line \(CD\). That means that \(x + y = 180\). Similarly, \(y + w = 180\). That means that both \(x\) and \(w\) are equal to \(180-y\), so they are equal to each other. Since angle \(AED\) and angle \(CEB\) have the same degree measure, they must be congruent.