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.
Supports accessibility for: Memory; Organization
Student Facing
Mentally evaluate all of the missing angle measures in each figure.
Figure A
Figure B
Figure C
Figure D
Student Response
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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?”
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:
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.
Supports accessibility for: Language; Organization
Student Facing
Here is a figure where ray \(r\) meets line \(\ell\). The dashed rays are angle bisectors.
- 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.
- Is the conjecture true? Explain your reasoning.
Student Response
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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.
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.
Supports accessibility for: Social-emotional skills; Organization; Language
Student Facing
Here are 2 intersecting lines that create 2 pairs of vertical angles:
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
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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.
- 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?
- 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?
- 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?
- 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.
- Test your conjecture with 6 points on a circle. How many regions do we get?
Student Response
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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)
Provide the tip: Look for vertical angles whenever two lines intersect.
19.4: Cool-down - Plead Your Case (5 minutes)
Cool-Down
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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:
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:
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.