Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
The purpose of this Warm-up is to remind students of angle relationships in parallel lines cut by a transversal. In this activity, students have an opportunity to notice and make use of structure (MP7) as they identify congruent angle pairs formed by parallel lines cut by a transversal.
Launch
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Activity
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Student Task Statement
Lines and are parallel. Find the value of in each figure.
Figure A
Figure B
Figure C
Figure D
Activity Synthesis
Ask students to share their strategies for each problem. Record and display their responses for all to see. Encourage students to use precise language to express their ideas. If students do not recall precise phrases, such as “alternate interior angles,” that language can wait until the Activity Synthesis of the subsequent activity.
Math Community
After the Warm-up, display the revised Math Community Chart created from student responses in Exercise 3. Tell students that today they are going to monitor for two things:
“Doing Math” actions from the chart that they see or hear happening.
“Doing Math” actions that they see or hear that they think should be added to the chart.
Provide sticky notes for students to record what they see and hear during the lesson.
During the lesson, students used transformations to create parallel lines and then observed which angles are congruent. But what if they started with parallel lines? Display this image:
Tell students that lines and are parallel, and line is a transversal that intersects them. Ask students to name a pair of corresponding angles. (Possible pairs: and , and , and , and .)
Students will have the opportunity to prove the following theorems in the Cool-down and the converses in the practice problems and section checkpoint. Converse statements are studied in a subsequent lesson. For now, point out that the two statements are related but have different given information.
Distribute new copies of the blackline master Blank Reference Chart. Inform students they will continue to need the first page along with this one, so they should keep the pages together.
Add the following theorems to the class reference chart, and ask students to add it to their reference charts:
Alternate Interior Angle Theorem: If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
Conversely, if two lines are cut by a transversal and alternate interior angles are congruent, then the lines have to be parallel.
(Theorem)
Corresponding Angle Theorem: If two parallel lines are cut by a transversal, then corresponding angles are congruent.
Conversely, if two lines are cut by a transversal and corresponding angles are congruent, then the lines have to be parallel.
(Theorem)
Math Community
Invite 2–3 students to share what “Doing Math” actions they noticed. Record and display their responses for all to see, such as by adding check marks to any already listed items or adding new items near the chart for the class to consider adding. Next, give students 1–2 minutes with a partner to discuss any changes or revisions they think the chart needs. Tell students they can suggest revisions during the Cool-down.
Student Lesson Summary
There are often several different ways to explain why statements are true. Comparing the different ways can lead to new insights or more flexible understanding. Consider the angles formed when 2 parallel lines and are cut by a transversal:
Suppose we want to explain why angle is congruent to angle . Label the midpoint of as . Rotating 180 degrees around takes angle to angle . Why? Well, and are equidistant from , so the rotation takes to . Also, it takes the transversal to itself, so it takes the ray to the ray . Finally, the rotation takes line onto line because 180-degree rotations take lines onto parallel lines, and is the only line parallel to that also goes through .
A different explanation can prove the same fact using a translation and the idea that vertical angles are congruent. Try thinking of that explanation yourself.
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In this activity, students translate one line in a pair of intersecting lines to create parallel lines cut by a transversal. Using the definition and properties of translations, students conclude that pairs of corresponding angles are congruent. Students work with a partner and trade roles explaining their thinking and listening, providing opportunities to explain their reasoning and critique the reasoning of others (MP3).
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Arrange students in groups of 2. Ask students to take turns marking angles as congruent: The first partner identifies a pair of congruent angles and explains why they think the angles are congruent while the other listens and works to understand. Then they switch roles.
Consider providing sentence starters like: Angle _____ is congruent to angle _____ because _____.
Activity
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Student Task Statement
Here are intersecting lines and :
Translate lines and by the directed line segment from to . Label the images of as .
What is true about lines and ? Explain your reasoning.
Take turns with your partner to identify congruent angles.
For each pair of congruent angles that you find, explain to your partner how you know the angles are congruent.
For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
Activity Synthesis
The purpose of discussion is to refine student explanations with more formal language.
Ask students to share their responses. As students share, record what they say by writing a congruence statement () and marking the figure. Insist that whenever the figure is marked with a congruence, students need to write a congruence statement and give a reason that references a definition or properties of translations. If students get stuck when justifying congruence statements, ask them to look for properties of translations in their reference charts for help.
If not mentioned by students, introduce the vocabulary of alternate interior angles and corresponding angles.
MLR1 Stronger and Clearer Each Time. Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their first draft response to “For one pair of congruent angles that you found, explain how you know the angles are congruent.” Invite listeners to ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing. Give students 3–5 minutes to revise their first draft based on the feedback they receive. Advances: Writing, Speaking, Listening
Engagement: Internalize Self-Regulation. Provide students an opportunity to self-assess and reflect on their own progress. For example, ask students how comfortable they are annotating a diagram. Supports accessibility for: Organization, Conceptual Processing
Materials
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Activity Narrative
In this activity, students rotate one line in a pair of intersecting lines by 180 degrees to create parallel lines cut by a transversal. Using the definition and properties of rotations, students conclude that pairs of corresponding angles are congruent. Students work with a partner and trade roles explaining their thinking and listening, providing opportunities to explain their reasoning and critique the reasoning of others (MP3).
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Tell students they will be looking at a similar set-up as in the previous activity, but they will be doing a 180-degree rotation instead. Emphasize that one important property of 180-degree rotations is that they take lines either to themselves if the center of rotation is on the line or to parallel lines if the center of rotation is off the line. Students can verify this experimentally by using tracing paper to rotate line by 180 degrees around various points on line , including , then translating along line until the line returns to where it began.
Add the following assertion to the class reference chart, and ask students to add it to their reference charts:
Rotation by 180 degrees takes lines to parallel lines or to themselves. (Assertion)
Activity Synthesis
The purpose of discussion is to refine student explanations that alternate interior angles are congruent with more formal language.
Ask for students to share their responses. As students share, record what they say by using congruence symbols and marking the figure. Insist that whenever the figure is marked with a congruence, students need to write a congruence statement and give a reason that refers to the definition and properties of translations.
Ask students how this activity is different from the previous activity. (The previous activity used translation, and this one uses rotation.)
Standards Alignment
Building On
8.G.A.5
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
Some students may have difficulty drawing a reasonably accurate image of the figure. Remind them of the tools in their geometry toolkits, such as tracing paper and a straightedge.
Activity
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Student Task Statement
Here are intersecting lines and :
Rotate line by 180 degrees around point . Label the images of as .
What is true about lines and ? Explain your reasoning.
Take turns with your partner to identify congruent angles.
For each pair of congruent angles that you find, explain to your partner how you know the angles are congruent.
For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
Student Response
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Building on Student Thinking
If students struggle to visually estimate the result of the 180-degree rotation, invite them to trace line onto tracing paper, and ask how they will know when they have rotated 180 degrees. Then they can trace the entire diagram and repeat the process.
