Unit 1, Lesson 21
One Hundred Eighty
Geometry
21.1
Warm-up
Instructional Routines
NoneMaterials
NoneActivity Narrative
The purpose of this activity is for students to analyze a proof and decide whether each step makes sense. By finding the error in this proof, students reinforce their understanding that corresponding angles are only congruent when the given lines are parallel.
Launch
Arrange students in groups of 4, and assign a different statement to each student in a group. Give students quiet work time to decide if their statement is true. Next, invite students to share their responses with their group before discussing the circumstances that make corresponding angles congruent. Follow with whole-class discussion.
Activity
NoneHere are 2 lines and that are not parallel and have been cut by a transversal.
Tyler thinks angle is congruent to angle because they are corresponding angles and a translation along the directed line segment from to would take one angle onto the other. Here are his reasons.
- The translation takes onto , so the image of is .
- The translation takes somewhere on ray because it would need to be translated by a distance greater than to land on the other side of .
- The image of has to land somewhere on line because translations take lines to parallel lines and line is the only line parallel to that goes through .
- The image of , call it , has to land on the right side of line or else line wouldn’t be parallel to the directed line segment from to .
- Your teacher will assign you one of Tyler’s statements to think about. Is the statement true? If not, explain your reasoning.
- In what circumstances are corresponding angles congruent? Be prepared to share your reasoning.
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of discussion is to highlight the fact that the Corresponding Angle Theorem is only true when the given lines are parallel. Ask students to pinpoint exactly where the argument breaks down in the situation where the two given lines are not parallel. (Tyler’s third statement) Ask students to share what circumstances ensure corresponding angles are congruent and explain their reasoning. (Only parallel lines with a transversal produce congruent corresponding angles, as we saw with rotation and transformation proofs.)
Lesson Synthesis
In this lesson, students explored two different proofs of the Triangle Angle Sum Theorem. Add the following theorem to the class reference chart, and ask students to add it to their reference charts:
Triangle Angle Sum Theorem: The three angle measures of any triangle always sum to 180 degrees.
(Theorem)
Here are some questions for discussion:
- “Look back at the two proofs of the Triangle Angle Sum Theorem. Are there any parts of the argument that depend on the particular measurements of a triangle, or would the same arguments have worked with other kinds of triangles?” (The same argument would work for any triangle. The particular measurements didn't matter.)
- “How are the two proofs of the Triangle Angle Sum Theorem different? How are they the same?” (Both activities place the three angles adjacent to each other to make a straight line. The proofs are different because the first proof used alternate interior angles and the second proof used translations.)
Tell students, “We proved alternate interior angles are congruent using transformations, so really both proofs use transformations. Once you’ve added a theorem to the reference chart, you can just use that theorem rather than restating the transformations every time.”
Student Lesson Summary
Using rotations and parallel lines, we can understand why the angles in a triangle always add to 180 degrees. Here is triangle .
Rotate triangle 180 degrees around the midpoint of segment , and label the image of as . Then rotate triangle 180 degrees around the midpoint of segment , and label the image of as .
Note that each 180-degree rotation takes line to a parallel line. So line is parallel to , and line is also parallel to . There is only one line parallel to that goes through point , so lines and are the same line. Since line is parallel to line , we know that alternate interior angles are congruent. That means that angle also measures , and angle also measures .
Since is a line, the 3 angle measures at point must sum to 180 degrees. So . This argument does not depend on the triangle we started with, so that proves the sum of the 3 angle measures of any triangle is always 180 degrees.