Lesson 21

One Hundred and Eighty

21.1: What Went Wrong? (10 minutes)

Warm-up

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.

Student Facing

Here are 2 lines \(\ell\) and \(m\) that are not parallel that have been cut by a transversal.

Lines AF labeled l and CD labeled m are not parallel. l and m are intersected by transversal line CE at points B and C.

Tyler thinks angle \(EBF\) is congruent to angle \(BCD\) because they are corresponding angles and a translation along the directed line segment from \(B\) to \(C\) would take one angle onto the other. Here are his reasons.

  • The translation takes \(B\) onto \(C\), so the image of \(B\) is \(C\).
  • The translation takes \(E\) somewhere on ray \(CB\) because it would need to be translated by a distance greater than \(BC\) to land on the other side of \(C\).
  • The image of \(F\) has to land somewhere on line \(m\) because translations take lines to parallel lines and line \(m\) is the only line parallel to \(\ell\) that goes through \(B’\).
  • The image of \(F\), call it \(F’\), has to land on the right side of line \(BC\) or else line \(FF’\) wouldn’t be parallel to the directed line segment from \(B\) to \(C\).
  1. Your teacher will assign you one of Tyler’s statements to think about. Is the statement true? Explain your reasoning.
  2. In what circumstances are corresponding angles congruent? Be prepared to share your reasoning.

Student Response

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

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.)

21.2: Triangle Angle Sum One Way (15 minutes)

Activity

In grade 8, students used informal arguments to prove the Triangle Angle Sum Theorem. This activity revisits that work and takes it further by using more rigorous definitions and careful reasoning. Students should make use of their reference charts.

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

Launch

Remind students of the meaning of the Triangle Angle Sum Theorem by displaying a large triangle on tracing paper with angle measures labeled \(a^\circ\), \(b^\circ\), and \(c^\circ\) for all to see. Tear off the angles and rearrange them to form a straight line. Ask what the sum of the three angle measures is, given that they form a straight line. (180 degrees.)

Representation: Internalize Comprehension. Begin the activity with concrete or familiar contexts. For example, provide students with a triangle that has known angle measurements such as \(50^\circ\), \(60^\circ\), and \(70^\circ\). Ask students to complete the task with this triangle. Then, ask students to consider the relationship between the angles formed by the parallel lines and each transversal.
Supports accessibility for: Conceptual processing; Memory

Student Facing

  1. Use technology to create a triangle. Use the Text tool to label the 3 interior angles as \(a\), \(b\), and \(c\).
  2. Mark the midpoints of 2 of the sides.
  3. Extend the side of the triangle without the midpoint in both directions to make a line.
  4. Use what you know about rotations to create a line parallel to the line you made that goes through the opposite vertex.
  5. What is the value of \(a+b+c\)? Explain your reasoning.

Student Response

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

Launch

Remind students of the meaning of the Triangle Angle Sum Theorem by displaying a large triangle on tracing paper with angle measures labeled \(a^\circ\), \(b^\circ\), and \(c^\circ\) for all to see. Tear off the angles and rearrange them to form a straight line. Ask what the sum of the three angle measures is, given that they form a straight line. (180 degrees.)

Representation: Internalize Comprehension. Begin the activity with concrete or familiar contexts. For example, provide students with a triangle that has known angle measurements such as \(50^\circ\), \(60^\circ\), and \(70^\circ\). Ask students to complete the task with this triangle. Then, ask students to consider the relationship between the angles formed by the parallel lines and each transversal.
Supports accessibility for: Conceptual processing; Memory

Student Facing

  1. Use a straightedge to create a triangle. Label the 3 angle measures as \(a^\circ\), \(b^\circ\), and \(c^\circ\).
  2. Use paper folding to mark the midpoints of 2 of the sides.
  3. Extend the side of the triangle without the midpoint in both directions to make a line.
  4. Use what you know about rotations to create a line parallel to the line you made that goes through the opposite vertex.
  5. What is the value of \(a+b+c\)? Explain your reasoning.

Student Response

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

Anticipated Misconceptions

Some students may get stuck finding the measures of the missing angles. Direct those students to their reference charts.

Activity Synthesis

The important idea is that, no matter what triangle students started with, the sum of the measures of the three angles is always 180 degrees. Select several student responses to display for all to see. Here are some questions for discussion:

  • “What is the same in these figures? What is different?” (The particular labelings of the points and angle measures may be different, but the result is always the same. The angles form a straight line and thus the angle measures sum to 180 degrees.)
  • “How did you create a second line that is parallel to the first?” (Rotating by 180 degrees around either of the midpoints created takes the first line to the second line.)
  • “Why does rotating 180 degrees around one midpoint create the same line as rotating around the other?” (Both lines are parallel to the given line, and both lines go through the vertex opposite the given line. The Parallel Postulate says that there is one unique line that is parallel to the given line that goes through the point in question, so the two lines must be the same line.)

Speaking, Representing: MLR 8 Discussion Supports. Use this routine to support whole-class discussion. After each student shares, provide the class with the following sentence frames to help them respond: "I agree because ….” or "I disagree because ….” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. 
Design Principle(s): Support sense-making

21.3: Triangle Angle Sum Another Way (10 minutes)

Activity

In this activity, students prove the Triangle Angle Sum theorem using translations.

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

Launch

Instruct students to create a triangle with sides extended to a line using this image as a guide.

Here is triangle \(ABC\) with angle measures \(a^\circ\), \(b^\circ\), and \(c^\circ\). Each side has been extended to a line.

Triangle ABC
Action and Expression: Internalize Executive Functions. Begin with a small-group or whole-class demonstration and think aloud of the first question to remind students how to translate a triangle along a directed line segment.  Keep the worked-out translation on display for students to reference as they work. 
Supports accessibility for: Memory; Conceptual processing

Student Facing

  1. Translate triangle \(ABC\) along the directed line segment from \(B\) to \(C\) to make triangle \(A’B’C’\). Label the measures of the angles in triangle \(A’B’C’\).
  2. Translate triangle \(A’B’C’\) along the directed line segment from \(A’\) to \(C\) to make triangle \(A’’B’’C’’\). Label the measures of the angles in triangle \(A’’B’’C’’\).
  3. Label the measures of the angles that meet at point \(C\). Explain your reasoning.
  4. What is the value of \(a+b+c\)? Explain your reasoning.

Student Response

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

Launch

Action and Expression: Internalize Executive Functions. Begin with a small-group or whole-class demonstration and think aloud of the first question to remind students how to translate a triangle along a directed line segment.  Keep the worked-out translation on display for students to reference as they work. 
Supports accessibility for: Memory; Conceptual processing

Student Facing

Here is triangle \(ABC\) with angle measures \(a^\circ\), \(b^\circ\), and \(c^\circ\). Each side has been extended to a line.

Triangle ABC
  1. Translate triangle \(ABC\) along the directed line segment from \(B\) to \(C\) to make triangle \(A’B’C’\). Label the measures of the angles in triangle \(A’B’C’\).
  2. Translate triangle \(A’B’C’\) along the directed line segment from \(A’\) to \(C\) to make triangle \(A’’B’’C’’\). Label the measures of the angles in triangle \(A’’B’’C’’\).
  3. Label the measures of the angles that meet at point \(C\). Explain your reasoning.
  4. What is the value of \(a+b+c\)? 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 it can help reveal what assertions were used. This can open up new areas to explore if we change those assumptions. For example, both of our proofs that the measures of the angles of a triangle sum to 180 degree were based on rigid transformations that take lines to parallel lines. If our assumptions about parallel lines changed, so would the consequences about triangle angle sums. Any study of geometry where these assumptions change is called non-Euclidean geometry.  In some non-Euclidean geometries, lines in the same direction may intersect while in others they do not. In spherical geometry, which studies curved surfaces like the surface of Earth, lines in the same direction always intersect. This has amazing consequences for triangles. Imagine a triangle connecting the north pole, a point on the equator, and a second point on the equator one quarter of the way around Earth from the first. What is the sum of the angles in this triangle?

Student Response

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

Anticipated Misconceptions

Some students may have difficulty drawing a reasonably accurate image of the figure under the translation. Remind them of the tools in their geometry toolkits, such as tracing paper, straightedges, and compasses. 

Some students may get stuck finding the measures of the missing angles. Direct those students to their reference charts.

Activity Synthesis

Here are some questions for discussion:

  • “How did you find the other angle measures at point \(C\)?” (Some of the angles are images of the original triangle after a rigid transformation. They must have the same measure as the original triangle. I can find the other angle measures using properties of vertical angles.)
  • “How do you know the point \(C’’\) lands on line \(AC\)?” (Translations takes lines to parallel lines and \(A’\) is taken to \(C\), so the segment \(A’C’’\) has to be taken to a parallel line that goes through \(C\). There’s only one such line, which is line \(AC\). So \(C’’\) lands somewhere on line \(AC\). In order for \(C’C’’\) to be parallel to line \(A’C\), it would have to land on line \(AC\) on the right side of \(C\).)

Lesson Synthesis

Lesson Synthesis

In this lesson, students explored two different proofs of the Triangle Angle Sum Theorem. Ask students to add this theorem to their reference charts as you add it to the class reference chart:

Triangle Angle Sum Theorem: The three angle measures of any triangle always sum to 180 degrees.

(Theorem)

\(a+b+c=180\)

Triangle with angles a, b, c. Dotted line parallel to side B C goes through A. Angles created at dotted line labeled b and c.

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 your 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 2 proofs of the Triangle Angle Sum Theorem different? How are they the same?” (Both activities use transformations to place the three angles adjacent to each other to make a straight line. The proofs are different because the first proof used 180 degree rotations whereas the second proof used translations.)

21.4: Cool-down - Triangle Angle Sum a Third Way (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

Using rotations and parallel lines, we can understand why the angles in a triangle always add to 180 degrees. Here is triangle \(ABC\).

Triangle ABC. Angle A is labeled a degrees, Angle B is labeled b degrees, Angle C is labeled c degrees.

Rotate triangle \(ABC\) 180 degrees around the midpoint of segment \(AB\) and label the image of \(C\) as \(D\). Then rotate triangle \(ABC\) 180 degrees around the midpoint of segment \(AC\) and label the image of \(B\) as \(E\).

Figure DAECB from 3 identical triangles: ABC, BAD and CEA. Angle BAC is labeled a degrees, angle ABC is b degrees and angle ACB is c degrees. The midpoints of AB and AC have a point.

Note that each 180 degree rotation takes line \(BC\) to a parallel line. So line \(DA\) is parallel to \(BC\) and line \(AE\) is also parallel to \(BC\). There is only one line parallel to \(BC\) that goes through point \(A\), so lines \(DA\) and \(AE\) are the same line. Since line \(DE\) is parallel to line \(BC\), we know that alternate interior angles are congruent. That means that angle \(BAD\) also measures \(b^\circ\) and angle \(CAE\) also measures \(c^\circ\).

Figure DAECB from 3 identical triangles: ABC, BAD and CEA. Angle BAC is labeled a degrees, angles DAB and ABC are b degrees, angles ACB and CAE are c degrees. The midpoints of AB and AC have a point.

Since \(DE\) is a line, the 3 angle measures at point \(A\) must sum to 180 degrees. So \(a+b+c=180\). 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.