Earlier in this unit, students learned that when a \(180^\circ\) rotation is applied to a line \(ℓ\), the resulting line is parallel to \(ℓ\). Here is a picture students worked with earlier in the unit:
The picture was created by applying \(180^\circ\) rotations to \(\triangle ABC\) with centers at the midpoints of segments \(AC\) and \(AB\). Notice that \(E\), \(A\), and \(D\) all lie on the same grid line that is parallel to line \(BC\). In this case, we have the structure of the grid to help see why this is true. This argument exhibits one aspect of MP7, using the structure of the grid to help explain why the three angles in this triangle add to 180 degrees.
In this lesson, students begin by examining the argument using grid lines described above. Then they examine a triangle off of a grid, \(PQR\). Here an auxiliary line plays the role of the grid lines: the line parallel to line \(PQ\) through the opposite vertex \(R\). The three angles sharing vertex \(R\) make a line and so they add to 180 degrees. Using what they have learned earlier in this unit (either congruent alternate interior angles for parallel lines cut by a transverse or applying rigid transformations explicitly), students argue that the sum of the angles in triangle \(PQR\) is the same as the sum of the angles meeting at vertex \(R\). This shows that the sum of the angles in any triangle is 180 degrees. The idea of using an auxiliary line in a construction to solve a problem is explicitly called out in MP7.
Lesson overview
 16.1 Warmup: True or False: Computational Relationships (5 minutes)

16.2 Activity: Angle Plus Two (15 minutes)
 There is a digital applet in this activity.

16.3 Activity: Every Triangle in the World (15 minutes)
 Includes "Are you Ready for More?" extension problem
 16.4 Optional Activity: Four Triangles Revisited (5 minutes)
 Lesson Synthesis
 16.5 Cooldown: Angle Sizes (5 minutes)
Learning goals:
 Create diagrams using 180degree rotations of triangles to justify (orally and in writing) that the measure of angles in a triangle sum up to 180 degrees.
 Generalize the Triangle Sum Theorem using rigid transformations or the congruence of alternate interior angles of parallel lines cut by a transversal.
Learning goals (student facing):
 Let’s see why the angles in a triangle add to 180 degrees.
Learning targets (student facing):
 I can explain using pictures why the sum of the angles in any triangle is 180 degrees.
Required materials:
 geometry toolkits
Glossary:
 Access the complete Grade 8 glossary.
Standards
 This lesson builds on the standards: CCSS.7.NS.ACCSS.8.G.A.1.bMS.8.G.1b
 This lesson builds towards the standard: CCSS.8.G.B.6MS.8.G.6
IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 20172019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/mathcurriculum/.
Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.
This site includes public domain images or openly licensed images that are copyrighted by their respective owners. Openly licensed images remain under the terms of their respective licenses.