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.

Monitor for students who start with different triangle types, such as scalene, equilateral, right, and obtuse.

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

The goal of this discussion is to solidify that, no matter what triangle students started with, the sum of the measures of the three angles is always 180 degrees. 
 
Display 2–3 diagrams from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the different representations. Here are some questions for discussion:

• “How are these diagrams the same? How are they 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 does the angles of the triangle summing to 180 degrees show up in each diagram?” (In all the diagrams there is a line which is split into three angles. Those angles have to sum to 180 degrees because a straight line is 180 degrees. Those angles are also the same as the angles in the triangle using alternate interior angles, so the angles of the triangle must sum to 180 degrees as well.)

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).

Here are some questions for discussion:

• “How did you find the other angle measures at point ?” (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 lands on line ?” (Translations take lines to parallel lines. So is parallel to . If the image of lands on the line, then the image of will too.)

Some students may get stuck connecting the interior angles of the triangle to the straight angle. Direct those students to their reference charts.

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.