Walking Around a Triangle

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SWBAT construct a simple argument to utilize the AA theorem of similarity and explain the exterior angle theorem of triangles.

Big Idea

In this investigation we open with AA criterion for triangles and help students discover that the exterior angles of a triangle (and any polygon) must always add to 360.

Introducing Today's Work

20 minutes

In this lesson, I start by reviewing a question or questions that connect back to our work with vertical angles, complementary and supplementary angles, transversals, and the interior angles of triangles. I will probably just draw a triangle, draw a line parallel to one of the sides and label the least number of angles necessary to find all the angles in the diagram. Something similar to this image titled triangle proof or this sketch in a google Doc, a triangle with one of the legs extended to create a problem that would also involve transversals. 

After giving the students a few moments to work and share approaches to the problem, I plan to ask the question, “How many degrees must be in the interior of a planar triangle?”  By this point, my students should be able to answer this question quickly. If so, I will follow with the question, “How many degrees must be on the exterior of a triangle?” Here, I expect I will have to define what constitutes an exterior angle (the angle between an extended leg of a triangle and the triangle itself).

For the investigation of the sum of the exterior angles of a triangle, I will encourage students to draw equilateral, isosceles and scalene triangles to test their conjectures (see triangle_image). As I circulate, I will ask students to look for patterns. Some prompts I may use include:

  • Do you notice how many different exterior angles can we draw off of any vertex? 
  • Do you notice what happens if we take three exterior angles from three different vertices and add them? 
  • What happens if we look at an exterior angle and two remote interior angles? 
  • What would happen if we looked a 4,5, or more sided polygon? 

Extensions and Scaffolds: My goal in launching this lesson is to get students to use what they know about interior angles to understand what must happen on the outside angles of triangles. In order to accomplish this, I will make sure to have the tools required for angle measure and pre-drawn triangles with exterior angles labeled for reference. I hope that all of my students will be able to construct viable arguments,  even if they struggle with the logic of a formal proof.



Student Work Time

35 minutes

As students work through their investigations I will circulate, push some of the patterns and questions mentioned above, and record interesting moments as they occur throughout the lesson. My goal is to find evidence of student discovery around the concepts of exterior angles.