In this activity, students create triangles with given angles using dried pasta. When comparing triangles constructed with just one angle measure in common, they notice that the triangles are all different, usually only sharing the one given angle. When comparing triangles constructed with two angle measures in common, they notice that the triangles are all similar (MP7).

This is true because a triangle is determined by the length of one side and the two angles made with that side. The length of that side can then be scaled with a dilation to give any particular similar triangle. This means that triangles with two pairs of congruent corresponding angles must be similar, commonly known as the angle-angle (AA) triangle similarity theorem. Students do not need to know the AA triangle similarity theorem by name at this time, but they will see this idea again in a later activity.

The goal of this discussion is to make sure students understand that if triangles share 2 pairs of congruent angles then they are similar. Discuss with students:

• “Did the angles in your triangles always add up to 180 degrees?” (Yes, but because the angles were rounded to the nearest 5 degrees sometimes they weren’t exactly 180 degrees.)

• “What did you notice in the first problem when you compared your triangle with other classmates who used the same angle?” (The triangles were all different.)

• “What did you notice in the problem where you had to use 2 angles to make a triangle?” (The triangles all looked the same but were different sizes. They were similar.)

• “Did you need all 3 angle measurements to build a triangle in the second problem?” (No, once we knew 2 angles, that was enough to build the pasta triangle, the third angle automatically being the right measure.)

• “How did you decide whether or not the sides of your triangle were proportional to other triangles?” (By dividing 2 side lengths of one triangle and comparing that to the quotient of the 2 corresponding sides in the other triangle. Measurement error may mean that quotients computed may not be exactly equal even if the triangles are similar.)

• “How did you check whether or not your triangle was similar to another?” (Lining up 1 angle and then dilating would tell me if the triangles were similar, and observing that the angles were different or the sides were not proportional would tell me they were not similar.)

This optional activity presents students with a complex figure and asks them to find triangles similar to a given triangle. This activity provides an additional opportunity for students to practice explaining why two triangles are similar (MP3). 

Monitor for students who use these methods to show that their chosen triangles are similar to triangle :

• Finding a sequence of translations, rotations, reflections, and dilations

• Checking that two (or three) corresponding angles are congruent

The goal of this discussion is to compare different strategies for showing that 2 triangles are similar. Invite students to share some of the triangles they found and explain how they determined the triangles to be similar.