While the Angle-Angle Triangle Similarity Theorem follows logically from the Angle-Side-Angle Triangle Congruence Theorem (dilate one triangle by a scale factor determined by the ratio of the sides between the known corresponding angle pairs, and the dilated triangle will be congruent to the target by the Angle-Side-Angle Triangle Congruence Theorem, so the original triangle must be similar to the target), it can still be unintuitive.

Students experienced the Angle-Angle Triangle Similarity Theorem briefly in a previous unit when they looked at examples of triangles constructed so that their angles were the same but their side lengths were undetermined. In middle school, students confirmed the Angle-Angle Triangle Similarity Theorem informally. However, their proofs until this point have tended to rely on knowing something about the proportionality of side lengths.

Students might be unsure that the same scale factor will work for all three pairs of corresponding sides when all they know is the measure of two angles of the triangle. It is a surprising result when you put it that way. This activity starts by building students’ intuition about the Angle-Angle Triangle Similarity Theorem, to motivate the proof of the theorem.

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

If students struggle with the proof of their conjecture, invite them to go back to what they discussed during the Warm-up. Here are some questions that can help progress the reasoning.

• "Is there a dilation you could use that would make those triangles congruent? How does that help you prove that they are similar?" (Yes, dilate the smaller triangle with a scale factor of 3. Then I can show there’s a dilation and sequence of rigid motions that takes one triangle onto the other, so they’re similar.)
• "How many sides did you need to have to figure out a dilation for the triangles with sides given? Why? Could you measure fewer sides?" (I only needed one side to figure out that I could scale the triangle by a scale factor of 3. I could measure just one pair of sides, and scale up by the right scale factor to make those sides match.)

The goal of this discussion is for students to be convinced that the Angle-Angle Triangle Similarity Theorem is a valid way to prove that two triangles are similar. 

Invite students to summarize the main points of the proof:

• Choose the pair of corresponding sides between the known congruent angles to generate a scale factor.
• Dilate one triangle using that scale factor.
• The triangles must be congruent now based on the Angle-Side-Angle Triangle Congruence Theorem.
• The triangles can be taken onto each other by dilation and rigid motions. Therefore, they are similar.

Add the following theorem to the class reference chart, and ask students to add it to their reference charts:

Angle-Angle Triangle Similarity Theorem: In two triangles, if two pairs of corresponding angles are congruent, then the triangles must be similar. (Theorem)

This activity expands upon the Angle-Angle Triangle Similarity Theorem by requiring the use of the Triangle Angle Sum Theorem to find two pairs of congruent angles. Once similarity is established, students use the scale factor to determine missing side lengths on the triangles.

This is the first time Math Language Routine 5: Co-Craft Questions is suggested in this course. In this routine, students are given a context or situation, often in the form of a problem stem (for example, a story, image, video, or graph) with or without numerical values. Students develop mathematical questions that can be asked about the situation. A typical prompt is: “What mathematical questions could you ask about this situation?" The purpose of this routine is to allow students to make sense of a context before feeling pressure to produce answers, and to develop students’ awareness of the language used in mathematics problems.

The purpose of this discussion is for students to recognize that there are some cases in which knowing two pairs of angles in a pair of triangles is enough to show that they are similar even if there is only one pair of congruent corresponding angles.

Invite groups to share their reasoning about why the triangles are similar. Listen for students to share that the third angle of each triangle can be found based on the sum of the angles. If it is not mentioned, ask students to refer to their reference chart to find the name of the theorem that shows this idea (Triangle Angle Sum Theorem).

Ask students what is special about the given angles that lets us know that triangles are similar when only two angles are given for each triangle. (The triangles are similar when the angles given are either:

• Two pairs of congruent, corresponding angles or
• One pair of congruent, corresponding angles and the other angles are not corresponding, but the sum of the two non-corresponding angles along with one of the given corresponding angles is 180 degrees.)

It is ok if students cannot precisely articulate the second condition, as long as they understand the process of using the Triangle Angle Sum Theorem to find the missing angles and then show that all pairs of corresponding angles are congruent.