If students are stuck on sketching a pair of triangles that fit all the characteristics, suggest that they start by drawing any triangle. Then they can measure it and use that information to create their second triangle.

The purpose of this discussion is to address sketching and rounding in this course as well as the process of generalizing conjectures from examples.

Invite a few students to demonstrate their transformations. Students will generalize these transformations in a subsequent activity, so it is not necessary to generalize the sequence at this point.

Display a pair of sketched triangles that generally look like scaled copies, but may not be exactly precise, such as the pair in the Student Response for this activity. Explain that, “For this pair of triangles , and , so the corresponding pairs of sides are not in the exact same proportion. Geometry technology was used to draw triangle as the image of triangle after a dilation, so they should be similar by our definition.”

Ask students, “Why do you think the proportions are not exactly the same?” (The image only shows one decimal place, so the numbers shown might have been rounded to the nearest tenth.)

Tell students that they will frequently encounter problems in which the answer isn't a whole number. As a guideline for this course, they should round side lengths to the nearest tenth and know that scale factors that are equivalent with rounding count as "the same.”

The purpose of this discussion is to generalize at least one sequence of transformations that can be used to show that triangles with all pairs of corresponding angles congruent and all pairs of corresponding side lengths in the same proportion must be similar.

Invite previously selected students to share their sequences of transformations. Write the sequences of transformations for all to see, and highlight the order of the transformations. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.

Connect the different responses to the learning goals by asking questions, such as:

• “How important is the center of the dilation for each of these strategies?” (When dilating first, it is not very important what the center of dilation is because the triangle will be moving with rigid transformations later. When dilating later, it is important to choose a good center so that any points that were lined up do not get misaligned.)
• “How did each strategy find the scale factor for dilation when there are no numbers given?” (If we divide the side length by itself, then multiply by the corresponding side length, the corresponding sides will be the same length.)
• “Why did each strategy need only a single dilation?” (Because all of the corresponding side lengths are in the same proportion, using the correct scale factor will produce an image of a triangle so that all of the corresponding sides are congruent.)

If no students dilated first, provide a sequence for one of the triangle pairs that does so. (For Card A: Dilate triangle with center and a scale factor of . Translate triangle along the directed line segment from to . Rotate triangle , using as the center, so that coincides with .) This sequence of transformations will be useful when proving the Angle-Angle Triangle Similarity Theorem in a later lesson.

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

If two triangles have all pairs of corresponding angles congruent and all pairs of corresponding side lengths in the same proportion, then the two triangles are similar. (Theorem)