Students sort different pairs of figures during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and to make connections (MP2, MP7).

Monitor for different ways groups choose to categorize the pairs of figures, but especially for categories that distinguish between congruent figures, similar figures, and neither congruent nor similar. 

After doing and discussing a card sort, students are assigned one pair of figures to figure out a sequence of rigid motions and dilations that takes one figure onto the other. Students use their prior knowledge of rigid transformations and their experience carrying out dilations to help them be successful in this task.

If students struggle to describe their transformations, remind them that they can use tracing paper to redraw the figures and add labels to the points.

If students struggle to name the scale factor for congruent figures, ask them to describe what they know about scale factors that make the image larger (the scale factors are greater than 1), then describe scale factors that make the image smaller (the scale factors are less than 1). Then, ask what scale factor might work if the image is the same size.

The purpose of this discussion is for students to describe sequences of rigid motions and dilations that show that two figures are either congruent or similar.

Record the scale factors for dilating from Figure to Figure , and from Figure to Figure , for each pair of figures except the rhombi. Include the right triangles and the trapezoids, which have a scale factor of 1. Ask students what they notice. (There’s a scale factor either way. The scale factors are multiplicative inverses. The congruent figures have a scale factor of 1.)

The goal of this activity is for students to use the definition of similarity, and to practice applying to similar figures some of the skills they learned when studying congruent figures. Students practice writing a sequence of rigid motions and dilations to take one figure onto another, write accurate similarity statements about the figures, and reason about what must be true about corresponding side lengths.

If students struggle to define sequences of rigid motions and dilations, review an example from the card sort. 

Ask students to determine if their sequence of rigid motions and dilations will always work for any pair of triangles that have all pairs of corresponding angles congruent and all pairs of corresponding side lengths in the same proportion. (Probably not, some figures would require rotations and dilations. But the general method of dilating and using rigid motions will always work.)

For students who dilated first, ask:

• "How do you choose the dilation to use?" (Dilate from any vertex, by using the scale factor given by the ratio of a pair of corresponding sides.)
• “Once you dilate, how do you know you’ll be able to use rigid motions to take one triangle to the other?“ (After dilation, the triangles are congruent because their corresponding angles remain congruent, and all the corresponding side lengths will be the same. Because they are congruent, there is always a sequence of rigid motions that takes one figure to the other. In this example, the lines are parallel and the triangles are oriented the same way, so only a translation is needed to align the figures.)

For students who used rigid motions first, ask:

• “How do you know rigid motion will line up parts of the figures?“ (Because corresponding angles are congruent and one pair of corresponding sides is parallel, we’ll be able to line up both of the rays that meet at the vertex with a translation, because you can always take an angle onto another congruent angle.)
• “How do you know that dilation will then line up the remaining two points?“ (Dilations keep angles the same measure, keep the center of the dilation in the same place, and move other points along the ray connecting the center to the point. Because the corresponding sides are all in the same proportion, if the right scale factor is used, then all of the vertex points should be in the same position so that the triangles line up.)