Lesson 7

The purpose of this warm-up is to elicit conjectures about similar triangles and dilation, which will be useful when students formalize and prove these conjectures in later activities in this lesson. While students may notice and wonder many things about these images, proportional lengths, congruent angles, and dilation are the important discussion points. This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is that the smaller triangle is a dilation of the larger triangle.

Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

Rewrite a few of the students’ contributions as conjectures. Select things students wondered about proportional lengths, congruent angles, and dilation, if available.

In previous lessons, students practiced coming up with sequences of rigid motions and dilations to take one figure onto another given similar figures. In this activity, students work to find a sequence of rigid motions and dilations that will take one triangle onto another for any pair of triangles with all pairs of corresponding angles congruent and all pairs of corresponding side lengths in the same proportion.

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

Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.

Emphasize to students that their sketches are just a tool to help them think about the transformations needed. Since the sketches don’t need to be super-precise, their transformations might not line everything up perfectly.

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.

Invite a few students to demonstrate their transformations. Students will be generalizing by playing Invisible Triangles in a subsequent activity, so it is not necessary to generalize the sequence at this point.

Students play a new version of the Invisible Triangles game to generalize, and then they work as a class to prove that they really have come up with a sequence of rigid motions and dilations that they can justify will take any pair of triangles with all pairs of corresponding angles congruent and all pairs of corresponding side lengths in the same proportion onto each other. 

Monitor for students who dilate first then use rigid motions as well as students who use rigid motions first and then dilate.

Arrange students in groups of 2. Distribute one transformer card and one set of three playing cards to each group. If feasible, have students use folders or books so that students can’t see each other’s desktops.

Invite students who dilated first to share their sequence:

• Ask students to justify why they knew that the dilation of one triangle would create at least one pair of congruent corresponding sides. (Since the lengths of the corresponding sides are in the same proportion, dilating by the right scale factor would make them congruent. Then we know a sequence of rigid motions exists to take one triangle to the other triangle.)

Invite students who lined up one angle first and then dilated to share their sequence:

• Ask students to justify why they knew it had to be possible to get corresponding angles to coincide using rigid motions. (The angles are congruent, so there must be a sequence of rigid motions to take one to the other.)
• Ask students to justify how they knew a dilation would get all pairs of corresponding vertices to coincide if they used rigid motions first and dilated second. (After we lined up a corresponding angle, since all the pairs of corresponding side lengths are in the same proportion, we knew the triangles were dilations of each other and dilating by the right scale factor would get them to line up exactly.)

Ask students why the proof is shorter if they dilate first. (Once we dilate to get at least one pair of corresponding sides to be congruent, we can see if we have the criteria to use one of the triangle congruence theorems.) If no student dilated first, outline the proof. This technique will come in very handy for proving the Angle-Angle Triangle Similarity Theorem.

Ask students to add this theorem to their reference charts as you add it to the class reference chart:

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)

\(\angle A \cong \angle C, \angle D \cong \angle B, \angle DEA \cong \angle BEC,\) \(\frac{AD}{CB}=\frac{DE}{BE}=\frac{EA}{EC}\) so \(\triangle DEA \sim \triangle BEC\)