If students are stuck with what to label on the diagram, suggest they look in the reference chart for properties of parallel lines.

If students are stuck starting their proof, ask about how we can prove that two triangles are similar. Suggest they look in the reference chart. (Find a sequence of rigid motions and a dilation that takes one triangle to the other. Use a dilation to make the triangles congruent, and then use one of the triangle congruence theorems to finish the proof. Use the Angle-Angle Triangle Similarity Theorem.)

The goal of this discussion is to connect similarity to the ability to calculate side lengths. Invite students to share their arguments for why the triangles are congruent. Students should be able to apply the Angle-Angle Triangle Similarity Theorem by finding two pairs of corresponding congruent angles. To find the congruent angles, students could use that angle is congruent to itself or use corresponding congruent angles formed by parallel lines with a transversal (or vertical angles and alternate interior angles if the parallel line does not intersect the original triangle like in the Are You Ready For More? example).

Ask students to name ratios that must be equivalent, now that we have proven that the triangles are similar. (, )

Display this theorem for all to see (although students do not need to copy it into their reference chart, as it is unlikely to be referred to later): “A line parallel to one side of a triangle creates a similar triangle.”

Students who struggle to visualize the similar triangles can trace each triangle separately onto tracing paper and label the corresponding sides using colored pencils.

The purpose of the discussion is for students to share their strategies for finding the values of the missing sides.

Invite previously selected students to share their strategies for finding the missing side lengths in the diagrams. 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:

• “Can any of these strategies be used for any pair of similar triangles?”
• “One strategy used the ratio and another used the ratio . Can you use algebra to rewrite one of these equations so that it looks like the other one?”

Make sure all students understand that we know that these triangles are similar because we know the information about the parallel sides. If we did not know the information about the segments being parallel, we would not have enough information to solve for the lengths of the sides in the second pair of triangles.

Encourage students to look for and discuss connections between finding a scale factor, using equivalent ratios across the two triangles, and using equivalent ratios within the two triangles. (Ratios of corresponding sides across triangles are all equivalent because the second component in each ratio is the scale factor times the first. A ratio within one triangle is equivalent to the corresponding ratio within the other triangle because both components get multiplied by the scale factor.)