Lesson 11

The purpose of this warm-up is to elicit the idea that lines parallel to the base of a triangle make interesting shapes, including similar triangles, which will be useful when students prove these conjectures in a later activity. While students may notice and wonder many things about these images, triangles and angles formed by parallel lines 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 given parallel lines, we can draw conclusions about angle measure, which can then lead to additional conclusions.

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.

The goal of this activity is for students to prove the converse of the theorem they developed in a previous lesson. In that lesson, they proved that a line that splits two sides of a triangle proportionally is parallel to the third side. In this activity, they prove that a line parallel to one side of a triangle creates similar triangles, and thus splits the other two sides proportionally.

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

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. Ask students to name ratios that must be equivalent, now that we have proven that the triangles are similar. (\(\frac{AM}{MB}=\frac{CN}{NB}\), \(\frac{AC}{MN}=\frac{CB}{NB}=\frac{AB}{MB}\))

Record these as theorems for all to see (though students do not need to copy them into their reference chart, as they are unlikely to be referred to later):

• A line parallel to one side of a triangle creates a similar triangle.
• A line parallel to one side of a triangle divides the other two sides proportionally.

Because the previous activity ended with students identifying equivalent ratios, this activity asks students to calculate using equivalent ratios. This builds a bridge from knowing that a segment is parallel to one side of a triangle to knowing that equivalent ratios can be written.

Monitor for students who:

• identify a scale factor from one triangle to the other 
• use the Pythagorean Theorem 
• use equivalent ratios based on ratios of corresponding sides across the two triangles (For example, the shorter sides in each triangle are 3 units long and 7.5 units long. The lengths of the medium sides should have the same ratio. The medium side in the small triangle is 4 units. \(3:7.5\) is equivalent to \(4:10\), so \(DE\) is 10.)
• use proportional relationships based on ratios of side lengths within one triangle (For example, the ratio of these sides in the smaller triangle is \(3:4\), so in the larger triangle, the ratio should also be \(3:4\). Since \(3:4\) is equivalent to \(7.5:10\), \(DE\) is 10.)

Ask students to think about what type of triangle they see if they forget to use the Pythagorean Theorem for the third side.

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

Select previously identified students to share these strategies in this order:

• using equivalent ratios based on ratios of corresponding sides across the two triangles 
• identifying a scale factor from one triangle to the other
• using the Pythagorean Theorem
• using ratios of side lengths within one triangle 

Make sure all students understand how the first strategy follows directly from the theorem proved in the previous activity, and that if line \(FG\) was not parallel to line \(BD\), we couldn’t solve the problem.

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.)