If students struggle to see which angles and sides correspond, encourage them to copy each triangle onto tracing paper and rotate them so they all have the same orientation. Colored pencils can also help students identify corresponding parts.

In this lesson, students examine two similar right triangles in a particular arrangement that will help them prove the Pythagorean Theorem in a later lesson. Students should be able to find the missing side lengths using the Pythagorean Theorem as well as properties of similar triangles such as a scale factor or equivalent ratios.

If students struggle to write similarity statements, have them also trace the large triangle without the altitude and then have them orient all three triangles in the same direction. Ask them to find corresponding angles and write their statements so that the letters of the corresponding angles are in the same position in the similarity statement.

Invite a student to share who:

• Used the Pythagorean Theorem to find the third side length.
• Used the scale factor they determined to find the third side length.
• Used equivalent ratios to find the third side length.

If no student used equivalent ratios, ask students to write several equivalent ratios for the three triangles. In a subsequent lesson, students will need to recognize, write, and manipulate equivalent ratios involving the same side lengths.

In this activity, the angle measures are not given, so students need to convince themselves that the triangles in question are similar. Students may be tempted to assume that they are similar simply because of the triangles in the previous diagram that looked like this, so monitor for students who are actively trying to convince themselves that the Angle-Angle Triangle Similarity Theorem applies in this case.

Monitor for students who use the strategies listed in order of precision:

• Students measuring angles (with tracing paper or protractors)
• Students measuring one or two angles and calculating the rest
• Students using algebraic or logical reasoning to convince themselves that the angles would be congruent

Students also get another chance to write equivalent ratios. One key thing they might notice is that a lot of the sides are in more than one ratio.

If students assume that the triangles are similar because the diagram looks like the previous diagrams, ask them how they know.

Invite previously selected students to share their reasoning for why the triangles are similar. 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:

• “What information did all of these methods use to show similarity?” (the Angle-Angle Similarity Theorem)
• “Which of these methods are the most precise?” (Using logical reasoning about the geometry of the triangles is probably more precise than using tracing paper or a protractor because there may be some error in measuring the angles in those ways.)

Finally, ask students to share equivalent ratios that they found. Record the equivalent ratios for all to see. Ask students how they might group the ratios that they found. If it is not suggested, point out that the ratios could be grouped into “within the same triangle” (such as , where each fraction represents the short leg divided by the hypotenuse) and “between two triangles” (such as   , where each fraction is a ratio of the corresponding sides in a pair of similar triangles).