Students will be able to extend their understanding of triangle similarity to prove the parallel and perpendicular slope criteria.

Students will be able to apply their understanding of triangle similarity in novel ways to write proofs and demonstrate their understanding on a group assessment.

10 minutes

In this Similar Triangles Warm-Up my students must determine whether the given pairs of triangles are similar and justify their reasoning with a triangle similarity statement. Since students have had a couple of lessons on similarity, I expect this warm-up to go rather quickly. To debrief the warm-up, I will project an answer key using the document camera, ask students to check their work, and facilitate a whole-class discussion around students’ questions.

20 minutes

In my constructions unit, students constructed parallel and perpendicular lines on a coordinate plane as a way to recall foundational ideas around slopes that are parallel or perpendicular. Now that we have learned the definition of similar figures, and we can justify whether triangles are similar, we are ready to prove the parallel and perpendicular slope criteria.

Like most notes, we start with a concrete example and then generalize our thinking (see example).

- For the
**parallel slope criteria**, we notice (1) corresponding angles are congruent and (2) all slope triangles are right triangles, which allows us to conclude that all slope triangles on parallel lines are similar by AA~, establishing that parallel lines have the same slope. - For the
**perpendicular slope criteria**, we notice that rotating any slope triangle 90 degrees will yield a slope triangle for the line perpendicular to a given line, establishing that perpendicular lines have slopes that are opposite reciprocals of one another.

50 minutes

Since triangle similarity is a foundation for right triangle trig, I want to make sure all of my students understand similarity at high levels before we progress to trigonometry. Today's Similar Polygons Group Quiz features a variety of problems that ask students to determine whether figures are similar, justify how they know, solve similarity problems, and write proofs (**MP1**, **MP3**).

For this assessment, students work together in groups of four. Each group is responsible for ensuring that every member of their group understands how to solve each problem. I say, "You need to prepared each member of your group to represent your group with clear explanations of high quality work."

During group assessments, each group is allowed to ask the teacher only one question. By limiting groups to one group question, it helps to ensure that the group has checked everyone’s thinking and understanding before seeking help. Before they begin, I remind students that **I will assign the group a group grade either by grading select problems from each member’s paper or by choosing one paper at random to grade **— this motivates individual accountability by engaging and involving every member in the group problem solving process.