In previous lessons, students practiced coming up with sequences of rigid motions and dilations to take one figure onto a given similar figure. In this lesson, students use rigid transformations and dilations to reason about and make generalizations about similarity of triangles and other types of figures. Students build on the work they did in earlier lessons, using their repeated reasoning to come up with a generalized sequence of rigid motions and dilations that they can justify will always take two triangles with congruent angles and proportional sides onto one another (MP8).
Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.
- Explain (in writing) using transformations why similar triangles have all corresponding pairs of angles congruent and all corresponding pairs of sides in proportion.
- Let’s describe similar triangles.
For the Invisible Triangles activity: Separate the transformer cards from the triangle cards and give each group one transformer card and one set of three triangle cards. If feasible, provide each group of 2 with a folder or other divider so students can’t see each other’s desktops.
- I know the relationships between corresponding sides and angles in similar triangles.
One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure onto the second.
Triangle \(A'B'C'\) is similar to triangle \(ABC\) because a rotation with center \(B\) followed by a dilation with center \(P\) takes \(ABC\) to \(A'B'C'\).