In the previous unit, students began justifying their responses. In this unit, students will work toward writing more rigorous proofs. The reason students write proofs is to use the resulting theorem in future work without having to repeat the argument. In this unit, proofs will begin with transformations. Once students have proven results, they no longer need to write out all the transformations and can use theorems and shortcuts instead.
In middle school, students learned that in a transformation, a part of the first figure and its image in the second figure are called corresponding parts. This lesson re-introduces students to the concept that if two figures are congruent, then each pair of corresponding parts of the figures are also congruent. They will use this concept throughout the unit as they prove triangles and other figures congruent. Students begin applying it in the very next activity in which students make conjectures about a quadrilateral they form by transforming a triangle and justify them. Writing proofs doesn’t only mean providing the reasons for someone else’s claim; constructing a viable argument includes writing conjectures (MP3).
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.
- Determine whether or not figures are congruent by reasoning about rigid transformations (in writing).
- Generate and comprehend congruence statements (orally and in writing) that establish corresponding parts.
- Let’s figure out what the corresponding sides and angles in figures have to do with congruence.
- I can identify corresponding parts from a congruence statement.
- I can use rigid transformations to figure out if figures are congruent.
- I can write a congruence statement.
For a rigid transformation that takes one figure onto another, a part of the first figure and its image in the second figure are called corresponding parts. We also talk about corresponding parts when we are trying to prove two figures are congruent and set up a correspondence between the parts to see if the parts are congruent.
In the figure, segment \(AB\) corresponds to segment \(DE\), and angle \(BCA\) corresponds to angle \(EFD\).