# Rigid Motions and Congruence

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## Objective

SWBAT define congruence in terms of rigid motion.

#### Big Idea

Students will build on prior knowledge to develop a definition of congruence in terms of rigid motion.

## Do Now

5 minutes

According to the CCSS, students first investigate congruence formally in the eighth grade. This do now is both a diagnostic and a review. Students are shown five triangles and asked which triangles are congruent to each other.  The triangles each have two given sides lengths and angle measures. Using the given information students identify which triangles have the three equal sides and three equal angles.

In order to identify which pairs of triangles are congruent, students need to build on prior knowledge about the sum of the measures of the interior angles in a triangle, isosceles triangles and then Pythagorean theorem.

The question in the Do Now asks students to identify congruent triangles, but not explain why the triangles are congruent. When we go over the Do Now, I ask students to explain how they know the triangles are congruent.

Some students may need a brief reminder of the definition of “congruent” and further explanation about how to find the missing side lengths and angle measures.

## Mini-Lesson

10 minutes

I begin the Mini-Lesson by asking a student to explain the definition of congruence. At this point in the lesson, most students will say, “Congruence is when two figures are exactly the same shape and size.” I ask, “Is there another way we can show two figures are congruent?” To continue the discussion, I remind students about the previous unit on transformations. We discuss the transformations that preserve congruence and define these transformations as rigid motions or isometries.

Students are then shown the diagram of pairs of triangles that can be proved congruent by rigid motions. We discuss which rigid motion can be used and then identify corresponding parts of the triangles that are congruent. This is only an introduction to standard GCO7 about CPCTC.

## Activity

15 minutes

To further investigate congruence through rigid motions, students are given a diagram with four congruent triangles. Students will work individually to Identify the rigid motion(s) that can be used to show congruence. They will then write congruency statements for corresponding parts of the triangles and label all congruent parts on the diagram.

For students who have difficulty identifying the rigid motions and the corresponding parts, I give them a second copy of the diagram to cut out and use as a manipulative.

## Summary

5 minutes

To summarize the lesson, we have a whole class discussion about the lesson activity. I ask the students to identify which parts of the small triangles are congruent to each other in relation to the whole diagram. In the activity, the students only compared two small triangles. I want them to make the connection between the part and the whole. This sets students up for performing further proofs of triangle congruence, i.e. transitive property, in later lessons.