# Lesson 11

Congruence

### Lesson Narrative

In this lesson, students find rigid transformations that show two figures are congruent and make arguments for why two figures are not congruent. They learn that, for many shapes, simply having corresponding side lengths that are equal will not guarantee the figures are congruent.

In a previous lesson, students defined what it means for two shapes to be congruent and started to apply the definition to determine if a pair of shapes are congruent. Now, they determine whether or not pairs of shapes are congruent with the structure of a grid. With this structure, students attend to precision (MP6) when describing translations, reflections, and rotations by using language like:

• “Translate 3 units down and 2 units to the left,” instead of “translate down and to the left.”
• “Reflect the shape over this vertical line,” instead of “reflect the shape.”

This lesson also asks students to consider more complex shapes with curved sides. The focus here is on the fact that the distance between any pair of corresponding points of congruent figures must be the same. Because there are too many pairs of points to consider, this is mainly a criterion for showing that two figures are not congruent: that is, if there is a pair of points on one figure that are a different distance apart than the corresponding points on another figure, then those figures are not congruent.

Throughout the lesson, students have to be careful how they name congruent shapes, making sure that corresponding points are listed in the proper order.

### Learning Goals

Teacher Facing

• Determine whether shapes are congruent by measuring corresponding points.
• Draw and label corresponding points on congruent figures.
• Justify (orally and in writing) that congruent figures have equal corresponding distances between pairs of points.
• Justify (orally and in writing) that two polygons on a grid are congruent using the definition of congruence in terms of transformations.

### Student Facing

Let’s decide if two figures are congruent.

### Student Facing

• I can decide using rigid transformations whether or not two figures are congruent.
• I can use distances between points to decide if two figures are congruent.

### Glossary Entries

• right angle

A right angle is half of a straight angle. It measures 90 degrees.