In this activity, students consider how to determine whether curved shapes are congruent. Unlike a polygon, a curved shape is not completely determined by a set of vertices. Students consider the features of ovals as a way to identify that two shapes are not congruent, such as the length of the widest part of each oval. In order to provide a viable argument that two curved shapes are congruent (MP3), students must apply the definition of congruence by identifying a rigid transformation so that one shape matches up exactly with the other.

Monitor for students who use precise language of transformations as they attempt to move one traced oval to match up perfectly with another. Select them to share their reasoning during the discussion. Also monitor for arguments based on specific features for why neither of the upper ovals can be congruent to either of the lower ones and select a few students to share during the whole-class discussion.

Invite selected students to share the rigid transformations they used to show which pairs of ovals are congruent. If no students suggest it, show this image for all to see:

Ask students to describe the rigid transformation from the top left oval to the top right oval using the marked point as the center of rotation, and demonstrate if needed.

Invite previously selected students to share the features they noticed that showed the top ovals could not be congruent to the bottom ovals.

Ask students, “How is showing that two ovals are congruent different than showing two polygons are congruent?” (There aren’t angle measures and side lengths to measure that can show the shapes are congruent. The only way is to use a transformation to take one shape to the other.)

In this activity, students compare corresponding parts of congruent figures with curved parts. Students have used the property that corresponding sides of congruent polygons have the same length in previous activities, but curved shapes like ovals do not have sides. Students connect corresponding side lengths on a polygon with corresponding points on a curved shape as they draw corresponding line segments on congruent figures. They use the property that translations, rotations, and reflections do not change distances between points to conclude that corresponding parts of the curved figures must stay the same distance apart.

Monitor for students who use these strategies to identify corresponding points on the two corresponding figures: 

• Looking for corresponding parts of the figures, including line segments and points
• Performing rigid transformations with tracing paper to match the figures to each other

Select students using these strategies and select them to share during the discussion.

Ask selected students to show how they determined the points corresponding to , , and , highlighting different strategies, such as identifying key features of the shapes and performing rigid motions. Ask students if these strategies would work for finding if it had not been marked. Performing rigid motions matches the shapes up perfectly, and so this method allows us to find the corresponding point for any point on the figure. 

While it is challenging to test “by eye” whether or not complex shapes like these are congruent, the mathematical meaning of the word “congruent” is the same as with polygons: Two shapes are congruent when there is a sequence of translations, reflections, and rotations that match up one shape exactly with the other. Any pair of corresponding segments in congruent figures will have the same length because translations, reflections, and rotations do not change distances between points.

If time allows, have students use tracing paper to make a new figure that is either congruent to the shape in the activity or slightly different. Display several for all to see and poll the class to see if students think the figure is congruent or not. Check to see how the class did by lining up the new figure with one of the originals. Work with these complex shapes is important because we tend to rely heavily on visual intuition to check whether or not two polygons are congruent. This intuition may not be reliable if the polygons are complex or have very subtle differences that cannot be easily seen. The meaning of congruence in terms of rigid transformations and our visual intuition of congruence can effectively reinforce one another:

• If shapes look congruent, then we can try to find the rigid transformation to demonstrate that they are congruent.

This activity is optional because it provides an opportunity for additional practice applying the property that all corresponding parts of congruent figures are congruent. In this activity, students examine two figures made out of disjoint pieces: the outline of a face, two eyes, and a mouth. The disjoint pieces are congruent to one another taken in isolation. But they are not congruent taken as a collective whole because the relative positions of the eyes, mouth, and face outline are different in the two images.

In order for two shapes to be congruent, all pairs of corresponding points in the two images must be the same distance apart. Strategically selecting corresponding points whose distance is not the same in the two figures requires a solid understanding of the meaning of congruence and corresponding parts.

Monitor for students who:

• Identify that the parts of the two faces are congruent, such as the face outline, eyes, and mouth.
• Identify distances between parts of each face that are not the same.

Select students to share these observations during the whole-class discussion.

Ask selected students to share which parts of the faces are congruent and their reasoning. Emphasize precise language that uses rigid transformations, such as a translation.

Ask selected students to share how they determined whether the two face images are congruent. If no students suggest it, ask, “What corresponding points were you able to identify that were not the same distance apart in the two faces?” (Distance between the eyes, from the eyes to the mouth, from the eyes to the face outline, or from the mouth to face outline.)

Even though the individual parts of the two faces are congruent, the two faces are not congruent. We could find one translation that takes the outline of one face to the outline of the other and similarly we could find a translation taking the left eye, right eye, and mouth of one figure to the left eye, right eye, and mouth of the other. But these translations are different. In order for two figures to be congruent, there must be one sequence of transformations that match all parts of one figure perfectly with the other.