Unit 1, Lesson 12
Congruent Polygons
Grade 8
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To Gather
Geometry toolkits
The purpose of this activity is for students to connect rigid transformations with congruent figures. In this activity, students identify which figures are images of an original triangle under a translation. They may notice features of figures under a translation, such as parallel corresponding segments, or the orientation of the figure staying the same. This will be useful in upcoming activities as students describe a sequence of transformations from one figure to another.
Provide access to geometry toolkits. Allow for 2 minutes of quiet work time followed by a whole-class discussion.
All of these triangles are congruent. Sometimes we can take one figure to another with a translation. Shade the triangles that are images of triangle
If any students assert that a triangle is a translation when it isn’t really, ask them to use tracing paper to demonstrate how to translate the original triangle to land on it. Inevitably, they need to rotate or flip the paper. Remind them that a translation consists only of sliding the tracing paper around without turning it or flipping it.
The purpose of this discussion is for students to articulate what features they can look for when they are identifying a translation. Ask students what they noticed about figures that were translations of triangle
If no students share these observations, suggest them now and ask students to discuss:
If time allows, choose a triangle that is not the image of triangle
MLR7: Compare and Connect
To Gather
Geometry toolkits
In this activity, students apply the definition of congruence to figures on a square grid. Students may use the structure of the grid to describe rigid transformations, including reflecting over a specified line or identifying coordinates of corresponding points. Students may also wish to use tracing paper to execute the transformations.
Students are given several pairs of shapes on grids and asked to determine if the shapes are congruent. The congruent shapes are deliberately chosen so that more than one transformation will likely be required to show the congruence. In these cases, students will likely find different ways to show the congruence. Monitor for students who use different sequences of transformations to show congruence. For example, for the first pair of quadrilaterals, some different ways are:
For the pairs of shapes that are not congruent, students need to identify a feature of one shape not shared by the other in order to argue that it is not possible to move one shape on top of another with rigid motions. At this stage, arguments can be informal. Monitor for students who use different features to show figures are not congruent:
Provide access to geometry toolkits. Allow for 5–10 minutes of quiet work time followed by a whole-class discussion.
Select students with different strategies, such as those described in the Activity Narrative, to share later.
For each of the following pairs of shapes, decide whether or not they are congruent. Explain your reasoning.
The goal of the discussion is for students to understand that when two shapes are congruent, there is a rigid transformation that matches one shape up perfectly with the other. Choosing the right sequence takes practice. Students should be encouraged to experiment using tools, such as tracing paper or technology when available. When two shapes are not congruent, there is no rigid transformation that matches one shape up perfectly with the other. It is not possible to perform every possible sequence of transformations in practice, so to show that one shape is not congruent to another, we identify a feature of one shape that is not shared by the other. For the shapes in this activity, students can focus on side lengths: For each pair of non-congruent shapes, one shape has a side length not shared by the other. Since rigid transformations do not change side lengths, this is enough to conclude that the two shapes are not congruent.
Display 2–3 approaches from previously selected students for all to see. Invite students to briefly describe their approaches to showing figures are congruent as well as not congruent. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:
To Gather
Geometry toolkits
In this partner activity, students take turns claiming that two given polygons are or are not congruent and explaining their reasoning. The partner's job is to listen for understanding and challenge their partner if their reasoning is incorrect or incomplete. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).
This activity continues to investigate congruence of polygons on a grid. Unlike in the previous activity, the non-congruent pairs of polygons share the same side lengths.
Arrange students in groups of 2, and provide access to geometry toolkits. Display this image for all to see.
Two figures, both 7 sided on a coordinate plane. The left figure has points A(0 comma 0), B(0 comma 3), C(1 comma 3), D(1 comma 2), E(2 comma 2) and F(2 comma 0). The right figure has points G(7 comma 3), H(4 comma 3), I(4 comma 4), J(5 comma 4), K(5 comma 5) and L(7 comma 5).
Demonstrate these steps of how to set up the activity after choosing a student as a partner:
Students work through this same process with their own partners on the questions in the activity.
Take turns with your partner to decide whether Shape A is congruent to Shape B.
To highlight student reasoning and language use, invite groups to respond to the following questions:
If two polygons have different sets of side lengths, they can’t be congruent.
For example, the figure on the left has side lengths 3, 2, 1, 1, 2, 1. The figure on the right has side lengths 3, 3, 1, 2, 2, 1. There is no way to make a correspondence between them where all corresponding sides have the same length.
If two polygons have the same side lengths, but not in the same order, the polygons can’t be congruent.
For example, rectangle
Two figures, A B C D and E F G H. Figure A B C D is a rectangle with side length 3 and base and top length 5. Figure E F G H has base length 5, top length is 3, left side length is 3 and right side length is 5.
If two polygons have the same side lengths, in the same order, but different corresponding angles, the polygons can’t be congruent.
For example, parallelogram
Help us improve by sharing suggestions or reporting issues.
Students may want to visually determine congruence each time or explain congruence by saying, “They look the same.” Encourage those students to explain congruence in terms of translations, rotations, reflections, and side lengths. For students who focus on features of the shapes such as side lengths and angles, ask them how they could show the side lengths or angle measures are the same or different using the grid or tracing paper.
For Part 5, students may be correct in saying the shapes are not congruent but for the wrong reason. They may say one is a 3-by-3 square and the other is a 2-by-2 square, counting the diagonal side lengths as one unit. If so, have them compare lengths by marking them on the edge of a card, or measuring them with a ruler.
In discussing congruence for Part 3, students may say that quadrilateral
Students may assume when building quadrilaterals with a set of objects of the same length that the resulting shapes are congruent. They may think that two shapes are congruent because they can physically manipulate them to make them congruent. Ask them to first build their quadrilateral and then compare it with their partner's. The goal is not to ensure that the two are congruent but to decide whether they have to be congruent.