Lesson 13Congruence

Let’s find ways to test congruence of interesting figures.

Learning Targets:

  • I can use distances between points to decide if two figures are congruent.

13.1 Not Just the Vertices

Trapezoids ABCD and A’B’C’D’ are congruent.

  • Draw and label the points on A’B’C’D’ that correspond to E and F .
  • Draw and label the points on ABCD that correspond to G’ and H’ .
  • Draw and label at least three more pairs of corresponding points.

13.2 Congruent Ovals

Are any of the ovals congruent to one another? Explain how you know.

Are you ready for more?

You can use 12 toothpicks to create a polygon with an area of five square toothpicks, like this:

Can you use exactly 12 toothpicks to create a polygon with an area of four square toothpicks?

13.3 Corresponding Points in Congruent Figures

Here are two congruent shapes with some corresponding points labeled.

  1. Draw the points corresponding to B , D , and E , and label them B’ , D’ , and E’ .

  2. Draw line segments AD and A’D’ and measure them. Do the same for segments BC and B’C’ and for segments AE and A’E’ . What do you notice?

  3. Do you think there could be a pair of corresponding segments with different lengths? Explain.

13.4 Astonished Faces

Here are two faces, Max and Frankie. Are these faces congruent? Explain your reasoning.

Lesson 13 Summary

To show two figures are congruent, you align one with the other by a sequence of rigid transformations. This is true even for figures with curved sides. Distances between corresponding points on congruent figures are always equal, even for curved shapes. For example, corresponding segments AB and A'B' on these congruent ovals have the same length:

Two congruent ovals on a square grid. In the first oval, two points on opposite sides of the oval are labeled A and B and are connected by a line segment. In the second oval, two points on opposite sides of the oval are labeled A prime and B prime and are connected by a line segment.

To show two figures are not congruent, you can find parts of the figures that should correspond but that have different measurements.

For example, these two ovals don’t look congruent.

On both, the longest distance is 5 units across, and the longest distance from top to bottom is 4 units. The line segment from the highest to lowest point is in the middle of the left oval, but in the right oval, it’s 2 units from the right end and 3 units from the left end. This proves they are not congruent.

Lesson 13 Practice Problems

  1. Which of these four figures are congruent to the top figure?
  2. These two figures are congruent, with corresponding points marked.

    1. Are angles ABC and A'B'C' congruent? Explain your reasoning.
    2. Measure angles ABC and A’B’C’ to check your answer.
  3. Here are two figures.

    Show, using measurement, that these two figures are not congruent.

  4. Each picture shows two polygons, one labeled Polygon A and one labeled Polygon B. Describe how to move Polygon A into the position of Polygon B using a transformation.