Congruent Polygons

Student Summary

How do we know if two figures are congruent?
  • If we copy one figure on tracing paper and move the paper so the copy covers the other figure exactly, then that suggests they are congruent.
  • If we can describe a sequence of translations, rotations, and reflections that move one figure onto the other so they match up exactly, they are congruent.
How do we know that two figures are not congruent?
  • If there is no correspondence between the figures where the parts have equal measure, that shows that the two figures are not congruent.

    • 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.

      Two figures on a grid. 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.

    • If two polygons have the same side lengths, but not in the same order, the polygons can’t be congruent.

      For example, rectangle ABCDABCD can’t be congruent to quadrilateral EFGHEFGH. Even though they both have two sides of length 3 and two sides of length 5, they don’t correspond in the same order.

      Two figures, A B C D and E F G H.
      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 JKLMJKLM can’t be congruent to rectangle ABCDABCD. Even though they have the same side lengths in the same order, the angles are different. All angles in ABCDABCD are right angles. In JKLMJKLM, angles JJ and LL are less than 90 degrees and angles KK and MM are more than 90 degrees.

      Parallelogram J K L M with base and top length 5 units and sides length 3 units.

Visual / Anchor Chart

Standards

Addressing
8.G.2

Know that a two-dimensional figure is congruent to another if the corresponding angles are congruent and the corresponding sides are congruent. Equivalently, two two-dimensional figures are congruent if one is the image of the other after a sequence of rotations, reflections, and translations. Given two congruent figures, describe a sequence that maps the congruence between them on the coordinate plane.