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Lesson 12Congruent Polygons

Let’s decide if two figures are congruent.

Learning Targets:

  • I can decide using rigid transformations whether or not two figures are congruent.

12.1 Translated Images

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 ABC under a translation.

Nine congruent triangles are shown

12.2 Congruent Pairs (Part 1)

For each of the following pairs of shapes, decide whether or not they are congruent. Explain your reasoning.

  1. Two congruent figures are shown on a graph.
  2. Two different figures are shown on a graph
  3. Two congruent triangles are on a graph
  4. Two different figures are shown on a graph

12.3 Congruent Pairs (Part 2)

For each pair of shapes, decide whether or not Shape A is congruent to Shape B. Explain how you know.

  1. Two congruent figures are shown on a graph.
  2. Two different figures are shown on a graph
  3. Two congruent figures are shown on a graph.
  4. Two different parallelograms are shown on a graph.
  5. Squares A and B are graphed on a grid.

Are you ready for more?

A polygon has 8 sides: five of length 1, two of length 2, and one of length 3. All sides lie on grid lines. (It may be helpful to use graph paper when working on this problem.)

  1. Find a polygon with these properties.

  2. Is there a second polygon, not congruent to your first, with these properties?

12.4 Building Quadrilaterals

Your teacher will give you a set of four objects.

  1. Make a quadrilateral with your four objects and record what you have made.

  2. Compare your quadrilateral with your partner’s. Are they congruent? Explain how you know.

  3. Repeat steps 1 and 2, forming different quadrilaterals. If your first quadrilaterals were not congruent, can you build a pair that is? If your first quadrilaterals were congruent, can you build a pair that is not? Explain.

Lesson 12 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.

  • We can prove that two figures are congruent by describing a sequence of translations, rotations, and reflections that move one figure onto the other so they match up exactly.

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 proves that the two figures are not congruent. In particular,

    • 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 different figures are shown
    • If two polygons have the same side lengths, but their orders can’t be matched as you go around each polygon, the polygons can’t be congruent. For example, rectangle ABCD can’t be congruent to quadrilateral EFGH . Even though they both have two sides of length 3 and two sides of length 5, they don’t correspond in the same order. In ABCD , the order is 3, 5, 3, 5 or 5, 3, 5, 3; in EFGH , the order is 3, 3, 5, 5 or 3, 5, 5, 3 or 5, 5, 3, 3.
      Two different figures are shown

    • 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 JKLM can’t be congruent to rectangle ABCD . Even though they have the same side lengths in the same order, the angles are different. All angles in ABCD are right angles. In JKLM , angles J and L are less than 90 degrees and angles K and M are more than 90 degrees.

      A parallelogram is shown with side lengths of 3, 3, 5, and 5.

Lesson 12 Practice Problems

    1. Show that the two pentagons are congruent.
    2. Find the side lengths of ABCDE and the angle measures of FGHIJ .
    Two congruent figures are shown

  2. For each pair of shapes, decide whether or not the two shapes are congruent. Explain your reasoning.

    1. Two congruent figures are shown
    2. Two figures are shown on a graph
    3. Circles A and C are shown on a graph.
    1. Draw segment PQ .
    2. When PQ is rotated 180^\circ around point R , the resulting segment is the same as PQ . Where could point R be located?

  4. Here is trapezoid ABCD .

    A quadrilateral is shown with angles of 60, 90, 90, and 120 degrees.

    Using rigid transformations on the trapezoid, build a pattern. Describe some of the rigid transformations you used.