Lesson 12Congruent Polygons

Learning Goal

Let’s decide if two figures are congruent.

Learning Targets

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

Lesson Terms

  • congruent
  • right angle

Warm Up: Translated Images

Problem 1

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 under a translation.

Blue triangle ABC and eight green triangles oriented in different ways.

Activity 1: Congruent Pairs (Part 1)

Problem 1

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

  1. A coordinate plane with shape ABCD in quadrant 1 and EFGH in quadrant 3.
  2. A coordinate plane with shape ABCDE in quadrant 2 and FGHIJ in quadrant 1.
  3. A coordinate plane with triangle ABC straddling quadrant 1 and 4 and DEF in quadrant 2.
  4. Two octagons, ABCDEFGH and a smaller version IJKLMNOP.

Activity 2: Congruent Pairs (Part 2)

Problem 1

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

  1. Shape A, GHIJKL and shape B, PQRSTU
  2. Shape A, GHIJKL and shape B, PQRSTU
  3. Trapezoid A, GHIJ in quadrant 2 and trapezoid B, PQRS in quadrant 1.
  4. Rhombus A, GHIJ straddling quadrants 2 and 3, and square B, PQRS in quadrant 1
  5. Square A, GHIJ in quadrant 2 and rhombus B, PQRS in quadrant 1.

Are you ready for more?

Problem 1

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?

Activity 3: Building Quadrilaterals

Problem 1

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 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 polygons that look similar but have different side lengths.

  • 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 can’t be congruent to quadrilateral . 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 , the order is 3, 5, 3, 5 or 5, 3, 5, 3; in , the order is 3, 3, 5, 5 or 3, 5, 5, 3 or 5, 5, 3, 3.

A rectangle, ABCD and polygon EFGH with the same side lengths but in different places.

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

Parallelogram JKLM with the same side lengths of previous shape ABCD.