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What Are Congruent Figures? - Definition & Examples

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  • 0:02 Congruent Figures
  • 0:37 Examples
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Lesson Transcript
Instructor
Joseph Vigil
Expert Contributor
Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, you'll learn what congruent figures are. You'll also see a few examples of congruent and non-congruent figures. Then, test your knowledge with a brief quiz.

Congruent Figures

You've found the poster that's just the right size for your bedroom wall - not too narrow, not too wide. In fact, you like it so much you think you want one the same exact size for your living room. A math person might say you're looking for a poster that's congruent to the one you already have.

If two figures are congruent, then they're exactly the same shape, and they're exactly the same size. They may appear different because one is shifted or rotated a certain way, but they're still the same shape, and all the sides of one are the same length as the corresponding sides of the other.

Examples of Congruency

Congruency is what two figures have if they're congruent. These shapes obviously show congruency because they're exactly the same:

Same shape, same size. They are congruent!
Two Circles of Same Shape and Same Size

But what about these shapes?

Two Triangles, Same Size, Different Orientations

One triangle points up, while the other points down. However, if you were to measure them, you'd discover that the corresponding sides of both triangles are exactly the same. In fact, the second triangle is the same as the first; it's just been rotated. So, these shapes are congruent.

It's important to remember that the directions shapes face don't affect their congruency. As long as they're the exact same shape and have the exact same measurements, they're congruent.

Let's look at squares now:

One Large Square, One Small Square

These are both squares, but one is much smaller than the other. Although they're the same shape, these two figures aren't the same size. Therefore, these squares are incongruent, or not congruent.

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Additional Activities

Identifying Congruent Figures in the Real World

Recall:

  • Two figures are congruent if and only if they have the exact same shape and the exact same size.
  • If the corresponding sides of two figures with the same shape have the same length, then the two figures are congruent.

Questions:

  1. A bicycle has two wheels, such that the distance around each wheel is the same. Are the two wheels congruent? Why or why not?
  2. Sherri has two rectangular notebooks, such that the paper in each one has the same length of 11 inches. The width of the paper in one of the notebooks is 8 inches, and the width of the paper in the other notebook is 7 inches. Is a piece of paper in one of the notebooks congruent to a piece of paper in the other notebook? Why or why not?
  3. The measure of the diameter of a U.S. quarter is 0.955 inches. If Mandy has two quarters, are they congruent? Why or why not? (Hint: circles with the same diameter have the same circumference, or distance around them).
  4. Come up with some of your own real-world examples of congruent figures, and explain why they are congruent.

Solutions:

  1. Yes; The two wheels are both circles, and the distance around them is the same. Therefore, they are both the same shape with corresponding sides of the same length, so they are congruent.
  2. No; Even though the paper in each of the notebooks are both rectangles, and they have the same length of 11 inches, their widths are different. Thus, all of the corresponding sides of the two pieces of paper do not have the same length, so they are not congruent.
  3. Yes; The two quarters are both circles, and they have the same diameter. Since they have the same diameter, they have the same circumference, so the distance around each quarter is the same. Therefore, they are two figures with the same shape and corresponding sides of the same length, so they are congruent.
  4. Answers will vary, but some examples could include the squares on a checkerboard, your reflection in the mirror, two standard sized stop signs, the stars on the American flag, or triangles created by the beams of a bridge that have the same corresponding side lengths.

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