Symmetric Property in Geometry: Definition & Examples

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  • 0:00 Symmetric Property of Equality
  • 1:40 Symmetric Property in Geometry
  • 2:24 Real-World Examples
  • 3:28 Lesson Summary
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Lesson Transcript
Instructor: 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.

The symmetric property shows up often in the world around us, so it is useful to be familiar with this property. This lesson will allow us to become comfortable with, see examples of, and work with this neat property.

Symmetric Property of Equality

It's no secret that 2 + 2 = 4. Most of us know this from elementary school when we learned how to add. But what if I switched the order of this and told you that 4 = 2 + 2? This is probably still an obvious statement to you, right? This fact, that if 2 + 2 = 4, then 4 = 2 + 2, is actually an illustration of the symmetric property of equality. In mathematics, the symmetric property of equality is really quite simple. This property states that if a = b, then b = a. That is, we can interchange the sides of an equation, and the equation is still a true statement.

For example, all of the following are demonstrations of the symmetric property:

  • If x + y = 7, then 7 = x + y
  • If 2c - d = 3e + 7f, then 3e + 7f = 2c - d
  • If apple = orange, then orange = apple

In all of these, we see that the symmetric property just says that we can switch the sides of an equation, and the equation is still true.

A good way to remember the symmetric property is to think of it as the mirror property. When you look in the mirror, your mirror image is looking back at you, and you and your mirror image are the same regardless of which side of the mirror you are on. Thinking of the symmetric property this way brings us to the symmetric property in geometry.

Symmetric Property in Geometry

As we said, the symmetric property can be thought of as the mirror property. In geometry, an image or object is said to be symmetric if both of its sides are the same. That is, the sides of a symmetric image or object are mirror images of each other, so the image's sides are symmetric with respect to a line. This line is called the axis of symmetry. For example, a rectangle is symmetric.

symmetric property 1

We can see from this image that if we were to fold the rectangle in half along the axis of symmetry, both sides of the rectangle would line up with each other. This is because both sides of the rectangle are the same.

Let's consider some real world examples of symmetry.

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