How to Prove Relationships in Figures using Congruence & Similarity

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  • 0:03 Congruent & Similar Figures
  • 1:31 Proving Relationships
  • 2:33 Examples
  • 4:22 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.

In this lesson, we'll look at similar and congruent figures and the properties that they hold. We will then look at how to use these properties to prove relationships in these figures in various examples.

Congruent & Similar Figures

When was the last time you went clothes shopping? If you are a woman, suppose you go to the store looking for a new skirt. As you're looking at the different styles you find two that you really like and see that they come in sizes small, medium, and large. Notice that the skirts are all the exact same shape, just different sizes.

Well guess what? This observation is actually a mathematical one, and speaking mathematically, we would say that these skirts are similar. In mathematics, two figures are similar figures when their only difference is their size. That is, one figure can be obtained from another figure simply by resizing the figure.

You grab a size small and a size medium for one of the skirts and a size small of the other skirt, then you head to the dressing room. You try on the small skirts and then discard them on the floor as you try on the medium skirt. You notice that the two small skirts are situated on the floor in such a way that if you were to slide one of the skirts over a couple of feet, it would be lying directly on top of the other skirt.

Once again, we can describe these two same-sized skirts using a mathematical term, and that term is congruent figures. Congruent figures are two figures of the same size and shape that can be obtained from one another by rotating, reflecting, or sliding one of the figures. Huh! Who knew that shopping for clothes involved so much mathematics!

Proving Relationships

You may be thinking that these terms for figures are neat and all, but so what? Well, as it turns out, when two figures are similar or congruent, they have certain properties, and these properties can be used to prove relationships between the figures.

When two figures are similar figures, they have the following properties:

  • Corresponding angles have equal measure
  • Sides of the shapes/figures are proportional

To illustrate this, let's look at the small and medium skirt again. Notice that the sides are proportional and the corresponding angles are equal.


Congruent figures have the following properties:

  • Corresponding angles have equal measure
  • Corresponding sides have equal length

Take a look at the two small skirts you laid on the dressing room floor again. Notice that corresponding angles and sides are equal.


These properties are extremely useful because, when we know that two figures are similar or congruent, we can use these properties to prove relationships between those figures. Let's take a look at doing just that.


Consider the right triangle shown in the image:


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