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Geometry: High School15 chapters | 160 lessons

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Lesson Transcript

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

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.

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!

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:

We can see that triangle *ABC* is similar to triangle *EDC*, and we want to prove that (*AC*)â‹…(*DC*) = (*EC*)â‹…(*BC*). Thankfully, we can do this using the properties of similar figures. Since the two triangles are similar, it must be the case that the lengths of their sides are proportional, so we have that:

(*AC*/*EC*) = (*BC*/*DC*) = (*AB*/*ED*)

Now, we have that (*AC*/*EC*) = (*BC*/*DC*) according to the properties of similar figures. From here, we can simply use cross multiplication to prove that (*AC*)â‹…(*DC*) = (*EC*)â‹…(*BC*).

We get that (*AC*)â‹…(*DC*) = (*EC*)â‹…(*BC*), which was what we had to prove. Well, that was pretty easy, and it's all thanks to the properties of similar figures!

Now, let's consider an example involving congruent figures. Consider the figure shown.

We are given that figure *ABEF* and figure *FECD* are congruent, and we want to prove that âˆ *A* + âˆ *B* = âˆ *f*2 + âˆ *e*2.

Since the two figures are congruent, we can use the properties of congruent figures to deduce the following facts:

âˆ *A* = âˆ *f*2

âˆ *B* = âˆ *e*2

Therefore, it must be the case that âˆ *A* + âˆ *B* = âˆ *f*2 + âˆ *e*2, which was what we had to prove.

Once again, we see how useful these properties are. Knowing that the two figures were congruent made this proof a breeze!

**Similar figures** are figures that are the same shape and only differ in their size. When two figures are similar, they have the following properties:

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

**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. When two figures are congruent, they have the following properties:

- Corresponding angles have equal measure
- Corresponding sides have equal length

When we know two figures are similar or congruent, we can use their properties to prove different relationships about those figures. As we've seen, these properties can make a seemingly difficult proof extremely simple.

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Geometry: High School15 chapters | 160 lessons

- Applications of Similar Triangles 6:23
- Triangle Congruence Postulates: SAS, ASA & SSS 6:15
- Congruence Proofs: Corresponding Parts of Congruent Triangles 5:19
- Converse of a Statement: Explanation and Example 5:09
- Similarity Transformations in Corresponding Figures 7:28
- How to Prove Relationships in Figures using Congruence & Similarity 5:14
- The AAS (Angle-Angle-Side) Theorem: Proof and Examples 6:31
- The HA (Hypotenuse Angle) Theorem: Proof, Explanation, & Examples 5:50
- The HL (Hypotenuse Leg) Theorem: Definition, Proof, & Examples 6:19
- Perpendicular Bisector Theorem: Proof and Example 6:41
- Angle Bisector Theorem: Proof and Example 6:12
- Congruency of Right Triangles: Definition of LA and LL Theorems 7:00
- Congruency of Isosceles Triangles: Proving the Theorem 4:51
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