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Geometry: High School15 chapters | 160 lessons

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Lesson Transcript

Instructor:
*Jeff Calareso*

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

When we have two triangles, how can we tell if they're congruent? They may look the same, but you can be certain by using one of several triangle congruence postulates, such as SSS, SAS or ASA.

This is Tricago.

Times are good in Tricago. Lots of new towers are being built. And each level in each tower is shaped like a triangle. That means that these triangles must by congruent. **Congruent Triangles** are *triangles with three congruent sides and three congruent angles*. If two of our triangle levels are congruent, we can always make them stack perfectly and neatly on top of each other.

But how do we know if they're congruent? Let's look at three statements that will tell us, based on what we know about the triangles' sides and angles.

First, there's the **side-side-side** postulate, or **SSS**. This states that if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. Should we see this in action? Okay, let's start with an equilateral triangle.

But what if we add a line from A to the midpoint of BC? Let's call this AD. We now have two smaller triangles: ABD and ACD.

Can we be sure? Well, since ABC is an equilateral triangle, we know angles B and C are each 60 degrees. And angles ADB and ADC are each 90 degrees. Then we split A into BAD and CAD, both of which are 30 degrees. So all three sides and all three angles match. So they are congruent!

So if we're trying to make sure the levels of our building will match, we can measure the three sides. Without even looking at the angles, if the sides all match, the SSS postulate tells us they must be congruent.

The SSS postulate is great, but measuring all three sides involves a lot of walking around big triangles. What if you're tired? Is there a shorter way? Yes! There's the **Side-Angle -Side** postulate, or **SAS**. This states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Here are two triangles.

That's two sides and the included angle. Using the SAS postulate, we know they're congruent. Note that the postulate stipulates the included angle. That means the angle formed by the two sides. If we only knew that angles A and D were equal, that wouldn't be enough. Fortunately, we stopped to measure angles B and E while we were walking around, so we're good.

Wait, Tricago is a city of threes, and that's only two postulates. Can we get a third one? It just wouldn't be Tricago if they didn't come in threes.

Okay, let's add the **angle-side-angle** postulate, or **ASA**. This states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. The ASA postulate is sort of the opposite of the SAS postulate.

Can we see this in action? Well, here's parallelogram ABCD.

To do that, draw line BD.

We also know that angles ABD and BDC are equal. Why? If AB and DC are parallel, then those angles are alternate interior angles. And guess what? ADB and DBC are also equal by the same rule. That's two angles and the included side. So these are congruent triangles! Thanks, ASA postulate.

Now, since it's a parallelogram, we also know that AD and BC are equal, as well as AB and DC. Oh, and angles A and C are equal, too. So, no question about it, Tricago is a city built on joined congruent triangles.

In summary, we learned a good lesson about respecting our neighbors in Tricago. You can't judge a city by its silly name. Oh, and we also learned all about identifying congruent triangles. These are triangles with three equal sides and three equal angles.

Since we love things in threes in Tricago, we focused on three postulates. First, there's the SSS postulate. This states that if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. Second, there's the SAS postulate. This states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. And third, there's the ASA postulate. This states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Once you have completed this lesson you should be able to state and apply the SAS, ASA, and SSS triangle congruency postulates.

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Geometry: High School15 chapters | 160 lessons

- Applications of Similar Triangles 6:23
- Triangle Congruence Postulates: SAS, ASA & SSS 6:15
- 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
- Practice Proving Relationships using Congruence & Similarity 6:16
- 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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