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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.

In this lesson, we'll learn about the hypotenuse angle theorem. With this theorem, we can prove two right triangles are congruent with just congruent hypotenuses and acute angles.

If you're a triangle, finding out that you're congruent to another triangle is a big deal. Imagine finding out one day that you have a twin that you didn't know about. How amazing would that be? It's like having a spare 'you' suddenly enter your life.

In geometry, we try to find triangle twins in any way we can. There are all kinds of methods, like side-side-side, angle-side-angle, side-angle-side and more. In the real world, it doesn't work that way. You can't just compare legs with a stranger to test for congruency.

With two right triangles, we already know that they have something in common - those right angles. So, it's like they're at least cousins. And we can prove they're congruent with the hypotenuse angle theorem.

The **hypotenuse angle theorem**, also known as the **HA theorem**, states that 'if the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and an acute angle of another right triangle, then the two triangles are congruent.'

How can we verify congruency with just a hypotenuse and an acute angle? It's like saying two people are twins because they have the same height and hair color. Let's look at a couple of triangles.

Here's triangle *ABC*:

Now along comes triangle *XYZ*:

Could they be twins? They're both right triangles. Angles *B* and *Y* are each 90 degrees. We're told that *AC* is congruent to *XZ*. So, that's one hypotenuse that's congruent to the other. And we're told that angle *A* is congruent to angle *X*. That's good, but it's not like a DNA test.

Or is it? In triangle *ABC*, what's the sum of the interior angles? 180. What about with triangle *XYZ*? It's also 180. So, if two angles are congruent, like *A* and *X*, and another two angles are congruent, like *B* and *Y*, then the other angles, *C* and *Z*, must also be congruent.

So, now we have angle *A*, side *AC* and angle *C* congruent to angle *X*, side *XZ* and angle *Z*. And that's angle-side-angle, or ASA. That means that the HA theorem is really just a simplification or variation of the ASA postulate that works with right triangles.

Let's try to find some twins with some proofs. You know, you're not twins without proof. Here are two triangles:

They're very close. Together, they look kinda like a kite, don't they? Maybe they like to fly kites together. But are they just really good friends, or are they twins?

We're given that angles *R* and *S* are right angles. And we're also given that angle *SQT* is congruent to angle *RQT*. Let's say we want to determine if *RT* is congruent to *ST*.

Let's start our proof by collecting DNA samples from each triangle. Wait, what? Two-dimensional polygons don't have DNA? Oh. Then I guess we'll need to do an ordinary proof. Okay, first, we know that angles *R* and *S* are right angles. We're given that. That means that triangles *QST* and *QRT* are right triangles. That's the definition of a right triangle.

Next, we know that angle *SQT* is congruent to angle *RQT*. That's given. And we know that *QT* is congruent to *QT* because of the reflexive property. Now we can say that triangle *QST* is congruent to *QRT* because of the HA theorem. So, they're not just kite buddies; they're twins!

That enables us to say that *RT* is congruent to *ST* due to CPCTC, or corresponding parts of congruent triangles are congruent. And we're done!

How about one more? Here are two triangles that are also close:

How close? They're practically joined at the vertex. Oh, triangle humor.

Anyway, we're given that *AC* is congruent to *CE* and that angles *B* and *D* are right angles. We want to know if *AB* is congruent to *DE*. First, we'll need to determine if the triangles are congruent.

Let's start by stating that angle *B* is a right angle. Next, angle *D* is a right angle. Okay, so *ABC* and *CDE* are right triangles. One right angle apiece and that's the definition of right triangles.

Now let's state that *AC* is congruent to *CE*. That's given. So, right triangles, and we know one hypotenuse is congruent to the other. That's not enough, is it? But wait. We can say that angle *ACB* is congruent to angle *DCE*. Why? They're vertical angles, and vertical angles are congruent.

Now it's time to bust out our HA theorem and state that triangles *ABD* and *CDE* are congruent. So, they are like conjoined twins! Now we can finish our proof by using CPCTC to state that *AB* is congruent to *DE*.

In summary, we learned a valuable lesson about twins. Well, maybe not human twins. But we did learn about right triangle twins. Specifically, we focused on the **hypotenuse angle theorem**, or the **HA theorem**. This theorem states that 'if the hypotenuse and one acute angle of a right triangle are congruent to the hypotenuse and one acute angle of another right triangle, then the triangles are congruent.' We saw how this is really just a variation of ASA, or angle-side-angle. We then used this theorem in a pair of proofs to help us demonstrate congruency. And all this without any DNA tests!

After this lesson, you'll have the ability to:

- Restate the hypotenuse angle theorem (HA theorem)
- Explain how the HA theorem is a variation of the angle-side-angle theorem
- Prove the HA theorem using examples

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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
- 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
- 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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