Back To Course

Geometry: High School15 chapters | 160 lessons

Watch short & fun videos
**
Start Your Free Trial Today
**

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over 70,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Free 5-day trial
Your next lesson will play in
10 seconds

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

To unlock this lesson you must be a Study.com Member.

Create your account

Already a member? Log In

BackDid you know… We have over 95 college courses that prepare you to earn credit by exam that is accepted by over 2,000 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
9 in chapter 5 of the course:

Back To Course

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
- Go to High School Geometry: Triangles, Theorems and Proofs

- NES Social Science: Help & Review
- Computer Science 311: Artificial Intelligence
- View High School: English 4
- View High School: English 3
- View High School: English 2
- The Evolution of National & State Governments
- Interpreting Economic Information
- Causes & Effects of the Great Depression
- Major U.S. Social Developments since 1945
- Current Environmental Problems
- FTCE Prep Product Comparison
- TExES Prep Product Comparison
- Study.com ASVAB Scholarship: Application Form & Information
- Study.com GED Scholarship: Application Form & Information
- Study.com GACE Scholarship: Application Form & Information
- Study.com CSET/CBEST Scholarship: Application Form & Information
- Study.com NES Scholarship: Application Form & Information

- Managing the Effects of Global Change on Organizations
- Tire Marks Forensic Examination: Methods & Purpose
- Using Concept Maps to Plan Instruction
- Substance Abuse & Juvenile Delinquency: Prevention & Correction Strategies
- Script, Process, Product & Audience as Elements of Theatre
- Health Outcomes for Older Persons with Multiple Chronic Conditions
- How to Give Feedback to a New Boss: Strategies & Examples for Employees
- Adsorption of Gases: Definition & Examples
- Quiz & Worksheet - Analyzing The Allure of Free
- Quiz & Worksheet - Rape Kits Utilization
- Quiz & Worksheet - Forensic Pathologists Duties
- Quiz & Worksheet - Importance of EBP
- Quiz & Worksheet - Development of Geometric Thought
- International Law & Global Issues Flashcards
- Foreign Policy, Defense Policy & Government Flashcards

- Middle School US History: Homework Help Resource
- FTCE Middle Grades General Science 5-9 (004): Test Practice & Study Guide
- AP Physics C: Homeschool Curriculum
- Biology for Teachers: Professional Development
- AP Physics C - Mechanics: Exam Prep
- Case Studies in Psychology
- AP Environmental Science - Glaciers: Tutoring Solution
- Quiz & Worksheet - Criminology's Crime Control & Due Process Models
- Quiz & Worksheet - Primary, Secondary, Tertiary & Quaternary Structures
- Quiz & Worksheet - Finished Goods Inventory
- Quiz & Worksheet - Asexual Plant Reproduction
- Quiz & Worksheet - Types of Circulatory Systems

- What Is the Economic Man? - Concept, Assumptions & Constraints
- Rituals: Definition, Types & Challenges
- What Will I Learn in an SAT Class?
- How to Use the GED Social Studies Prep Course
- Cover Letter Lesson Plan
- What's in Common Core Standards Appendix C?
- Scientific Method Experiments for Kids
- AP English Book List Example
- Context Clues Lesson Plan
- Do Private Schools Take Standardized Tests?
- What is the GMAT test?
- Science Picture Books

Browse by subject