The Hypotenuse-Leg (HL) Theoremis a crucial concept in geometry... Show more
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
5
0
Katie Whitson
9/21/2023
Geometry
Congruence in Right Triangles
67
•
Sep 21, 2023
•
Easy Hypotenuse Leg Theorem and Right Triangle Proofs for Kids
1 / 3
Applying the HL Theorem in Proofs
This page expands on the application of the Hypotenuse-Leg Theorem by presenting more complex proofs and introducing related concepts.
The first example builds on the previous proof, introducing additional given information:
- PS ≅ PT
- PRS ≅ PRT
The goal is to prove that ΔPRS ≅ ΔPRT. The proof utilizes the Isosceles Triangle Theorem in conjunction with the HL Theorem to establish congruence.
Vocabulary: The Isosceles Triangle Theorem states that the base angles of an isosceles triangle are congruent.
A more elaborate proof is then presented, involving a bisected line segment and perpendicular lines:
Given:
- BE bisects AD at C
- AB ⊥ BC
- DE ⊥ EC
- AB ≅ DE
Prove: ΔABC ≅ ΔDEC
This proof demonstrates how to combine multiple geometric concepts, including bisectors, perpendicular lines, and right angles, to set up the conditions for applying the HL Theorem.
Highlight: The HL Theorem can be used in conjunction with other geometric theorems and concepts to prove congruence in more complex scenarios.
Advanced Applications of the HL Theorem
The final page presents a sophisticated proof that combines the Hypotenuse-Leg Theorem with the concept of perpendicular bisectors.
Given:
- CD ≅ EA
- AD is the perpendicular bisector of CE
Prove: ΔCBD ≅ ΔEBA
The proof systematically uses the given information and the properties of perpendicular bisectors to establish the conditions necessary for applying the HL Theorem.
Vocabulary: A perpendicular bisector is a line that passes through the midpoint of another line segment at a right angle.
The proof demonstrates how to:
- Identify the hypotenuse (CD ≅ EA)
- Use the properties of perpendicular bisectors to establish leg congruence (CB ≅ EB)
- Confirm right angles using the definition of perpendicular lines
- Apply the HL Theorem to conclude triangle congruence
Highlight: This advanced proof showcases how the HL Theorem can be applied in complex geometric scenarios, particularly when combined with properties of perpendicular bisectors.
This example reinforces the versatility and power of the Hypotenuse-Leg Theorem in proving congruence for right triangles, even in more intricate geometric configurations.
Congruence in Right Triangles: Hypotenuse-Leg (HL) Theorem
The Hypotenuse-Leg (HL) Theorem is introduced as a fundamental concept for proving congruence in right triangles. This theorem states that if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
Definition: The Hypotenuse-Leg Theorem states that if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
The page provides a visual representation of the theorem using triangles PQR and XYZ, demonstrating how the theorem is applied when the hypotenuse and one leg of each triangle are congruent.
Example: Given right triangles PQR and XYZ, if PR ≅ XZ (hypotenuse) and PQ ≅ XY (leg), then ΔPQR ≅ ΔXYZ.
An example proof is presented to demonstrate the application of the HL Theorem:
Given:
- ∠PRS and ∠RPQ are right angles
- SP ≅ OR
Prove: ΔPRS ≅ ΔRPQ
The proof proceeds step-by-step, using the definition of right angles and the given information to apply the HL Theorem and conclude that the triangles are congruent.
Highlight: The HL Theorem is a powerful tool for proving congruence in right triangles when the hypotenuse and one leg of each triangle are known to be congruent.
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
4.9/5
App Store
4.8/5
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
Easy Hypotenuse Leg Theorem and Right Triangle Proofs for Kids
The Hypotenuse-Leg (HL) Theoremis a crucial concept in geometry for proving congruence in right triangles. This theorem states that if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another... Show more
Applying the HL Theorem in Proofs
This page expands on the application of the Hypotenuse-Leg Theorem by presenting more complex proofs and introducing related concepts.
The first example builds on the previous proof, introducing additional given information:
- PS ≅ PT
- PRS ≅ PRT
The goal is to prove that ΔPRS ≅ ΔPRT. The proof utilizes the Isosceles Triangle Theorem in conjunction with the HL Theorem to establish congruence.
Vocabulary: The Isosceles Triangle Theorem states that the base angles of an isosceles triangle are congruent.
A more elaborate proof is then presented, involving a bisected line segment and perpendicular lines:
Given:
- BE bisects AD at C
- AB ⊥ BC
- DE ⊥ EC
- AB ≅ DE
Prove: ΔABC ≅ ΔDEC
This proof demonstrates how to combine multiple geometric concepts, including bisectors, perpendicular lines, and right angles, to set up the conditions for applying the HL Theorem.
Highlight: The HL Theorem can be used in conjunction with other geometric theorems and concepts to prove congruence in more complex scenarios.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Advanced Applications of the HL Theorem
The final page presents a sophisticated proof that combines the Hypotenuse-Leg Theorem with the concept of perpendicular bisectors.
Given:
- CD ≅ EA
- AD is the perpendicular bisector of CE
Prove: ΔCBD ≅ ΔEBA
The proof systematically uses the given information and the properties of perpendicular bisectors to establish the conditions necessary for applying the HL Theorem.
Vocabulary: A perpendicular bisector is a line that passes through the midpoint of another line segment at a right angle.
The proof demonstrates how to:
- Identify the hypotenuse (CD ≅ EA)
- Use the properties of perpendicular bisectors to establish leg congruence (CB ≅ EB)
- Confirm right angles using the definition of perpendicular lines
- Apply the HL Theorem to conclude triangle congruence
Highlight: This advanced proof showcases how the HL Theorem can be applied in complex geometric scenarios, particularly when combined with properties of perpendicular bisectors.
This example reinforces the versatility and power of the Hypotenuse-Leg Theorem in proving congruence for right triangles, even in more intricate geometric configurations.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Congruence in Right Triangles: Hypotenuse-Leg (HL) Theorem
The Hypotenuse-Leg (HL) Theorem is introduced as a fundamental concept for proving congruence in right triangles. This theorem states that if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
Definition: The Hypotenuse-Leg Theorem states that if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
The page provides a visual representation of the theorem using triangles PQR and XYZ, demonstrating how the theorem is applied when the hypotenuse and one leg of each triangle are congruent.
Example: Given right triangles PQR and XYZ, if PR ≅ XZ (hypotenuse) and PQ ≅ XY (leg), then ΔPQR ≅ ΔXYZ.
An example proof is presented to demonstrate the application of the HL Theorem:
Given:
- ∠PRS and ∠RPQ are right angles
- SP ≅ OR
Prove: ΔPRS ≅ ΔRPQ
The proof proceeds step-by-step, using the definition of right angles and the given information to apply the HL Theorem and conclude that the triangles are congruent.
Highlight: The HL Theorem is a powerful tool for proving congruence in right triangles when the hypotenuse and one leg of each triangle are known to be congruent.
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Mock Exam
Quiz
Flashcards
Essay
Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
4.9/5
App Store
4.8/5
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user