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Updated Mar 2, 2026
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Learn Triangle Congruence with SSS, SAS, and CPCTC!
A
Anwita
@anwita_jvda
1 / 6
Page 2: Practical Proofs Application
This page demonstrates practical applications of triangle congruence proofs using SSS and SAS postulates. Two detailed examples are presented with step-by-step solutions.
Example: First proof shows triangle EFG congruent to triangle GHE using SSS postulate, utilizing given conditions EF ≅ GH and FG ≅ HE, along with the reflexive property for GE.
Highlight: Second proof demonstrates triangle DRA congruent to triangle DRG using SAS postulate, incorporating right angles and the reflexive property.
Page 3: Advanced Proofs and Hypotenuse-Leg Theorem
This page introduces the Hypotenuse-Leg (HL) Congruence Theorem and continues with more complex proof examples.
Definition: The Hypotenuse-Leg Congruence Theorem states that if the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent.
Example: A proof involving triangles NOT and RSP demonstrates the application of given conditions and theorem principles.
Page 4: ASA and AAS Congruence
This page explores two additional triangle congruence theorems: Angle-Side-Angle (ASA) and Angle-Angle-Side (AAS).
Definition: ASA Congruence states that if two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
Definition: AAS Congruence establishes that if two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
Page 5: CPCTC Theorem Application
This page focuses on the CPCTC theorem and its practical application in geometric proofs.
Definition: CPCTC (Corresponding Parts of Congruent Triangles are Congruent) is a theorem stating that when two triangles are proven congruent, all their corresponding parts are also congruent.
Example: A detailed proof showing JK = JL using angle bisector JM and the ASA theorem, followed by CPCTC application.
Page 6: Extended Proofs
This page provides additional practice with extended proofs, incorporating various concepts learned throughout the unit.
Highlight: The proof combines multiple concepts including vertical angles, midpoint properties, and the CPCTC theorem.
Example: A complex proof involving parallel lines, midpoints, and triangle congruence is presented with detailed steps and reasons.
Page 1: Triangle Congruence Fundamentals
This page introduces the fundamental concepts of triangle congruence, focusing on the SSS and SAS postulates. The content establishes the groundwork for understanding how to prove triangles are congruent using different methods.
Definition: Triangle congruence occurs when all six pairs of corresponding parts between two triangles are equal.
Highlight: Two primary postulates involve sides (SSS and SAS), while one theorem specifically applies to right triangles.
Vocabulary: SSS - A postulate stating that if all three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
Example: The reflexive property of congruence is demonstrated when there is a shared side between triangles, and vertical angles are shown to be equal.
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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.
Most popular content in Geometry
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Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
4.6/5
App Store
4.7/5
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 Knowunity AI. 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 Knowunity AI. 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
Learn Triangle Congruence with SSS, SAS, and CPCTC!
A
Anwita
@anwita_jvda
Triangle Congruence and Geometric Proofs - A comprehensive guide exploring triangle congruence SSS SAS proofs, congruence theorems, and geometric relationships.
• The guide covers essential triangle congruence postulates including SSS (Side-Side-Side) and SAS (Side-Angle-Side)
• Introduces the CPCTC theorem... Show more
Page 2: Practical Proofs Application
This page demonstrates practical applications of triangle congruence proofs using SSS and SAS postulates. Two detailed examples are presented with step-by-step solutions.
Example: First proof shows triangle EFG congruent to triangle GHE using SSS postulate, utilizing given conditions EF ≅ GH and FG ≅ HE, along with the reflexive property for GE.
Highlight: Second proof demonstrates triangle DRA congruent to triangle DRG using SAS postulate, incorporating right angles and the reflexive property.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Page 3: Advanced Proofs and Hypotenuse-Leg Theorem
This page introduces the Hypotenuse-Leg (HL) Congruence Theorem and continues with more complex proof examples.
Definition: The Hypotenuse-Leg Congruence Theorem states that if the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent.
Example: A proof involving triangles NOT and RSP demonstrates the application of given conditions and theorem principles.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Page 4: ASA and AAS Congruence
This page explores two additional triangle congruence theorems: Angle-Side-Angle (ASA) and Angle-Angle-Side (AAS).
Definition: ASA Congruence states that if two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
Definition: AAS Congruence establishes that if two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
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Page 5: CPCTC Theorem Application
This page focuses on the CPCTC theorem and its practical application in geometric proofs.
Definition: CPCTC (Corresponding Parts of Congruent Triangles are Congruent) is a theorem stating that when two triangles are proven congruent, all their corresponding parts are also congruent.
Example: A detailed proof showing JK = JL using angle bisector JM and the ASA theorem, followed by CPCTC application.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Page 6: Extended Proofs
This page provides additional practice with extended proofs, incorporating various concepts learned throughout the unit.
Highlight: The proof combines multiple concepts including vertical angles, midpoint properties, and the CPCTC theorem.
Example: A complex proof involving parallel lines, midpoints, and triangle congruence is presented with detailed steps and reasons.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Page 1: Triangle Congruence Fundamentals
This page introduces the fundamental concepts of triangle congruence, focusing on the SSS and SAS postulates. The content establishes the groundwork for understanding how to prove triangles are congruent using different methods.
Definition: Triangle congruence occurs when all six pairs of corresponding parts between two triangles are equal.
Highlight: Two primary postulates involve sides (SSS and SAS), while one theorem specifically applies to right triangles.
Vocabulary: SSS - A postulate stating that if all three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
Example: The reflexive property of congruence is demonstrated when there is a shared side between triangles, and vertical angles are shown to be equal.
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.
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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 Knowunity AI. 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 Knowunity AI. 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