Subjects
Biology
Chemistry
AP Chemistry
English
AP English Literature
Geometry
Dec 13, 2025
25
9 pages
calista ๐ชป @urstrulycalista
Triangle congruence is a fundamental concept in geometry that helps us prove when two triangles are identical in... Show more
Triangle congruence means that two triangles have exactly the same shape and size. When triangles are congruent, all their corresponding parts (sides and angles) match perfectly.
This concept forms the foundation for many geometric proofs and problem-solving techniques. You'll soon learn there are several ways to prove triangles are congruent without having to check every single side and angle.
Remember When we say triangles are congruent, we use the symbol โ , as in โณABC โ โณDEF.
An included angle is the angle formed between two sides of a triangle. This concept is critical for understanding several congruence postulates.
For example, if you're looking at sides AB and BC, the included angle would be angle B, since it's formed by those two specific sides meeting at point B.
When identifying included angles, always look for the angle created at the point where your two sides intersect. This will be important when we work with the SAS congruence postulate.
Quick Tip The included angle is always at the "shared" vertex of the two sides you're considering.
When two triangles are congruent, all their corresponding parts are also congruent. This principle is known as CPCTC (Corresponding Parts of Congruent Triangles are Congruent).
However, you don't need to check all six parts (3 sides and 3 angles) to prove triangles are congruent. You can use shortcuts like two sides and the included angle, two angles and the included side, or just three sides.
For example, if โณABC โ โณDEF, then side AB equals side DE, angle A equals angle D, and so on for all corresponding parts. This makes CPCTC a powerful tool for solving geometric problems.
Important Once you prove triangles are congruent, you can use CPCTC to state that any corresponding parts are equal!
There are five ways to prove triangles are congruent. Here are the first three
SSS If all three sides of one triangle are equal to the corresponding sides of another triangle, the triangles are congruent.
SAS If two sides and the included angle of one triangle are equal to the corresponding sides and included angle of another triangle, the triangles are congruent.
ASA If two angles and the included side of one triangle are equal to the corresponding angles and included side of another triangle, the triangles are congruent.
Learning Tip Try creating a mental image for each postulate - SSS means matching all sides, SAS means two sides sandwiching an angle, and ASA means two angles sandwiching a side.
Let's continue with the remaining congruence rules
AAS If two angles and a non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.
HL If the hypotenuse and one leg of a right triangle are equal to the hypotenuse and the corresponding leg of another right triangle, the triangles are congruent.
The HL Theorem is special because it only works for right triangles. For example, if AC = EF and BC = GF in right triangles ABC and EGF, then the triangles are congruent.
Remember The HL Theorem is essentially a special case of the SSS postulate that only works for right triangles!
The SSS postulate requires that all three sides of one triangle equal the corresponding sides of another triangle.
When looking for SSS congruence, watch for shared sides between triangles. A shared side is equal to itself by the reflexive property (like QS โก QS).
To apply SSS correctly, make sure you have three pairs of congruent sides. Look for matching tick marks that indicate equal sides. If you only have two pairs of equal sides, you can't use SSS.
Practice Tip When checking for SSS, count the tick marks carefully. You need exactly three pairs of matching sides to use this postulate.
The ASA postulate requires two angles and the included side to be congruent.
When working with ASA problems, look for
Watch for vertical angles (opposite angles at an intersection) as they're always equal. Also remember that shared sides between triangles are congruent to themselves by the reflexive property.
Problem-Solving Tip If you have two angles but the side isn't between them, you're using AAS, not ASA. Don't confuse these similar-sounding postulates!
The SAS postulate requires two sides and the included angle to be congruent. The angle must be formed by the two sides you're using.
When applying SAS, pay attention to
The Vertical Angle Postulate can help identify equal angles at intersections. Remember that vertical angles are always congruent, which can give you the angle you need for SAS.
Watch out! If the angle isn't between the two sides, you can't use SAS. The "included" part is crucial for this postulate to work.
When identifying an included angle, remember it's formed by two sides meeting at a vertex. For sides JK and KL, the included angle would be angle K since it's where those sides meet.
When determining which postulate proves triangles are congruent, analyze what information you have
Test Tip Always look for the reflective property when sides are shared between trianglesโthis gives you a congruent pair automatically!
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.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help โ all at your fingertips.
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Transform this note into: โ 50+ Practice Questions โ Interactive Flashcards โ Full Mock Exam โ Essay Outlines
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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 itโs very much worth it and youโll 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 donโt get it!!! In the end I think itโs worth it ๐๐ โ ๏ธAlso DID I MENTION ITS FREEE YOU DONโT HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLYโ๏ธโ๏ธโ ๏ธ
Thomas R
iOS user
Knowunity is the BEST app Iโve 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 help๐ค๐ค
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.๐๐ฉท๐
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 ๐๐๐ฒ๐ค๐โจ๐๐ฎ
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 Iโve 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 itโs very much worth it and youโll 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 donโt get it!!! In the end I think itโs worth it ๐๐ โ ๏ธAlso DID I MENTION ITS FREEE YOU DONโT HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLYโ๏ธโ๏ธโ ๏ธ
Thomas R
iOS user
Knowunity is the BEST app Iโve 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 help๐ค๐ค
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.๐๐ฉท๐
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 ๐๐๐ฒ๐ค๐โจ๐๐ฎ
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 Iโve seen! Geometry too!
Paul T
iOS user
calista ๐ชป
@urstrulycalista
Triangle congruence is a fundamental concept in geometry that helps us prove when two triangles are identical in shape and size. Understanding these congruence rules gives you powerful tools to solve geometric problems and develop mathematical proofs.
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Triangle congruence means that two triangles have exactly the same shape and size. When triangles are congruent, all their corresponding parts (sides and angles) match perfectly.
This concept forms the foundation for many geometric proofs and problem-solving techniques. You'll soon learn there are several ways to prove triangles are congruent without having to check every single side and angle.
Remember: When we say triangles are congruent, we use the symbol โ , as in โณABC โ โณDEF.
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
An included angle is the angle formed between two sides of a triangle. This concept is critical for understanding several congruence postulates.
For example, if you're looking at sides AB and BC, the included angle would be angle B, since it's formed by those two specific sides meeting at point B.
When identifying included angles, always look for the angle created at the point where your two sides intersect. This will be important when we work with the SAS congruence postulate.
Quick Tip: The included angle is always at the "shared" vertex of the two sides you're considering.
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Improve your grades
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By signing up you accept Terms of Service and Privacy Policy
When two triangles are congruent, all their corresponding parts are also congruent. This principle is known as CPCTC (Corresponding Parts of Congruent Triangles are Congruent).
However, you don't need to check all six parts (3 sides and 3 angles) to prove triangles are congruent. You can use shortcuts like: two sides and the included angle, two angles and the included side, or just three sides.
For example, if โณABC โ โณDEF, then side AB equals side DE, angle A equals angle D, and so on for all corresponding parts. This makes CPCTC a powerful tool for solving geometric problems.
Important: Once you prove triangles are congruent, you can use CPCTC to state that any corresponding parts are equal!
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There are five ways to prove triangles are congruent. Here are the first three:
SSS : If all three sides of one triangle are equal to the corresponding sides of another triangle, the triangles are congruent.
SAS : If two sides and the included angle of one triangle are equal to the corresponding sides and included angle of another triangle, the triangles are congruent.
ASA : If two angles and the included side of one triangle are equal to the corresponding angles and included side of another triangle, the triangles are congruent.
Learning Tip: Try creating a mental image for each postulate - SSS means matching all sides, SAS means two sides sandwiching an angle, and ASA means two angles sandwiching a side.
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Let's continue with the remaining congruence rules:
AAS : If two angles and a non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.
HL : If the hypotenuse and one leg of a right triangle are equal to the hypotenuse and the corresponding leg of another right triangle, the triangles are congruent.
The HL Theorem is special because it only works for right triangles. For example, if AC = EF and BC = GF in right triangles ABC and EGF, then the triangles are congruent.
Remember: The HL Theorem is essentially a special case of the SSS postulate that only works for right triangles!
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
The SSS postulate requires that all three sides of one triangle equal the corresponding sides of another triangle.
When looking for SSS congruence, watch for shared sides between triangles. A shared side is equal to itself by the reflexive property (like QS โก QS).
To apply SSS correctly, make sure you have three pairs of congruent sides. Look for matching tick marks that indicate equal sides. If you only have two pairs of equal sides, you can't use SSS.
Practice Tip: When checking for SSS, count the tick marks carefully. You need exactly three pairs of matching sides to use this postulate.
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The ASA postulate requires two angles and the included side to be congruent.
When working with ASA problems, look for:
Watch for vertical angles (opposite angles at an intersection) as they're always equal. Also remember that shared sides between triangles are congruent to themselves by the reflexive property.
Problem-Solving Tip: If you have two angles but the side isn't between them, you're using AAS, not ASA. Don't confuse these similar-sounding postulates!
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
The SAS postulate requires two sides and the included angle to be congruent. The angle must be formed by the two sides you're using.
When applying SAS, pay attention to:
The Vertical Angle Postulate can help identify equal angles at intersections. Remember that vertical angles are always congruent, which can give you the angle you need for SAS.
Watch out! If the angle isn't between the two sides, you can't use SAS. The "included" part is crucial for this postulate to work.
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
When identifying an included angle, remember it's formed by two sides meeting at a vertex. For sides JK and KL, the included angle would be angle K since it's where those sides meet.
When determining which postulate proves triangles are congruent, analyze what information you have:
Test Tip: Always look for the reflective property when sides are shared between trianglesโthis gives you a congruent pair automatically!
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.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help โ all at your fingertips.
0
Smart Tools NEW
Transform this note into: โ 50+ Practice Questions โ Interactive Flashcards โ Full Mock Exam โ Essay Outlines
App Store
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 itโs very much worth it and youโll 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 donโt get it!!! In the end I think itโs worth it ๐๐ โ ๏ธAlso DID I MENTION ITS FREEE YOU DONโT HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLYโ๏ธโ๏ธโ ๏ธ
Thomas R
iOS user
Knowunity is the BEST app Iโve 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 help๐ค๐ค
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.๐๐ฉท๐
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 ๐๐๐ฒ๐ค๐โจ๐๐ฎ
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 Iโve 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 itโs very much worth it and youโll 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 donโt get it!!! In the end I think itโs worth it ๐๐ โ ๏ธAlso DID I MENTION ITS FREEE YOU DONโT HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLYโ๏ธโ๏ธโ ๏ธ
Thomas R
iOS user
Knowunity is the BEST app Iโve 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 help๐ค๐ค
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.๐๐ฉท๐
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 ๐๐๐ฒ๐ค๐โจ๐๐ฎ
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 Iโve seen! Geometry too!
Paul T
iOS user
Calculus 1
Algebra 1
Overige