Triangle congruence by Angle-Side-Angle (ASA) and Angle-Angle-Side (AAS)are essential... Show more
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Feb 14, 2026
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Triangle Congruence: ASA and AAS Theorems Explained with Worksheets and Examples
1 / 5
Writing a Proof Using ASA
This page focuses on applying the ASA theorem in practical scenarios and writing formal proofs. It presents a real-world example involving the design of a miniature golf course.
Example: A teen organization is building a miniature golf course with triangular bumpers. Students are asked to prove that the bumpers meet the design conditions using the ASA theorem.
The proof is structured step-by-step, demonstrating how to use given information to establish congruence between two triangles. This example shows the practical application of geometric theorems in real-world design problems.
Highlight: The proof emphasizes the importance of identifying the given information, stating what needs to be proved, and logically applying the ASA theorem to reach the conclusion.
The page also includes a "Got It?" section, providing another example for students to practice applying the ASA theorem in a proof.
Vocabulary: Right angles are introduced in the proof, reminding students of the importance of recognizing special angle relationships in geometric proofs.
Writing a Proof Using AAS
This page transitions to the application of the Angle-Angle-Side (AAS) Theorem in proofs. It provides a structured example of how to write a proof using AAS.
Example: A proof is presented using the AAS theorem, demonstrating how to establish triangle congruence when two angles and a non-included side are known to be congruent.
The proof follows a logical sequence, starting with the given information, identifying congruent parts, and applying the AAS theorem to conclude that the triangles are congruent.
Highlight: The proof introduces the use of alternate interior angles, showcasing how different geometric concepts can be combined in a single proof.
The page includes a "Got It?" section, offering another example for students to practice applying the AAS theorem in a proof scenario.
Vocabulary: The term "bisector" is used in the practice example, introducing students to the concept of angle bisection in the context of triangle congruence.
Determining Whether Triangles Are Congruent
This page focuses on applying the knowledge of triangle congruence theorems to determine whether given triangles are congruent. It emphasizes critical thinking and the ability to justify answers.
Example: A diagram is presented with a question asking whether two triangles are congruent. Students are required to analyze the given information and select the best justification for their answer.
The example highlights the importance of understanding corresponding sides in triangle congruence, demonstrating that congruence cannot be established if the sides do not correspond correctly.
Highlight: This section emphasizes the importance of not just identifying congruence but also explaining why triangles are or are not congruent.
The page includes a "Got It?" section with more complex examples, requiring students to determine congruence and justify their answers using the appropriate theorem or postulate.
Vocabulary: Terms like "midpoint" and "vertical angles" are introduced, showing how these concepts can be used in conjunction with congruence theorems.
Practice Problems and Applications
This final page provides a series of practice problems to reinforce understanding of ASA and AAS congruence. It includes various scenarios where students must apply their knowledge of triangle congruence theorems.
Example: One problem presents a complex diagram with given information about angles and line segments. Students are asked to prove that two triangles are congruent using the information provided.
The problems vary in complexity, some requiring the application of multiple geometric concepts to establish congruence.
Highlight: These exercises emphasize the importance of identifying the correct theorem (ASA, AAS, or others) based on the given information.
The page also includes problems that integrate other geometric concepts, such as right angles and midpoints, with triangle congruence theorems.
Vocabulary: Terms like "vertical angles" and "alternate interior angles" are used in the problems, reinforcing the interconnectedness of various geometric concepts.
This practice section serves as a comprehensive review of the triangle congruence theorems covered in the lesson, particularly focusing on ASA and AAS congruence.
Triangle Congruence by ASA & AAS
This page introduces two important concepts in triangle congruence: the Angle-Side-Angle (ASA) Postulate and the Angle-Angle-Side (AAS) Theorem. These are fundamental tools for proving triangles congruent.
Definition: The ASA Postulate states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
Definition: The AAS Theorem states that if two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.
The page provides visual representations of both ASA and AAS congruence, showing the corresponding parts that need to be congruent for each theorem to apply.
Example: An example is given asking students to identify which two triangles are congruent by ASA, emphasizing the importance of the included side in ASA congruence.
Highlight: Understanding the difference between included and non-included sides is crucial for correctly applying ASA and AAS theorems.
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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
Triangle Congruence: ASA and AAS Theorems Explained with Worksheets and Examples
Triangle congruence by Angle-Side-Angle (ASA) and Angle-Angle-Side (AAS) are essential concepts in geometry. These theorems provide methods to prove triangles congruent without measuring all sides and angles. ASA theoremstates that two triangles are congruent if two angles and the... Show more
Writing a Proof Using ASA
This page focuses on applying the ASA theorem in practical scenarios and writing formal proofs. It presents a real-world example involving the design of a miniature golf course.
Example: A teen organization is building a miniature golf course with triangular bumpers. Students are asked to prove that the bumpers meet the design conditions using the ASA theorem.
The proof is structured step-by-step, demonstrating how to use given information to establish congruence between two triangles. This example shows the practical application of geometric theorems in real-world design problems.
Highlight: The proof emphasizes the importance of identifying the given information, stating what needs to be proved, and logically applying the ASA theorem to reach the conclusion.
The page also includes a "Got It?" section, providing another example for students to practice applying the ASA theorem in a proof.
Vocabulary: Right angles are introduced in the proof, reminding students of the importance of recognizing special angle relationships in geometric proofs.
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Writing a Proof Using AAS
This page transitions to the application of the Angle-Angle-Side (AAS) Theorem in proofs. It provides a structured example of how to write a proof using AAS.
Example: A proof is presented using the AAS theorem, demonstrating how to establish triangle congruence when two angles and a non-included side are known to be congruent.
The proof follows a logical sequence, starting with the given information, identifying congruent parts, and applying the AAS theorem to conclude that the triangles are congruent.
Highlight: The proof introduces the use of alternate interior angles, showcasing how different geometric concepts can be combined in a single proof.
The page includes a "Got It?" section, offering another example for students to practice applying the AAS theorem in a proof scenario.
Vocabulary: The term "bisector" is used in the practice example, introducing students to the concept of angle bisection in the context of triangle congruence.
Sign up to see the contentIt's free!
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Determining Whether Triangles Are Congruent
This page focuses on applying the knowledge of triangle congruence theorems to determine whether given triangles are congruent. It emphasizes critical thinking and the ability to justify answers.
Example: A diagram is presented with a question asking whether two triangles are congruent. Students are required to analyze the given information and select the best justification for their answer.
The example highlights the importance of understanding corresponding sides in triangle congruence, demonstrating that congruence cannot be established if the sides do not correspond correctly.
Highlight: This section emphasizes the importance of not just identifying congruence but also explaining why triangles are or are not congruent.
The page includes a "Got It?" section with more complex examples, requiring students to determine congruence and justify their answers using the appropriate theorem or postulate.
Vocabulary: Terms like "midpoint" and "vertical angles" are introduced, showing how these concepts can be used in conjunction with congruence theorems.
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Practice Problems and Applications
This final page provides a series of practice problems to reinforce understanding of ASA and AAS congruence. It includes various scenarios where students must apply their knowledge of triangle congruence theorems.
Example: One problem presents a complex diagram with given information about angles and line segments. Students are asked to prove that two triangles are congruent using the information provided.
The problems vary in complexity, some requiring the application of multiple geometric concepts to establish congruence.
Highlight: These exercises emphasize the importance of identifying the correct theorem (ASA, AAS, or others) based on the given information.
The page also includes problems that integrate other geometric concepts, such as right angles and midpoints, with triangle congruence theorems.
Vocabulary: Terms like "vertical angles" and "alternate interior angles" are used in the problems, reinforcing the interconnectedness of various geometric concepts.
This practice section serves as a comprehensive review of the triangle congruence theorems covered in the lesson, particularly focusing on ASA and AAS congruence.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Triangle Congruence by ASA & AAS
This page introduces two important concepts in triangle congruence: the Angle-Side-Angle (ASA) Postulate and the Angle-Angle-Side (AAS) Theorem. These are fundamental tools for proving triangles congruent.
Definition: The ASA Postulate states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
Definition: The AAS Theorem states that if two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.
The page provides visual representations of both ASA and AAS congruence, showing the corresponding parts that need to be congruent for each theorem to apply.
Example: An example is given asking students to identify which two triangles are congruent by ASA, emphasizing the importance of the included side in ASA congruence.
Highlight: Understanding the difference between included and non-included sides is crucial for correctly applying ASA and AAS theorems.
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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4.6/5
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4.7/5
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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