CPCTC Proofs: A Comprehensive Guide to Triangle Congruence
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Paylee
7/10/2025
Geometry
Triangle Proofs CPCT
1,051
•
Jul 10, 2025
•
CPCTC Proofs Worksheet with Answers - Easy Steps to Prove Triangle Congruence
Advanced Applications of CPCTC
This page presents three more sophisticated examples of CPCTC proofs, showcasing its versatility in solving complex geometric problems.
Example 1 involves proving angle congruence when a point is the midpoint of two line segments. The proof uses the ASA theorem and CPCTC.
Example 2 demonstrates how to prove that an angle bisector creates two congruent segments. This proof also utilizes the ASA theorem before applying CPCTC.
Vocabulary: Angle bisector - a line that divides an angle into two equal parts.
Example 3 is a more complex proof involving parallel lines and midpoints. It uses the AAS theorem and CPCTC to prove segment congruence.
Highlight: These examples show how CPCTC can be combined with various triangle congruence theorems to solve diverse geometric problems.
CPCTC in Practice: Proof Exercises
This page provides two practice exercises for applying CPCTC in geometric proofs, along with their solutions.
Exercise 1: Given: ∠BAC ≅ ∠EDC, BC ≅ EC Prove: ∠ABC ≅ ∠DEC
The solution uses the AAS theorem to prove triangle congruence, then applies CPCTC to conclude angle congruence.
Exercise 2: Given: JK ≅ LM, ∠JKM ≅ ∠LMK Prove: JM ≅ LK
This proof employs the ASA theorem to establish triangle congruence before using CPCTC to prove side congruence.
Tip: When using corresponding parts of congruent triangles in proofs, always clearly state which congruence theorem you're using before applying CPCTC.
Advanced CPCTC Problems and Solutions
This final page presents three challenging CPCTC proof problems, demonstrating advanced applications of the concept.
Problem 1 involves proving angle congruence when a point is the midpoint of a line segment and two other segments are congruent. The solution uses the SSS theorem and CPCTC.
Problem 2 requires proving side congruence in a quadrilateral with parallel sides. This proof combines alternate interior angles, the ASA theorem, and CPCTC.
Problem 3 involves parallel lines and congruent angles. The solution uses the ASA theorem and CPCTC to prove angle congruence.
Highlight: These problems showcase how CPCTC can be used in conjunction with various geometric concepts like parallel lines, midpoints, and different triangle congruence theorems.
Tip: When solving complex CPCTC problems, break down the proof into smaller steps and identify which congruence theorem to use before applying CPCTC.
Understanding CPCTC in Triangle Proofs
This page introduces the concept of CPCTC and its application in geometric proofs. It provides two detailed examples of CPCTC proofs.
Definition: CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent.
Highlight: CPCTC is used to prove parts of triangles are congruent, but only after proving the triangles themselves are congruent.
The first example uses the SSS triangle congruence theorem to prove that ∠BCA = ∠DCA. The proof follows these steps:
- Establish given information
- Prove triangle congruence using SSS
- Apply CPCTC to conclude angle congruence
The second example employs the AAS theorem to prove that PQRS is a parallelogram. This proof demonstrates how CPCTC can be used in more complex geometric situations.
Example: Given: PS || QR, ∠OPS ≅ ∠SRQ, ∠QPS ≅ ∠SRQ Prove: PQRS is a parallelogram The proof uses AAS to establish triangle congruence, then applies CPCTC to show opposite sides are parallel and congruent.
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Can't find what you're looking for? Explore other subjects.
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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
CPCTC Proofs Worksheet with Answers - Easy Steps to Prove Triangle Congruence
CPCTC Proofs: A Comprehensive Guide to Triangle Congruence
This guide explores the concept of CPCTC (Corresponding Parts of Congruent Triangles are Congruent) in geometry proofs. It covers various triangle congruence theorems, provides CPCTC examples, and offers step-by-step instructions... Show more
Advanced Applications of CPCTC
This page presents three more sophisticated examples of CPCTC proofs, showcasing its versatility in solving complex geometric problems.
Example 1 involves proving angle congruence when a point is the midpoint of two line segments. The proof uses the ASA theorem and CPCTC.
Example 2 demonstrates how to prove that an angle bisector creates two congruent segments. This proof also utilizes the ASA theorem before applying CPCTC.
Vocabulary: Angle bisector - a line that divides an angle into two equal parts.
Example 3 is a more complex proof involving parallel lines and midpoints. It uses the AAS theorem and CPCTC to prove segment congruence.
Highlight: These examples show how CPCTC can be combined with various triangle congruence theorems to solve diverse geometric problems.
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
CPCTC in Practice: Proof Exercises
This page provides two practice exercises for applying CPCTC in geometric proofs, along with their solutions.
Exercise 1: Given: ∠BAC ≅ ∠EDC, BC ≅ EC Prove: ∠ABC ≅ ∠DEC
The solution uses the AAS theorem to prove triangle congruence, then applies CPCTC to conclude angle congruence.
Exercise 2: Given: JK ≅ LM, ∠JKM ≅ ∠LMK Prove: JM ≅ LK
This proof employs the ASA theorem to establish triangle congruence before using CPCTC to prove side congruence.
Tip: When using corresponding parts of congruent triangles in proofs, always clearly state which congruence theorem you're using before applying CPCTC.
Advanced CPCTC Problems and Solutions
This final page presents three challenging CPCTC proof problems, demonstrating advanced applications of the concept.
Problem 1 involves proving angle congruence when a point is the midpoint of a line segment and two other segments are congruent. The solution uses the SSS theorem and CPCTC.
Problem 2 requires proving side congruence in a quadrilateral with parallel sides. This proof combines alternate interior angles, the ASA theorem, and CPCTC.
Problem 3 involves parallel lines and congruent angles. The solution uses the ASA theorem and CPCTC to prove angle congruence.
Highlight: These problems showcase how CPCTC can be used in conjunction with various geometric concepts like parallel lines, midpoints, and different triangle congruence theorems.
Tip: When solving complex CPCTC problems, break down the proof into smaller steps and identify which congruence theorem to use before applying CPCTC.
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
Understanding CPCTC in Triangle Proofs
This page introduces the concept of CPCTC and its application in geometric proofs. It provides two detailed examples of CPCTC proofs.
Definition: CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent.
Highlight: CPCTC is used to prove parts of triangles are congruent, but only after proving the triangles themselves are congruent.
The first example uses the SSS triangle congruence theorem to prove that ∠BCA = ∠DCA. The proof follows these steps:
- Establish given information
- Prove triangle congruence using SSS
- Apply CPCTC to conclude angle congruence
The second example employs the AAS theorem to prove that PQRS is a parallelogram. This proof demonstrates how CPCTC can be used in more complex geometric situations.
Example: Given: PS || QR, ∠OPS ≅ ∠SRQ, ∠QPS ≅ ∠SRQ Prove: PQRS is a parallelogram The proof uses AAS to establish triangle congruence, then applies CPCTC to show opposite sides are parallel and 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?
You have the choice, we offer a 100% secure payment via PayPal or an Amazon gift card. There is no limit to how much you can earn.
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