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105
1
Paylee
7/8/2025
Geometry
Segment Proofs
880
•
Jul 8, 2025
•
How to Complete Segment Proofs in Geometry - Step by Step, With Angles, and Worksheets with Answers
Page 2: Advanced Segment Proofs
This page presents more complex segment proofs, building on the concepts introduced in the previous page.
The fourth proof demonstrates how to prove that KL ≅ LM given that K is the midpoint of JL, M is the midpoint of LN, and JK ≅ MN. This proof combines the definition of midpoint with the transitive and symmetric properties of congruence.
Vocabulary: The symmetric property of congruence states that if A ≅ B, then B ≅ A.
The fifth proof shows how to prove that XZ = TV given that XY ≅ UV and YZ ≅ TU. This proof utilizes the segment addition property and substitution property to establish the equality of the segments.
The sixth proof demonstrates how to prove that XZ ≅ VW given that WY ≅ YZ and XY ≅ VY. This proof combines the segment addition postulate with the transitive property to establish the congruence of the segments.
Example: In the proof XY + YZ = XZ, VY + YW = VW, the segment addition postulate is applied to show that the sum of two adjacent segments is equal to the whole segment.
Page 3: Midpoint Theorems and Congruence Properties
This page focuses on proofs involving midpoint theorems and congruence properties.
The seventh proof demonstrates how to prove that DE ≅ AE given that E is the midpoint of AC and DE = EC. This proof uses the definition of midpoint and the transitive property of equality to establish the congruence.
The eighth proof shows how to prove that S is the midpoint of RT given that RS = 1/2RT. This proof employs the multiplication property of equality and the segment addition postulate to establish that S divides RT into two equal parts.
Definition: The multiplication property of equality states that if you multiply both sides of an equation by the same number, the equation remains true.
The ninth proof demonstrates how to prove that LM ≅ NO given that M is the midpoint of LN and N is the midpoint of MO. This proof combines the definition of midpoint with the transitive property of equality to establish the congruence.
Highlight: The transitive property of equality states that if a = b and b = c, then a = c.
Page 4: Advanced Applications of Segment Proofs
This page presents more complex applications of segment proofs, incorporating various geometric properties and postulates.
The tenth proof demonstrates how to prove that Q is the midpoint of PR given that 2PQ = PR. This proof uses the segment addition postulate and the subtraction property to establish that PQ = QR, which defines Q as the midpoint of PR.
The eleventh proof shows how to prove that AD = CE given that AB ≅ CD and BD ≅ DE. This proof combines the segment addition postulate with the transitive property to establish the equality of AD and CE.
Example: In the proof AB + BD = AD, CD + DE = CE, the segment addition postulate is applied twice to show that the sum of two adjacent segments is equal to the whole segment on both sides of the equation.
The twelfth proof demonstrates how to prove that HI ≅ JK given that GI ≅ JL and GH ≅ KL. This proof employs the segment addition property, substitution property, and subtraction property of equality to establish the congruence of HI and JK.
Vocabulary: The subtraction property of equality states that if you subtract the same quantity from both sides of an equation, the equation remains true.
These advanced proofs reinforce the importance of logical reasoning and the application of multiple geometric properties in solving complex segment problems.
Page 1: Introduction to Segment Proofs
This page introduces the concept of segment proofs in geometry and provides several examples to illustrate the process.
The first proof demonstrates how to prove that 2DE = DF given that E is the midpoint of DF. The proof uses the definition of midpoint, the addition property of equality, and the segment addition postulate to establish the relationship.
Definition: A midpoint is a point that divides a line segment into two equal parts.
The second proof shows how to prove that L is the midpoint of KM given that KL ≅ LN and LM ≅ LN. This proof utilizes the transitive property of congruence to establish that KL ≅ LM, which leads to the conclusion that L is the midpoint of KM.
Example: In the proof KL ≅ LN, LM ≅ LN → KL ≅ LM, the transitive property of congruence is applied to show that two segments are congruent to a common segment, therefore they are congruent to each other.
The third proof demonstrates how to prove that PS ≅ TU given that PQ ≅ TQ and UQ ≅ QS. This proof employs the segment addition postulate and the definition of congruence to establish the relationship between the segments.
Highlight: The segment addition postulate states that the length of a whole segment is equal to the sum of the lengths of its parts.
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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 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
How to Complete Segment Proofs in Geometry - Step by Step, With Angles, and Worksheets with Answers
How to complete segment proofs in geometry step by step: A comprehensive guide to mastering segment proofs in geometry, including midpoint theorems, segment addition postulates, and congruence properties.
Key points:
- Understand the given information and what needs to be... Show more
Page 2: Advanced Segment Proofs
This page presents more complex segment proofs, building on the concepts introduced in the previous page.
The fourth proof demonstrates how to prove that KL ≅ LM given that K is the midpoint of JL, M is the midpoint of LN, and JK ≅ MN. This proof combines the definition of midpoint with the transitive and symmetric properties of congruence.
Vocabulary: The symmetric property of congruence states that if A ≅ B, then B ≅ A.
The fifth proof shows how to prove that XZ = TV given that XY ≅ UV and YZ ≅ TU. This proof utilizes the segment addition property and substitution property to establish the equality of the segments.
The sixth proof demonstrates how to prove that XZ ≅ VW given that WY ≅ YZ and XY ≅ VY. This proof combines the segment addition postulate with the transitive property to establish the congruence of the segments.
Example: In the proof XY + YZ = XZ, VY + YW = VW, the segment addition postulate is applied to show that the sum of two adjacent segments is equal to the whole segment.
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Page 3: Midpoint Theorems and Congruence Properties
This page focuses on proofs involving midpoint theorems and congruence properties.
The seventh proof demonstrates how to prove that DE ≅ AE given that E is the midpoint of AC and DE = EC. This proof uses the definition of midpoint and the transitive property of equality to establish the congruence.
The eighth proof shows how to prove that S is the midpoint of RT given that RS = 1/2RT. This proof employs the multiplication property of equality and the segment addition postulate to establish that S divides RT into two equal parts.
Definition: The multiplication property of equality states that if you multiply both sides of an equation by the same number, the equation remains true.
The ninth proof demonstrates how to prove that LM ≅ NO given that M is the midpoint of LN and N is the midpoint of MO. This proof combines the definition of midpoint with the transitive property of equality to establish the congruence.
Highlight: The transitive property of equality states that if a = b and b = c, then a = c.
Page 4: Advanced Applications of Segment Proofs
This page presents more complex applications of segment proofs, incorporating various geometric properties and postulates.
The tenth proof demonstrates how to prove that Q is the midpoint of PR given that 2PQ = PR. This proof uses the segment addition postulate and the subtraction property to establish that PQ = QR, which defines Q as the midpoint of PR.
The eleventh proof shows how to prove that AD = CE given that AB ≅ CD and BD ≅ DE. This proof combines the segment addition postulate with the transitive property to establish the equality of AD and CE.
Example: In the proof AB + BD = AD, CD + DE = CE, the segment addition postulate is applied twice to show that the sum of two adjacent segments is equal to the whole segment on both sides of the equation.
The twelfth proof demonstrates how to prove that HI ≅ JK given that GI ≅ JL and GH ≅ KL. This proof employs the segment addition property, substitution property, and subtraction property of equality to establish the congruence of HI and JK.
Vocabulary: The subtraction property of equality states that if you subtract the same quantity from both sides of an equation, the equation remains true.
These advanced proofs reinforce the importance of logical reasoning and the application of multiple geometric properties in solving complex segment 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
Page 1: Introduction to Segment Proofs
This page introduces the concept of segment proofs in geometry and provides several examples to illustrate the process.
The first proof demonstrates how to prove that 2DE = DF given that E is the midpoint of DF. The proof uses the definition of midpoint, the addition property of equality, and the segment addition postulate to establish the relationship.
Definition: A midpoint is a point that divides a line segment into two equal parts.
The second proof shows how to prove that L is the midpoint of KM given that KL ≅ LN and LM ≅ LN. This proof utilizes the transitive property of congruence to establish that KL ≅ LM, which leads to the conclusion that L is the midpoint of KM.
Example: In the proof KL ≅ LN, LM ≅ LN → KL ≅ LM, the transitive property of congruence is applied to show that two segments are congruent to a common segment, therefore they are congruent to each other.
The third proof demonstrates how to prove that PS ≅ TU given that PQ ≅ TQ and UQ ≅ QS. This proof employs the segment addition postulate and the definition of congruence to establish the relationship between the segments.
Highlight: The segment addition postulate states that the length of a whole segment is equal to the sum of the lengths of its parts.
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