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Dec 13, 2025
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Essential Postulates and Theorems in Geometry
C
Claire 1218
@claire1218_hfue
1 / 3
Math Toolbox Essentials
Ever felt stuck on a math problem? Your first step should be checking what tools can help! For algebra, you can solve systems of equations using elimination or substitution when you have two variables and need to find their intersection point.
Got a quadratic equation? You have multiple approaches: factoring, completing the square, or using the quadratic formula: . For graphing, remember that circles follow the equation , with (h,k) as the center.
When dealing with triangles, the Pythagorean theorem $a^2 + b^2 = c^2$ is your best friend. For finding distances between points, use the distance formula: .
💡 Quick Tip: When approaching any geometry problem, first identify the shape and what you're looking for, then select the appropriate formula from your toolbox!
Proof Fundamentals: Definitions
When writing proofs, you need to know what terms actually mean! A midpoint divides a segment into two equal parts . An angle bisector divides an angle into two equal parts.
Perpendicular lines form 90° angles when they meet. Understanding these basic definitions is crucial because they're the building blocks of more complex proofs.
Remember that complementary angles add up to 90°, while supplementary angles sum to 180°. A linear pair consists of adjacent angles that are supplementary.
Proof Fundamentals: Properties & Postulates
The properties of equality are the backbone of algebraic proofs. If you add, subtract, multiply, or divide both sides of an equation by the same value, the equality remains true. The substitution property lets you replace equal values with each other.
Postulates are statements we accept as true without proof. The Segment Addition Postulate states that if point B is between points A and C, then AB + BC = AC. Similarly, the Angle Addition Postulate works the same way for angles.
When dealing with parallel lines cut by a transversal, remember that corresponding angles are congruent. This is a powerful tool for proving lines are parallel or finding missing angles.
🔑 Remember: Postulates are your starting points in proofs - they don't need to be proven themselves but can be used to prove other statements!
Important Theorems for Proofs
Theorems are proven statements you can use in your proofs. The Midpoint Theorem tells us that if M is the midpoint of AB, then AM = ½AB. The Vertical Angles Theorem (VAT) states that vertical angles are congruent.
When working with parallel lines cut by a transversal, alternate interior angles are congruent, and same-side interior angles are supplementary. These relationships give you powerful tools for solving angle problems.
If two lines are perpendicular, they form congruent adjacent angles. Conversely, if two lines form congruent adjacent angles, then they must be perpendicular.
Advanced Theorems and Applications
Triangle angles always add up to 180°—this simple fact is incredibly useful! Additionally, an exterior angle of a triangle equals the sum of the two remote interior angles, giving you another way to find missing angles.
For polygons, the angle sum formula is 180°, where n is the number of sides. The sum of exterior angles (one at each vertex) of any convex polygon is always 360°—a beautiful mathematical pattern!
Lines and slopes have special relationships too. Two non-vertical lines are parallel if and only if their slopes are equal. They're perpendicular if and only if the product of their slopes is -1.
🌟 Test Prep Alert: The triangle angle sum (180°) and polygon angle sum formulas frequently appear on exams! Know these cold!
Corollaries and Special Cases
Corollaries are direct results of theorems. If two angles of one triangle are congruent to two angles of another triangle, then their third angles must also be congruent. This is because triangle angles always sum to 180°.
An equiangular triangle (where all angles are equal) must have three 60° angles. In any triangle, there can be at most one right angle or one obtuse angle—never more.
Right triangles have special properties: their acute angles are always complementary (they add to 90°). This makes solving for missing angles much simpler when you know one of them.
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.
Most popular content: Geometric Theorems
Most popular content in Geometry
Most popular content
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
Essential Postulates and Theorems in Geometry
C
Claire 1218
@claire1218_hfue
Need a quick reference for geometry and algebra? This study guide packs all the essential formulas, definitions, and proof strategies you'll need for math class. From solving systems of equations to understanding geometric theorems, this is your go-to resource for... Show more
Math Toolbox Essentials
Ever felt stuck on a math problem? Your first step should be checking what tools can help! For algebra, you can solve systems of equations using elimination or substitution when you have two variables and need to find their intersection point.
Got a quadratic equation? You have multiple approaches: factoring, completing the square, or using the quadratic formula: . For graphing, remember that circles follow the equation , with (h,k) as the center.
When dealing with triangles, the Pythagorean theorem $a^2 + b^2 = c^2$ is your best friend. For finding distances between points, use the distance formula: .
💡 Quick Tip: When approaching any geometry problem, first identify the shape and what you're looking for, then select the appropriate formula from your toolbox!
Proof Fundamentals: Definitions
When writing proofs, you need to know what terms actually mean! A midpoint divides a segment into two equal parts . An angle bisector divides an angle into two equal parts.
Perpendicular lines form 90° angles when they meet. Understanding these basic definitions is crucial because they're the building blocks of more complex proofs.
Remember that complementary angles add up to 90°, while supplementary angles sum to 180°. A linear pair consists of adjacent angles that are supplementary.
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
Proof Fundamentals: Properties & Postulates
The properties of equality are the backbone of algebraic proofs. If you add, subtract, multiply, or divide both sides of an equation by the same value, the equality remains true. The substitution property lets you replace equal values with each other.
Postulates are statements we accept as true without proof. The Segment Addition Postulate states that if point B is between points A and C, then AB + BC = AC. Similarly, the Angle Addition Postulate works the same way for angles.
When dealing with parallel lines cut by a transversal, remember that corresponding angles are congruent. This is a powerful tool for proving lines are parallel or finding missing angles.
🔑 Remember: Postulates are your starting points in proofs - they don't need to be proven themselves but can be used to prove other statements!
Important Theorems for Proofs
Theorems are proven statements you can use in your proofs. The Midpoint Theorem tells us that if M is the midpoint of AB, then AM = ½AB. The Vertical Angles Theorem (VAT) states that vertical angles are congruent.
When working with parallel lines cut by a transversal, alternate interior angles are congruent, and same-side interior angles are supplementary. These relationships give you powerful tools for solving angle problems.
If two lines are perpendicular, they form congruent adjacent angles. Conversely, if two lines form congruent adjacent angles, then they must be perpendicular.
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
Advanced Theorems and Applications
Triangle angles always add up to 180°—this simple fact is incredibly useful! Additionally, an exterior angle of a triangle equals the sum of the two remote interior angles, giving you another way to find missing angles.
For polygons, the angle sum formula is 180°, where n is the number of sides. The sum of exterior angles (one at each vertex) of any convex polygon is always 360°—a beautiful mathematical pattern!
Lines and slopes have special relationships too. Two non-vertical lines are parallel if and only if their slopes are equal. They're perpendicular if and only if the product of their slopes is -1.
🌟 Test Prep Alert: The triangle angle sum (180°) and polygon angle sum formulas frequently appear on exams! Know these cold!
Corollaries and Special Cases
Corollaries are direct results of theorems. If two angles of one triangle are congruent to two angles of another triangle, then their third angles must also be congruent. This is because triangle angles always sum to 180°.
An equiangular triangle (where all angles are equal) must have three 60° angles. In any triangle, there can be at most one right angle or one obtuse angle—never more.
Right triangles have special properties: their acute angles are always complementary (they add to 90°). This makes solving for missing angles much simpler when you know one of them.
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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4.8/5
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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 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