Geometry gets real when we start working with parallel and... Show more
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Proving Lines Parallel
Ever wondered how mathematicians can be so certain that lines will never intersect? It's all about the angles!
When two lines are cut by a transversal (a line crossing both), we can use special angle relationships to prove the lines are parallel. The key theorem to remember is: if alternate interior angles are congruent, then the lines are parallel.
Let's see this in action with a proof:
- Given: ∠2 ≅ ∠3 (congruent alternate interior angles)
- Prove: lines l and m are parallel
- Since ∠2 ≅ ∠3 (given), and ∠1 ≅ ∠2 (by Vertical Angles Theorem), we can use the Transitive Property to conclude that ∠1 ≅ ∠3
- Therefore, l ∥ m by the Converse of Corresponding Angles Theorem
Quick Tip: Whenever you see congruent angles in specific positions (corresponding, alternate interior, or alternate exterior), think "parallel lines" right away!
Identifying Parallel Lines
You'll often need to determine which lines are parallel based on given angle relationships. Here's how to approach these problems:
When examining a diagram with multiple lines and angles, look for these key relationships:
- Corresponding angles that are congruent (like ∠1 ≅ ∠5) tell us the lines are parallel
- Alternate interior angles that are congruent (like ∠3 ≅ ∠13) also indicate parallel lines
- Same-side interior angles that are supplementary work too
In a proof, you'll often use these angle relationships along with the transitive property to establish parallelism. For example, if we know g ∥ h and ∠1 ≅ ∠11, we can prove j ∥ k by showing that ∠11 ≅ ∠15 (a corresponding angle with j and k).
Sometimes you'll need to build a chain of angle relationships. If ∠1 ≅ ∠2 and ∠1 ≅ ∠3, then by the Transitive Property, ∠2 ≅ ∠3, which proves the lines are parallel by the Converse of Alternate Interior Angles Theorem.
Proving Perpendicular Lines
Right angles are everywhere in geometry! Being able to prove lines are perpendicular is a crucial skill.
A perpendicular bisector is a special line that's perpendicular to a segment and passes through its midpoint. This concept comes up frequently in constructions and proofs. Remember that a true perpendicular bisector must satisfy both conditions - being perpendicular AND passing through the midpoint.
When finding the shortest distance from a point to a line, you always draw a perpendicular line from the point to the line. This is a fundamental principle used in many geometric applications.
Remember this! If a pair of angles forms a linear pair (adjacent angles on a straight line) and they're congruent, then the lines forming those angles are perpendicular.
Important theorems about perpendicular lines:
- If one line is perpendicular to one of two parallel lines, it's perpendicular to both
- Two lines perpendicular to the same line are parallel to each other
Perpendicular Line Proofs
Proving lines are perpendicular often requires combining multiple theorems about parallel and perpendicular lines.
Let's work through an example proof:
- Given: r ∥ s and ∠1 ≅ ∠2
- Prove: r ⊥ t
- Since r ∥ s, we know ∠2 ≅ ∠3 (Corresponding Angles Theorem)
- Using the Transitive Property, we can say ∠1 ≅ ∠3
- Since ∠1 and ∠3 form a linear pair and are congruent, each angle must be 90°
- Therefore, r ⊥ t
Remember these key relationships for tests:
- If a linear pair of angles are congruent, the lines are perpendicular
- If one line is perpendicular to one of two parallel lines, it's perpendicular to both
- If two lines are each perpendicular to the same line, those two lines are parallel
These relationships form a powerful toolkit for solving complex geometric problems and constructing rigorous proofs.
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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.
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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
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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
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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❗️❗️⚠️
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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
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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
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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
Understanding Lines and Transversals
Geometry gets real when we start working with parallel and perpendicular lines. These relationships form the backbone of geometric proofs and constructions you'll use throughout high school math and beyond. Let's explore how to prove lines are parallel or perpendicular... Show more
Proving Lines Parallel
Ever wondered how mathematicians can be so certain that lines will never intersect? It's all about the angles!
When two lines are cut by a transversal (a line crossing both), we can use special angle relationships to prove the lines are parallel. The key theorem to remember is: if alternate interior angles are congruent, then the lines are parallel.
Let's see this in action with a proof:
- Given: ∠2 ≅ ∠3 (congruent alternate interior angles)
- Prove: lines l and m are parallel
- Since ∠2 ≅ ∠3 (given), and ∠1 ≅ ∠2 (by Vertical Angles Theorem), we can use the Transitive Property to conclude that ∠1 ≅ ∠3
- Therefore, l ∥ m by the Converse of Corresponding Angles Theorem
Quick Tip: Whenever you see congruent angles in specific positions (corresponding, alternate interior, or alternate exterior), think "parallel lines" right away!
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Identifying Parallel Lines
You'll often need to determine which lines are parallel based on given angle relationships. Here's how to approach these problems:
When examining a diagram with multiple lines and angles, look for these key relationships:
- Corresponding angles that are congruent (like ∠1 ≅ ∠5) tell us the lines are parallel
- Alternate interior angles that are congruent (like ∠3 ≅ ∠13) also indicate parallel lines
- Same-side interior angles that are supplementary work too
In a proof, you'll often use these angle relationships along with the transitive property to establish parallelism. For example, if we know g ∥ h and ∠1 ≅ ∠11, we can prove j ∥ k by showing that ∠11 ≅ ∠15 (a corresponding angle with j and k).
Sometimes you'll need to build a chain of angle relationships. If ∠1 ≅ ∠2 and ∠1 ≅ ∠3, then by the Transitive Property, ∠2 ≅ ∠3, which proves the lines are parallel by the Converse of Alternate Interior Angles Theorem.
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Proving Perpendicular Lines
Right angles are everywhere in geometry! Being able to prove lines are perpendicular is a crucial skill.
A perpendicular bisector is a special line that's perpendicular to a segment and passes through its midpoint. This concept comes up frequently in constructions and proofs. Remember that a true perpendicular bisector must satisfy both conditions - being perpendicular AND passing through the midpoint.
When finding the shortest distance from a point to a line, you always draw a perpendicular line from the point to the line. This is a fundamental principle used in many geometric applications.
Remember this! If a pair of angles forms a linear pair (adjacent angles on a straight line) and they're congruent, then the lines forming those angles are perpendicular.
Important theorems about perpendicular lines:
- If one line is perpendicular to one of two parallel lines, it's perpendicular to both
- Two lines perpendicular to the same line are parallel to each other
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Access to all documents
Improve your grades
Join milions of students
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Perpendicular Line Proofs
Proving lines are perpendicular often requires combining multiple theorems about parallel and perpendicular lines.
Let's work through an example proof:
- Given: r ∥ s and ∠1 ≅ ∠2
- Prove: r ⊥ t
- Since r ∥ s, we know ∠2 ≅ ∠3 (Corresponding Angles Theorem)
- Using the Transitive Property, we can say ∠1 ≅ ∠3
- Since ∠1 and ∠3 form a linear pair and are congruent, each angle must be 90°
- Therefore, r ⊥ t
Remember these key relationships for tests:
- If a linear pair of angles are congruent, the lines are perpendicular
- If one line is perpendicular to one of two parallel lines, it's perpendicular to both
- If two lines are each perpendicular to the same line, those two lines are parallel
These relationships form a powerful toolkit for solving complex geometric problems and constructing rigorous proofs.
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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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