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Updated Apr 28, 2026
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Fun with Geometry: Undefined Terms & Angle Adventures
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Mika
@ika_kali
1 / 6
1.4 Segments & Their Measure
This section delves into line segments, their properties, and the Segment Addition Postulate. It also introduces the concept of congruent segments and the Ruler Postulate.
Definition: The Segment Addition Postulate states that if point B is between points A and C on a line, then AB + BC = AC.
Vocabulary:
- Congruent segments: Segments that have the same length
- Midpoint: A point that divides a segment into two congruent segments
The section provides several examples of applying the Segment Addition Postulate to solve problems involving segment lengths.
Example: If MP is 47 units long, and MN = 5x + 1 and NP = 11x - 2, find the values of MN and NP.
Highlight: The Ruler Postulate establishes a one-to-one correspondence between points on a line and real numbers, allowing us to measure distances between points.
1.5 Angles & Their Measure
This section introduces angles, their components, and classification based on their measures. It also covers the Angle Addition Postulate and provides examples of angle calculations.
Definition: An angle consists of two different rays with a common endpoint called the vertex. The interior of an angle contains all points between its two sides.
Vocabulary:
- Acute angle: An angle measuring less than 90°
- Right angle: An angle measuring exactly 90°
- Obtuse angle: An angle measuring more than 90° but less than 180°
- Straight angle: An angle measuring exactly 180°
Example: The Angle Addition Postulate states that if P is in the interior of ∠ABC, then m∠ABC = m∠ABP + m∠PBC.
Highlight: Understanding different types of angle classifications is essential for solving geometric problems and constructing proofs.
1.6 Angle Pairs
This section explores various types of angle pairs and their relationships. It introduces important concepts such as adjacent angles, vertical angles, and complementary and supplementary angles.
Vocabulary:
- Adjacent angles: Two angles that share a common side and vertex with no interior points in common
- Vertical angles: Two angles whose sides form opposite rays
- Linear pair: Two adjacent angles whose non-common sides are opposite rays
- Complementary angles: Two angles whose measures have a sum of 90°
- Supplementary angles: Two angles whose measures have a sum of 180°
- Angle bisector: A ray that divides an angle into two congruent adjacent angles
Example: If two angles form a linear pair and one angle measures 69°, while the other is expressed as 5x - 46°, find the value of x.
Highlight: Recognizing and understanding these angle relationships is crucial for solving more complex geometric problems and proofs.
1.7 Midpoint and Distance Formulas
This section introduces formulas for finding midpoints on a number line and in the coordinate plane, as well as the distance formula for calculating the distance between two points.
Definition: The midpoint formula for a line segment AB with endpoints (x₁, y₁) and (x₂, y₂) is: Midpoint =
Example: Find the midpoint of a line segment with endpoints at (3, -2) and (10, -6).
The distance formula is also introduced:
Definition: The distance between two points A(x₁, y₁) and B(x₂, y₂) is: d = √
Highlight: These formulas are essential tools for solving problems in coordinate geometry and are frequently used in more advanced geometric concepts.
Distance Formula Applications
This final section focuses on applying the distance formula to solve various geometric problems in the coordinate plane.
Example: Find the distance between the points (6, -4) and (11, -5).
The section provides step-by-step solutions to distance problems, reinforcing the application of the formula in different scenarios.
Highlight: Mastering the distance formula is crucial for analyzing geometric figures in the coordinate plane and solving real-world problems involving distances between points.
1.3 Points, Lines, & Planes
This section introduces fundamental geometric terms and undefined concepts in geometry. It covers the basic building blocks of geometric figures and their properties.
Definition: Points, lines, and planes are considered undefined terms in geometry as they are the most basic concepts from which all other geometric ideas are built.
Vocabulary:
- Coplanar points: Points that lie on the same plane
- Colinear points: Points that lie on the same line
The section also introduces important geometric postulates:
- Through any two points, there is exactly one line.
- If two distinct lines intersect, they intersect at exactly one point.
- If two distinct planes intersect, they intersect in exactly one line.
- Through any three noncollinear points, there is exactly one plane.
Example: A point has no dimension but has a location. A line extends infinitely in both directions and has one dimension (length). A plane extends infinitely in two dimensions (length and width) without thickness.
Highlight: Understanding these undefined concepts in geometry is crucial for building a strong foundation in geometric reasoning and 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.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
Fun with Geometry: Undefined Terms & Angle Adventures
M
Mika
@ika_kali
A comprehensive guide to fundamental geometric concepts, focusing on undefined terms in geometry, points, lines, planes, segments, angles, and coordinate geometry. The material covers essential postulates, measurements, and formulas crucial for understanding basic geometry.
- Introduces the 3 undefined terms... Show more
1.4 Segments & Their Measure
This section delves into line segments, their properties, and the Segment Addition Postulate. It also introduces the concept of congruent segments and the Ruler Postulate.
Definition: The Segment Addition Postulate states that if point B is between points A and C on a line, then AB + BC = AC.
Vocabulary:
- Congruent segments: Segments that have the same length
- Midpoint: A point that divides a segment into two congruent segments
The section provides several examples of applying the Segment Addition Postulate to solve problems involving segment lengths.
Example: If MP is 47 units long, and MN = 5x + 1 and NP = 11x - 2, find the values of MN and NP.
Highlight: The Ruler Postulate establishes a one-to-one correspondence between points on a line and real numbers, allowing us to measure distances between points.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
1.5 Angles & Their Measure
This section introduces angles, their components, and classification based on their measures. It also covers the Angle Addition Postulate and provides examples of angle calculations.
Definition: An angle consists of two different rays with a common endpoint called the vertex. The interior of an angle contains all points between its two sides.
Vocabulary:
- Acute angle: An angle measuring less than 90°
- Right angle: An angle measuring exactly 90°
- Obtuse angle: An angle measuring more than 90° but less than 180°
- Straight angle: An angle measuring exactly 180°
Example: The Angle Addition Postulate states that if P is in the interior of ∠ABC, then m∠ABC = m∠ABP + m∠PBC.
Highlight: Understanding different types of angle classifications is essential for solving geometric problems and constructing proofs.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
1.6 Angle Pairs
This section explores various types of angle pairs and their relationships. It introduces important concepts such as adjacent angles, vertical angles, and complementary and supplementary angles.
Vocabulary:
- Adjacent angles: Two angles that share a common side and vertex with no interior points in common
- Vertical angles: Two angles whose sides form opposite rays
- Linear pair: Two adjacent angles whose non-common sides are opposite rays
- Complementary angles: Two angles whose measures have a sum of 90°
- Supplementary angles: Two angles whose measures have a sum of 180°
- Angle bisector: A ray that divides an angle into two congruent adjacent angles
Example: If two angles form a linear pair and one angle measures 69°, while the other is expressed as 5x - 46°, find the value of x.
Highlight: Recognizing and understanding these angle relationships is crucial for solving more complex geometric problems and proofs.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
1.7 Midpoint and Distance Formulas
This section introduces formulas for finding midpoints on a number line and in the coordinate plane, as well as the distance formula for calculating the distance between two points.
Definition: The midpoint formula for a line segment AB with endpoints (x₁, y₁) and (x₂, y₂) is: Midpoint =
Example: Find the midpoint of a line segment with endpoints at (3, -2) and (10, -6).
The distance formula is also introduced:
Definition: The distance between two points A(x₁, y₁) and B(x₂, y₂) is: d = √
Highlight: These formulas are essential tools for solving problems in coordinate geometry and are frequently used in more advanced geometric concepts.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Distance Formula Applications
This final section focuses on applying the distance formula to solve various geometric problems in the coordinate plane.
Example: Find the distance between the points (6, -4) and (11, -5).
The section provides step-by-step solutions to distance problems, reinforcing the application of the formula in different scenarios.
Highlight: Mastering the distance formula is crucial for analyzing geometric figures in the coordinate plane and solving real-world problems involving distances between points.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
1.3 Points, Lines, & Planes
This section introduces fundamental geometric terms and undefined concepts in geometry. It covers the basic building blocks of geometric figures and their properties.
Definition: Points, lines, and planes are considered undefined terms in geometry as they are the most basic concepts from which all other geometric ideas are built.
Vocabulary:
- Coplanar points: Points that lie on the same plane
- Colinear points: Points that lie on the same line
The section also introduces important geometric postulates:
- Through any two points, there is exactly one line.
- If two distinct lines intersect, they intersect at exactly one point.
- If two distinct planes intersect, they intersect in exactly one line.
- Through any three noncollinear points, there is exactly one plane.
Example: A point has no dimension but has a location. A line extends infinitely in both directions and has one dimension (length). A plane extends infinitely in two dimensions (length and width) without thickness.
Highlight: Understanding these undefined concepts in geometry is crucial for building a strong foundation in geometric reasoning and 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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4.6/5
App Store
4.7/5
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 Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user