Triangles are fundamental shapes in geometry with properties that make...
Geometry55 views·Updated Aug 22, 2026·3 pagesChapter 5: Exploring Geometry
Sebasbg13@sebastianbg13
1 / 3
1
of 3
Classifying Triangles
Ever wonder how many different types of triangles exist? Triangles can be classified in two main ways: by their sides and by their angles.
When looking at sides, we have scalene triangles (no congruent sides), isosceles triangles (at least 2 congruent sides), and equilateral triangles (all 3 sides congruent). For angles, triangles are classified as acute triangles (all angles less than 90°), right triangles (one 90° angle), obtuse triangles (one angle greater than 90°), or equiangular triangles (all angles equal).
To classify a triangle, you can calculate the lengths of its sides using the distance formula: d = √. For example, if triangle OPQ has coordinates O(0,0), P, and Q(6,3), you would calculate each side length and compare them to determine if any are equal.
Quick Tip: When a triangle has no congruent sides, it's a scalene triangle. You can verify this by calculating each side length using the distance formula and comparing the results.
2
of 3
Finding Angle Measures
The angles inside a triangle hide some powerful relationships! First, let's distinguish between interior angles (the angles inside the triangle) and exterior angles (formed when you extend the sides outward).
The Triangle Sum Theorem states that the sum of interior angles in any triangle equals 180°. This means if you know two angles, you can always find the third by subtracting their sum from 180°. For example, if two angles measure 70° and 55°, the third angle must be 180° - = 55°.
The Exterior Angle Theorem gives us another useful relationship: an exterior angle equals the sum of the two non-adjacent interior angles. This helps solve problems where you need to find missing angles based on given information.
Remember: When solving for missing angles, set up an equation using either the Triangle Sum Theorem (interior angles add to 180°) or the Exterior Angle Theorem (exterior angle equals sum of non-adjacent interior angles).
3
of 3
Right Triangle Properties
Right triangles have special properties that make them particularly useful. The most important is that the two non-right angles are always complementary, meaning they add up to 90°.
This property comes directly from the Triangle Sum Theorem. Since one angle is 90° and the total must be 180°, the remaining two angles must sum to 90°. This is called the Corollary to the Triangle Sum Theorem.
When solving problems with right triangles, you can use this complementary relationship to find missing angles. For example, if one acute angle in a right triangle is 32°, the other acute angle must be 90° - 32° = 58°. Setting up an equation like 2x° + ° + 90° = 180° can help you find the value of x and determine the measures of all angles.
Problem-Solving Strategy: When working with a right triangle, always remember that the two acute angles sum to 90°. This gives you a quick way to find one angle when you know the other!
We thought you’d never ask...
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.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Similar Content
Calculus 1Most popular content in Geometry
9GeometryGeometry Basics: Points, Lines, and Planes
Learn the fundamental concepts of geometry, including points, lines, and planes, and their properties and relationships.
9th3388
GeometryBasic Geometry Vocab
Just a couple of terms that are useful to know in Geometry
10th1500
GeometryParallel Lines cut by a Transveral
transversal, corresponding angles, alternate interior angles, alternate interior angles, alternate exterior angles, supplementary angles, vertical angles, consecutive interior angles/same side interior
10th811
GeometryIntroductory Geo Notes
Notes from first geometry class, meant for begginer students
8th611
GeometryUnit 10: Circles Homework 2: Central Angles & Arc Measures
Geometry Homework, 100%
10th1,9018
GeometryGeometry - Angles Notes
This tackles different angles in geometry and how to solve them.
7th27810
GeometryTrapezoids and lines
Trapezoid’s, isosceles trapezoid, mid segment of a trapezoid
862
GeometryGeometry Terms
Basic geometry terms and definitions
1537
GeometryGeometry Notes 9th Grade - Unit 1 A-1
Geometry Notes for Basic Geometry Vocabulary
9th720
Most popular content
9AlgebraFAR NOTES- CPM
Compiled by CPM
College2,11847
AP Physics 1Newton's Second Law of Motion
A detailed explanation of Concept of Newton's Second Law of Motion, Examples, Formulas, and Sample Problems with solution.
43614
AlgebraFAR NOTES-HERCULES
LAST MINUTE NOTES BY HERCULES CPA
College1,19818
AlgebraFAR NOTES - KUYA WOWOWIE
Far notes by Kuya Wowowie
College95118
EnglishIntroduction to linguistics
Introduction to linguistics exam revision notes. Structure of language, typologies of language, parts of speech, language families, Chomsky, Hockett, semantic triangle, Prague Linguistic Circle, writing systems, acquisition and learning
1,28153
AlgebraRFBT NOTES- HERCULES
LAST MINUTE NOTES BY HERCULES CPA
College92216
Fun StuffDMV Practice Test 1
First set of Questions from DMV Handbook
1,383131
AlgebraAFAR NOTES- CPM
Compiled by CPM
College1,31817
PhysicsTranslational Motion and Rotational Motion
Applications of Translational and Rotational Motion
11th1811
Students love us — and so will you.
4.6/5App Store
4.7/5Google 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 SiOS 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 KlichAndroid 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.
AnnaiOS user
Geometry55 views·Updated Aug 22, 2026·3 pagesChapter 5: Exploring Geometry
Sebasbg13@sebastianbg13
Triangles are fundamental shapes in geometry with properties that make them both fascinating and practical. Understanding how to classify triangles and work with their angles helps you solve many geometric problems. Let's explore the key concepts about triangle angles and...
1
of 3
Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
Classifying Triangles
Ever wonder how many different types of triangles exist? Triangles can be classified in two main ways: by their sides and by their angles.
When looking at sides, we have scalene triangles (no congruent sides), isosceles triangles (at least 2 congruent sides), and equilateral triangles (all 3 sides congruent). For angles, triangles are classified as acute triangles (all angles less than 90°), right triangles (one 90° angle), obtuse triangles (one angle greater than 90°), or equiangular triangles (all angles equal).
To classify a triangle, you can calculate the lengths of its sides using the distance formula: d = √. For example, if triangle OPQ has coordinates O(0,0), P, and Q(6,3), you would calculate each side length and compare them to determine if any are equal.
Quick Tip: When a triangle has no congruent sides, it's a scalene triangle. You can verify this by calculating each side length using the distance formula and comparing the results.
2
of 3Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
Finding Angle Measures
The angles inside a triangle hide some powerful relationships! First, let's distinguish between interior angles (the angles inside the triangle) and exterior angles (formed when you extend the sides outward).
The Triangle Sum Theorem states that the sum of interior angles in any triangle equals 180°. This means if you know two angles, you can always find the third by subtracting their sum from 180°. For example, if two angles measure 70° and 55°, the third angle must be 180° - = 55°.
The Exterior Angle Theorem gives us another useful relationship: an exterior angle equals the sum of the two non-adjacent interior angles. This helps solve problems where you need to find missing angles based on given information.
Remember: When solving for missing angles, set up an equation using either the Triangle Sum Theorem (interior angles add to 180°) or the Exterior Angle Theorem (exterior angle equals sum of non-adjacent interior angles).
3
of 3Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
Right Triangle Properties
Right triangles have special properties that make them particularly useful. The most important is that the two non-right angles are always complementary, meaning they add up to 90°.
This property comes directly from the Triangle Sum Theorem. Since one angle is 90° and the total must be 180°, the remaining two angles must sum to 90°. This is called the Corollary to the Triangle Sum Theorem.
When solving problems with right triangles, you can use this complementary relationship to find missing angles. For example, if one acute angle in a right triangle is 32°, the other acute angle must be 90° - 32° = 58°. Setting up an equation like 2x° + ° + 90° = 180° can help you find the value of x and determine the measures of all angles.
Problem-Solving Strategy: When working with a right triangle, always remember that the two acute angles sum to 90°. This gives you a quick way to find one angle when you know the other!
We thought you’d never ask...
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.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Similar Content
Most popular content in Geometry
9GeometryGeometry Basics: Points, Lines, and Planes
Learn the fundamental concepts of geometry, including points, lines, and planes, and their properties and relationships.
9th3388
GeometryBasic Geometry Vocab
Just a couple of terms that are useful to know in Geometry
10th1500
GeometryParallel Lines cut by a Transveral
transversal, corresponding angles, alternate interior angles, alternate interior angles, alternate exterior angles, supplementary angles, vertical angles, consecutive interior angles/same side interior
10th811
GeometryIntroductory Geo Notes
Notes from first geometry class, meant for begginer students
8th611
GeometryUnit 10: Circles Homework 2: Central Angles & Arc Measures
Geometry Homework, 100%
10th1,9018
GeometryGeometry - Angles Notes
This tackles different angles in geometry and how to solve them.
7th27810
GeometryTrapezoids and lines
Trapezoid’s, isosceles trapezoid, mid segment of a trapezoid
862
GeometryGeometry Terms
Basic geometry terms and definitions
1537
GeometryGeometry Notes 9th Grade - Unit 1 A-1
Geometry Notes for Basic Geometry Vocabulary
9th720
Most popular content
9AlgebraFAR NOTES- CPM
Compiled by CPM
College2,11847
AP Physics 1Newton's Second Law of Motion
A detailed explanation of Concept of Newton's Second Law of Motion, Examples, Formulas, and Sample Problems with solution.
43614
AlgebraFAR NOTES-HERCULES
LAST MINUTE NOTES BY HERCULES CPA
College1,19818
AlgebraFAR NOTES - KUYA WOWOWIE
Far notes by Kuya Wowowie
College95118
EnglishIntroduction to linguistics
Introduction to linguistics exam revision notes. Structure of language, typologies of language, parts of speech, language families, Chomsky, Hockett, semantic triangle, Prague Linguistic Circle, writing systems, acquisition and learning
1,28153
AlgebraRFBT NOTES- HERCULES
LAST MINUTE NOTES BY HERCULES CPA
College92216
Fun StuffDMV Practice Test 1
First set of Questions from DMV Handbook
1,383131
AlgebraAFAR NOTES- CPM
Compiled by CPM
College1,31817
PhysicsTranslational Motion and Rotational Motion
Applications of Translational and Rotational Motion
11th1811
Students love us — and so will you.
4.6/5App Store
4.7/5Google 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 SiOS 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 KlichAndroid 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.
AnnaiOS user