Triangles are fundamental shapes in geometry with properties that make...
Geometry55 views·Updated Aug 1, 2026·3 pagesChapter 5: Exploring Geometry
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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.
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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).
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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!
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Geometry55 views·Updated Aug 1, 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...
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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
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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).
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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.
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