The six main types of triangles are classified based on their side lengths and angle measures: isosceles, equilateral, scalene, acute, obtuse, and right triangles. Understanding these classifications helps in various geometric and mathematical applications.
Triangles can be categorized in two primary ways: by the lengths of their sides or by the measures of their angles. This provides a comprehensive system for identifying and working with different triangular forms.
Classification by Side Lengths
Triangles are categorized into three types based on the relationships between their side lengths:
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Isosceles Triangle:
- An isosceles triangle is a triangle with two congruent sides (sides of equal length) and one unique side.
- Consequently, the angles opposite the two congruent sides are also congruent.
- Example: A triangle with sides measuring 5 cm, 5 cm, and 7 cm.
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Equilateral Triangle:
- An equilateral triangle is a triangle with three congruent sides (all sides are of equal length).
- As a result, all three angles are also congruent, each measuring exactly 60 degrees.
- Key Feature: It is a special type of isosceles triangle.
- Example: A triangle where all sides are 10 cm long.
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Scalene Triangle:
- A scalene triangle is a triangle in which all three sides have different lengths.
- Due to the unequal side lengths, all three angles also have different measures.
- Example: A triangle with sides measuring 3 cm, 4 cm, and 5 cm.
Classification by Angle Measures
Triangles are also categorized into three types based on the measures of their interior angles:
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Acute Triangle:
- An acute triangle is a triangle where all three interior angles are acute angles, meaning each angle measures less than 90 degrees.
- Example: A triangle with angles measuring 60°, 70°, and 50°.
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Obtuse Triangle:
- An obtuse triangle is a triangle that has one obtuse angle, meaning one interior angle measures greater than 90 degrees.
- A triangle can only have one obtuse angle, as the sum of all angles must be 180 degrees.
- Example: A triangle with angles measuring 110°, 40°, and 30°.
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Right Triangle:
- A right triangle is a triangle that has one right angle, meaning one interior angle measures exactly 90 degrees.
- The side opposite the right angle is called the hypotenuse, and it is always the longest side.
- Example: A triangle with angles measuring 90°, 45°, and 45°.
Summary Table of Triangle Types
| Classification Category | Triangle Type | Definition | Key Characteristics | Learn More |
|---|---|---|---|---|
| By Side Lengths | Isosceles | Two congruent sides and one unique side. | Two equal sides; two equal angles. | Isosceles Triangle |
| Equilateral | Three congruent sides and three congruent angles. | All sides equal; all angles 60°. | Equilateral Triangle | |
| Scalene | All three sides have different lengths. | All sides different; all angles different. | Scalene Triangle | |
| By Angle Measures | Acute | All three interior angles are acute (less than 90°). | All angles < 90°. | Acute Triangle |
| Obtuse | One interior angle is obtuse (greater than 90°). | One angle > 90°. | Obtuse Triangle | |
| Right | One interior angle is exactly 90°. | One angle = 90°. | Right Triangle |
Understanding these fundamental classifications is crucial for further studies in geometry, trigonometry, and various fields of engineering and architecture.
