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Introduction & Overview
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In this section, we learn about triangles categorized by their sides—equilateral, isosceles, and scalene—as well as by their angles—acute, right, and obtuse. Additionally, essential properties such as the angle sum property, exterior angle theorem, and triangle inequality theorem are discussed.
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Types of Triangles by Sides
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● Types by Sides:
- Equilateral Triangle: All sides equal
- Isosceles Triangle: Two sides equal
- Scalene Triangle: All sides unequal
Detailed Explanation
Triangles can be classified based on the lengths of their sides into three distinct types:
- Equilateral Triangle: This type has all three sides of equal length. Because all sides are equal, all three angles are also equal, each measuring 60°.
- Isosceles Triangle: This triangle has two sides that are of equal length. The angles opposite those equal sides are also equal.
- Scalene Triangle: In this triangle, all three sides have different lengths. Consequently, all three angles are also different.
Examples & Analogies
Imagine a slice of pizza. If all slices are the same, like an equilateral triangle, they look identical. If two slices are longer than one, like an isosceles triangle, those two are equal, but the third is shorter. If all slices are different lengths, it's like a scalene triangle – no two slices are identical.
Types of Triangles by Angles
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● Types by Angles:
- Acute-Angled Triangle: All angles < 90°
- Right-Angled Triangle: One angle = 90°
- Obtuse-Angled Triangle: One angle > 90°
Detailed Explanation
Triangles can also be categorized based on their angles:
- Acute-Angled Triangle: All three angles are less than 90 degrees. This type of triangle appears 'sharp' and 'pointy'.
- Right-Angled Triangle: One angle is exactly 90 degrees, making this triangle useful in various applications, especially in geometry and trigonometry.
- Obtuse-Angled Triangle: Here, one angle is greater than 90 degrees, resulting in a 'wide' appearance on one side.
Examples & Analogies
Consider a door. When a door is opened to about 90 degrees, it forms a right angle. Now, if someone pushed the door wide open (more than 90 degrees), you've got an obtuse angle. If it's barely opened, and all angles are less than 90 degrees, that's like an acute triangle.