Triangles in Geometry

Understanding Triangles

Triangles are fundamental shapes in geometry with three sides and three angles. They form the basis for many geometric concepts and real-world applications.

What You'll Learn:

Types of triangles (by sides and angles)
Triangle properties and theorems
Perimeter and area calculations
Congruence and similarity
Special triangles (right, isosceles, equilateral)
Pythagorean theorem

Types of Triangles

By Sides

Equilateral Triangle

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All three sides are equal

All angles = 60°

Isosceles Triangle

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Two sides are equal

Two base angles are equal

Scalene Triangle

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All sides are different

All angles are different

By Angles

Right Triangle

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One angle = 90°

Pythagorean theorem applies

Acute Triangle

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All angles < 90°

Obtuse Triangle

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One angle > 90°

Important Properties

Sum of Angles

The sum of all interior angles in a triangle is always 180°

∠A + ∠B + ∠C = 180°

Triangle Inequality

The sum of any two sides must be greater than the third side

a + b > c, b + c > a, a + c > b

Exterior Angle

An exterior angle equals the sum of the two non-adjacent interior angles

Area and Perimeter

Perimeter

P = a + b + c

Sum of all three sides

Area (Base × Height)

A = ½ × base × height

Most common formula

Heron's Formula

A = √[s(s-a)(s-b)(s-c)]

Where s = (a+b+c)/2

Practice Problems

Problem 1: Find the Missing Angle

In a triangle, two angles measure 45° and 60°. What is the third angle?

Problem 2: Triangle Area

Find the area of a triangle with base 8 cm and height 6 cm.

Special Triangles

30-60-90 Triangle

Side ratios: 1 : √3 : 2

Used in trigonometry

45-45-90 Triangle

Side ratios: 1 : 1 : √2

Isosceles right triangle

3-4-5 Triangle

Pythagorean triple

Common in applications