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Exploring the fundamentals of triangle geometry, this overview covers the angle sum property, area and perimeter calculations, and classifications by sides and angles. It delves into the Pythagorean theorem for right-angled triangles and their practical applications in various fields. Understanding these concepts is key to solving geometric problems and has implications in architecture, engineering, and construction.

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## Definition of a Triangle

### Three-sided polygon

A triangle is a three-sided polygon with three vertices and corresponding angles

### Origin of the term "triangle"

The term "triangle" comes from the Latin word "triangulus," combining "tri" (three) and "angulus" (angle)

### Properties and types of triangles

A comprehensive understanding of triangles, including their properties and types, is essential for exploring geometric concepts and solving related mathematical problems

## Properties of Triangles

### Angle sum property

The sum of the interior angles of a triangle is always 180 degrees, which is instrumental in determining unknown angles

### Area of a triangle

The area of a triangle is calculated using the formula \( \frac{1}{2} \times \text{base} \times \text{height} \) and has practical applications in fields such as architecture and engineering

### Perimeter of a triangle

The perimeter of a triangle is the sum of the lengths of its sides and is essential in various disciplines, including geometry, construction, and land surveying

## Classification of Triangles

### Equilateral triangle

An equilateral triangle has all three sides and angles equal, with each angle measuring 60 degrees

### Isosceles triangle

An isosceles triangle has two sides and two angles of equal measure

### Scalene triangle

A scalene triangle has sides and angles of differing lengths and measures, respectively

### Right-angled triangle

A right-angled triangle features one 90-degree angle and can be isosceles or scalene but not equilateral

## Pythagorean Theorem

### Definition and principle

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides

### Practical applications

The Pythagorean theorem is a powerful tool for calculating unknown side lengths in right-angled triangles and leads to the concept of Pythagorean triples

### Example problems

Applying the principles of triangle geometry can be demonstrated through example problems, such as determining if a triangle is right-angled or finding missing angles in an isosceles triangle

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