# Course 3 Chapter 5: Triangles and the Pythagorean Theorem

Triangles are one of the most common and fundamental shapes in mathematics. They have three sides and three angles and are the basis of many geometric shapes, forming the building blocks of the natural world. The Pythagorean Theorem is a mathematical statement that was discovered by the ancient Greek mathematician Pythagoras, and it is an essential tool for measuring the length of any side of a right triangle.

## What is a Triangle? A triangle is a polygon with three sides and three angles. The sides of a triangle are connected by three angles. The sum of the three angles of a triangle is always 180°. The sides of a triangle can be equal, meaning they all have the same length, or they can have different lengths. Triangles have three distinct categories: equilateral, isosceles, and scalene.

## The Pythagorean Theorem The Pythagorean Theorem is a mathematical statement that states the relationship between the sides of a right triangle. In a right triangle, the longest side is referred to as the hypotenuse and is opposite the right angle. The Pythagorean Theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In other words, if you know two of the three sides of a right triangle, you can use the Pythagorean Theorem to calculate the length of the remaining side.

## Uses of the Pythagorean Theorem The Pythagorean Theorem can be used to calculate the length of the sides of any right triangle. It can also be used to calculate the area and perimeter of a triangle. Additionally, it can be used to solve problems involving angles, such as finding the measure of an angle in a right triangle. The theorem is also useful in solving problems involving circles and other geometric shapes.

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Triangles and the Pythagorean Theorem are fundamental elements of mathematics. Triangles are three-sided polygons with three angles, and the Pythagorean Theorem is used to calculate the length of the sides of a right triangle. The theorem is an important tool for solving problems involving angles, circles, and other geometric shapes. Understanding triangles and the Pythagorean Theorem is essential for understanding mathematics.