Definition and Basic Properties
A triangle is a polygon with three sides and three angles. Its fundamental properties include the fact that the sum of its interior angles is always 180 degrees, it is formed by connecting three non-collinear points, and the sides must satisfy the triangle inequality theorem, where the sum of any two sides must be greater than the third side.
Classifications by Sides and Angles
Triangles are classified by sides as equilateral (all sides equal), isosceles (two sides equal), or scalene (all sides different). By angles, they are acute (all angles less than 90 degrees), right (one 90-degree angle), or obtuse (one angle greater than 90 degrees). These classifications determine unique properties, such as equal base angles in isosceles triangles.
Practical Example: Right Triangle
Consider a right triangle with sides 3, 4, and 5 units, where the right angle is between the sides of 3 and 4 units. This satisfies the Pythagorean theorem (3² + 4² = 5²), illustrating how properties like perpendicular sides and hypotenuse enable calculations for distances in architecture or navigation.
Importance and Applications
Triangles underpin geometry and trigonometry, used in engineering for structural stability, in surveying for land measurement, and in computer graphics for rendering 3D models. Understanding their properties prevents errors in design and ensures precise problem-solving in physics and mathematics.