A triangle is one of the most fundamental shapes in geometry. It is a polygon formed by connecting three non-collinear points, creating three sides and three vertices. We label the vertices as A, B, and C, the sides opposite to these vertices as a, b, and c respectively, and the interior angles as alpha, beta, and gamma. Triangles can have different orientations and sizes, but they all share these basic structural elements.
Triangles can be systematically classified in two ways. First, by their side lengths: equilateral triangles have all three sides equal, isosceles triangles have exactly two sides equal, and scalene triangles have all sides different. Second, by their angles: acute triangles have all angles less than 90 degrees, right triangles have exactly one 90-degree angle, and obtuse triangles have one angle greater than 90 degrees. These classifications help us understand and work with different types of triangles in geometry.
Triangles have several fundamental properties. The angle sum theorem states that the sum of all interior angles in any triangle equals 180 degrees. This can be proven using parallel lines. The triangle inequality theorem states that the sum of any two sides must be greater than the third side for a valid triangle to exist. The exterior angle theorem shows that any exterior angle equals the sum of the two non-adjacent interior angles. These properties help us understand the relationships between sides and angles in triangles.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides: a squared plus b squared equals c squared. This can be visualized by drawing squares on each side of the triangle. The converse is also true: if three sides satisfy this relationship, the triangle must be a right triangle. Special right triangles include the 45-45-90 triangle with sides in ratio 1 to 1 to square root of 2, and the 30-60-90 triangle with sides in ratio 1 to square root of 3 to 2.
Triangle calculations involve finding areas, missing sides, and angles. The basic area formula is one-half base times height. For triangles where all sides are known, we can use Heron's formula involving the semi-perimeter. For finding missing parts, we use trigonometric ratios in right triangles, or the Law of Sines and Law of Cosines for general triangles. The Law of Sines relates sides to opposite angles, while the Law of Cosines generalizes the Pythagorean theorem. These tools allow us to solve real-world problems like finding building heights or measuring distances.