A triangle is a polygon with three sides, three vertices or corners, and three angles. The vertices are typically labeled with capital letters like A, B, and C. The sides are labeled with lowercase letters a, b, and c, and the angles are often represented by Greek letters alpha, beta, and gamma. Triangles are the simplest polygon and form the basis for many geometric concepts.
Triangles can be classified in different ways. Based on their sides, triangles can be equilateral with all sides equal, isosceles with two sides equal, or scalene with no sides equal. Based on their angles, triangles can be acute with all angles less than 90 degrees, right with one angle exactly 90 degrees, or obtuse with one angle greater than 90 degrees. These classifications help us understand the properties and relationships within different types of triangles.
Triangles have several important properties. First, the sum of all interior angles in any triangle is always 180 degrees. The sum of exterior angles is 360 degrees. The Triangle Inequality Theorem states that the length of any side must be less than the sum of the other two sides. The area of a triangle can be calculated using the formula: one-half times base times height. And the perimeter is simply the sum of all three sides. These properties are fundamental in geometry and have many applications in mathematics and real-world problem solving.
Triangles have several special lines and points that help us understand their properties. A median connects a vertex to the midpoint of the opposite side. An altitude is a perpendicular line from a vertex to the opposite side. An angle bisector divides an angle into two equal parts. The centroid, which is the intersection of all three medians, is the triangle's center of gravity. The orthocenter is where all three altitudes intersect. And the incenter, formed by the intersection of angle bisectors, is the center of the triangle's inscribed circle. These special elements reveal important geometric relationships within triangles.
To summarize what we've learned about triangles: A triangle is a polygon with three sides, three vertices, and three angles. Triangles can be classified by their sides as equilateral, isosceles, or scalene. They can also be classified by their angles as acute, right, or obtuse. Key properties of triangles include the fact that their interior angles always sum to 180 degrees and the triangle inequality theorem. Triangles also have special points such as the centroid, orthocenter, and incenter, which help us understand their geometric properties. Understanding triangles is fundamental to geometry and has many practical applications in mathematics, engineering, and architecture.