A triangle is a polygon with three sides, three vertices or corners, and three angles. The sides are typically labeled with lowercase letters a, b, and c, while the vertices are labeled with uppercase letters A, B, and C. The angles are often represented by Greek letters alpha, beta, and gamma. The sum of all interior angles in any triangle is always 180 degrees.
Triangles can be classified in different ways. Based on their sides, triangles can be equilateral, where all three sides are equal; isosceles, where two sides are equal; or scalene, where no sides are equal. Based on their angles, triangles can be acute, where all angles are less than 90 degrees; right, where one angle is exactly 90 degrees; or obtuse, where one angle is greater than 90 degrees. The sum of all interior angles in any triangle is always 180 degrees.
Triangles have several important properties. First, the sum of all interior angles in any triangle is always 180 degrees. This means that if you know two angles, you can easily find the third. The sum of exterior angles is 360 degrees. The area of a triangle can be calculated using the formula: one-half times the base times the height. The perimeter is simply the sum of all three sides. Another important property is the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side. This is why not every three lengths can form a triangle.
Triangles have several special lines and points that are important in geometry. Medians connect each vertex to the midpoint of the opposite side. All three medians intersect at a single point called the centroid, which is the center of mass of the triangle. Altitudes are perpendicular lines drawn from each vertex to the opposite side. They intersect at the orthocenter. Angle bisectors divide each angle into two equal parts and intersect at the incenter, which is the center of the inscribed circle. These special points and lines have many interesting properties and applications in geometry and physics.
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, and by their angles as acute, right, or obtuse. The sum of interior angles in any triangle is always 180 degrees. The area of a triangle can be calculated using the formula: one-half times the base times the height. Triangles have special points including the centroid, orthocenter, and incenter, which have important properties in geometry. Understanding triangles is fundamental to many areas of mathematics, engineering, and design.