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 often labeled with lowercase letters a, b, and c. And the angles are commonly denoted using Greek letters alpha, beta, and gamma.
Triangles can be classified in different ways. Based on their sides, triangles can be equilateral with all sides equal, isosceles with two equal sides, or scalene with all sides different. 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.
Triangles have several important properties. First, the sum of all angles in a triangle is always 180 degrees. The area of a triangle can be calculated as one-half times the base times the height. The perimeter is simply the sum of all three sides. And according to the Triangle Inequality Theorem, the sum of the lengths of any two sides must be greater than the length of the third side, which is why not all combinations of three lengths can form a triangle.
Triangles have several special lines and points. 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. And a perpendicular bisector is perpendicular to a side at its midpoint. These special lines intersect at important points inside or outside the triangle, such as the centroid, orthocenter, incenter, and circumcenter, each with unique properties.
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. Key properties include the fact that angles sum to 180 degrees, the area formula is one-half times base times height, and the triangle inequality theorem. Special lines in triangles include medians, altitudes, angle bisectors, and perpendicular bisectors. Triangles are fundamental shapes used extensively in geometry, engineering, and architecture.