A triangle is a polygon with three edges and three vertices. Each edge connects two vertices, forming three interior angles. The sum of these interior angles is always 180 degrees. Triangles are the simplest polygon and form the building blocks for many geometric shapes and structures.
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. The classification helps us understand the properties and relationships within each type of triangle.
Triangles have several important properties. First, the sum of the interior angles in any 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. 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. For right triangles specifically, the Pythagorean Theorem states that the square of the hypotenuse equals the sum of the squares of the other two sides.
Triangles have several special lines and points. Medians are lines drawn from a 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 a vertex to the opposite side. The three altitudes 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 lines and points have important properties in geometry and are used in various mathematical proofs and applications.
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. Triangles have special lines like medians, altitudes, and angle bisectors, which intersect at special points like the centroid, orthocenter, and incenter. Triangles are fundamental shapes in geometry and have numerous applications in architecture, engineering, and navigation.