Definitions
Triangle
A polygon with three edges and three vertices.
Vertex
A point where two sides of a polygon meet.
Edge
A line segment between two vertices in a polygon.
Interior Angle
An angle formed by two adjacent sides inside the polygon.
Types of Triangles
By Sides
Triangles can be classified based on their side lengths into the following types:
Equilateral Triangle
An equilateral triangle has all three sides of equal length and all three angles of equal measure, each being 60 degrees.
Isosceles Triangle
An isosceles triangle has two sides of equal length and two equal angles opposite these sides.
Scalene Triangle
A scalene triangle has all three sides of different lengths and all three angles of different measures.
By Angles
Triangles can also be classified based on their angles into the following types:
Acute Triangle
In an acute triangle, all three interior angles are less than 90 degrees.
Right Triangle
A right triangle has one interior angle that is exactly 90 degrees.
Obtuse Triangle
An obtuse triangle has one interior angle that is greater than 90 degrees.
Triangle Inequality Theorem
The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Altitude, Median, and Angle Bisector
In a triangle, an altitude is a line segment from a vertex to the line containing the opposite side and perpendicular to it. A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. An angle bisector of a triangle is a line that splits an angle into two equal angles.
Properties of Triangles
Triangles exhibit several unique properties, such as the sum of interior angles always being 180 degrees, and various congruence rules like SSS, SAS, ASA, AAS, and RHS pertaining to triangle congruency.
To remember :
Triangles, fundamental to geometry, are classified by their sides or angles into types such as equilateral, isosceles, scalene, acute, right, and obtuse. They possess important properties like the sum of interior angles being 180 degrees and the triangle inequality theorem. Other key aspects include altitudes, medians, and angle bisectors, critical in various geometrical applications.
