Basic Geometrical Ideas - Triangles

Definition: A triangle is a simple closed curve made up of three line segments. It is considered the polygon with the least number of sides. Visually, it consists of three vertices connected by three edges.

Elements of a Triangle: A triangle has six elements: three sides and three angles. In a triangle $\Delta ABC$, the three sides are $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$. The three angles are $\angle BAC$, $\angle ABC$, and $\angle BCA$. The points $A$, $B$, and $C$ are called the vertices.

Interior and Exterior: A triangle divides a plane into three parts: the interior (the region inside the boundary), the boundary (the sides of the triangle), and the exterior (the region outside the triangle). Points located within the space enclosed by the sides are said to be in the interior of the triangle.

Classification by Sides: Triangles are categorized based on their side lengths. A Scalene triangle has all three sides of different lengths. An Isosceles triangle has at least two sides of equal length. An Equilateral triangle has all three sides equal in length, visually appearing perfectly symmetrical.

Classification by Angles: Triangles are also categorized by the measure of their angles. An Acute-angled triangle has all angles measuring less than $90^{\circ}$. A Right-angled triangle has exactly one angle equal to $90^{\circ}$ (forming an 'L' shape at one vertex). An Obtuse-angled triangle has one angle greater than $90^{\circ}$.

Angle Sum Property: The sum of the measures of the three interior angles of any triangle is always constant and equal to $180^{\circ}$. This is a fundamental property used to find missing angle measures.

Triangle Inequality Property: For any triangle, the sum of the lengths of any two sides must be strictly greater than the length of the third side. For sides $a$, $b$, and $c$, this means $a + b > c$, $b + c > a$, and $c + a > b$.