Shape and Space - Properties and Classification of Triangles

A triangle is a two-dimensional polygon with three sides, three vertices, and three interior angles. Visually, it is the simplest closed shape made of straight line segments.

The Interior Angle Sum Theorem states that the sum of the three interior angles in any triangle is always $180^\circ$. Imagine tearing the three corners of any paper triangle and placing them side-by-side; they will form a perfect straight line.

Triangles are classified by their side lengths: an Equilateral triangle has three equal sides and three $60^\circ$ angles; an Isosceles triangle has at least two equal sides and two equal base angles; and a Scalene triangle has no equal sides or angles.

Triangles are also classified by their internal angles: an Acute triangle has three angles less than $90^\circ$; a Right-angled triangle has exactly one $90^\circ$ angle (often marked with a small square in the corner); and an Obtuse triangle has one angle greater than $90^\circ$.

The Triangle Inequality Theorem dictates that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side ($a + b > c$). Visually, if one side is too long relative to the others, the two shorter sides will not be able to connect to form a closed shape.

The height (altitude) of a triangle is the perpendicular distance from a vertex to the opposite side, which acts as the base. In a right-angled triangle, the two sides forming the $90^\circ$ angle can serve as the base and height. In an obtuse triangle, the height might fall outside the triangle's body.

The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the two opposite interior angles. If you extend one side of a triangle outward, the angle formed between the extension and the adjacent side is the exterior angle.