Triangles are among the most studied shapes in geometry, and they can be sorted in two main ways. Classified by side length, an equilateral triangle has all three sides equal, an isosceles triangle has exactly two sides equal, and a scalene triangle has no equal sides. Each side classification carries angle consequences as well. Classified by angle, an acute triangle has all three angles less than 90°, a right triangle has exactly one 90° angle, and an obtuse triangle has one angle greater than 90°. Every triangle, regardless of type, obeys the Triangle Angle Sum Theorem: the three interior angles always add up to exactly 180°. This single fact is the key to finding any unknown angle in a triangle when the other two are known.
Beyond classification, triangles can be compared with one another through congruence and similarity. Two triangles are congruent if they have exactly the same size and shape, and there are four standard ways to prove this. The Side-Side-Side (SSS) postulate says that if all three pairs of sides are equal, the triangles are congruent. The Side-Angle-Side (SAS) postulate requires two sides and the angle between them to match. The Angle-Side-Angle (ASA) postulate requires two angles and the side between them to match. The Angle-Angle-Side (AAS) theorem requires two angles and a non-included side. Similar triangles, on the other hand, have the same shape but may differ in size: their corresponding angles are equal and their corresponding sides are in proportion. By the Angle-Angle (AA) similarity postulate, matching two pairs of angles is enough to guarantee similarity.
Two more important theorems govern triangle behavior. The Triangle Inequality Theorem states that the sum of any two side lengths must be greater than the third side, and this must hold for all three combinations. If a, b, and c are the sides, then a + b > c, a + c > b, and b + c > a; otherwise, the three lengths cannot form a triangle. The Exterior Angle Theorem describes the relationship between an exterior angle and the two non-adjacent interior angles (called remote interior angles): the exterior angle equals the sum of those two remote interior angles. Together, these tools let us determine whether triangles can exist, find missing measurements, and compare triangles precisely.