Lines, Angles, and Triangles

Geometry starts from a few undefined terms — a point (a location, no size), a line (a straight path extending forever in both directions), and a plane (a flat surface extending forever in every direction) — and builds everything else on top of them. A segment is the piece of a line between two endpoints; a ray starts at one point and extends forever in a single direction. When three points A, B, and C lie on a line in that order, the Segment Addition Postulate says the two shorter pieces add up to the whole: AB + BC = AC.

An angle is formed by two rays that share an endpoint, called the vertex. Every angle falls into one of five types by its degree measure: acute (under 90°), right (exactly 90°), obtuse (between 90° and 180°), straight (exactly 180° — its two rays form a single line), and reflex (over 180°). Just like segments, angles have their own addition rule: if a ray splits a larger angle into two smaller ones, the two pieces add up to the whole — the Angle Addition Postulate. Drag the angle in the first tab below to see all five types, plus each angle's complement and supplement, update live.

Two angles are complementary if they add to 90°, and supplementary if they add to 180° — either pairing can happen between angles that are nowhere near each other. Adjacent angles specifically share a vertex and a side without overlapping; when two adjacent angles' unshared sides form a straight line, that pair is called a linear pair, and a linear pair is always supplementary. Vertical angles — the pair formed directly across an intersection of two lines — are always equal, whether or not any other lines are involved. Two lines that meet at a right angle are perpendicular; a line that passes through a segment’s midpoint at a right angle is its perpendicular bisector, and every point on it is the same distance from both of the segment’s endpoints.

Whenever a transversal cuts two parallel lines, only two angle measures ever show up — call them θ and 180° − θ — repeated in a predictable pattern. Same-colored angles above are always equal to each other, no matter which of the two intersections they’re at; any two angles that share a single line are supplementary, always adding to 180°.

A few specific relationships are worth naming: vertical angles (directly across one intersection) are always equal, with or without parallel lines. Corresponding angles (matching position at each intersection) are equal exactly because the lines are parallel. Alternate interior angles (between the two lines, on opposite sides of the transversal) are also equal. Same-side interior angles are the one pairing that’s not equal — they’re supplementary instead.

Every triangle’s interior angles sum to 180°, which is where the exterior angle theorem comes from: an exterior angle equals the sum of the two interior angles that aren’t next to it. In an isosceles triangle, the two angles opposite the equal sides are themselves equal — a shortcut that often skips a step of algebra entirely.