The Language and Notation of Geometry

Geometry has its own shorthand, and it starts with capital letters. A single capital letter names a point, which is an exact position with no length and no width, so we call this one point A. Two letters name something passing through two points, and the word in front tells you how far it goes. A line BC is straight and never stops in either direction, so it carries arrowheads at both ends. An interval DE is only the piece between two endpoints, so it has dots and it stops. A ray FG starts at F, passes through G and keeps going one way for ever. Same pair of letters, three different objects, and the word you choose is what makes your meaning exact.

An angle is the amount of turn between two arms that meet at a corner. The corner is called the vertex, and it is the vertex letter that does the real work in the name. Here the arms run from B out to A and out to C, so the angle is written as angle ABC, with B, the vertex, sitting in the middle. You may read it backwards as angle CBA, because both describe the same opening. If only one angle sits at that corner you can shorten it to angle B. What you cannot do is write angle BAC and mean this one, because that name puts A in the middle, which points at a completely different corner.

Angles are sorted by how much turn they contain, and there are six names to learn. An acute angle is less than ninety degrees, so it is a narrow opening. A right angle is exactly ninety, the square corner you see on the edge of a page. An obtuse angle is wider than a right angle but still less than a straight line, so it lies between ninety and one hundred and eighty. A straight angle is exactly one hundred and eighty, where the two arms make one straight line. Past that comes a reflex angle, anywhere between one hundred and eighty and three hundred and sixty. A full turn back to the start is three hundred and sixty degrees, called a revolution.

Diagrams carry marks that tell you facts the picture alone cannot. Matching tick marks, the little dashes drawn across two sides, mean those sides are exactly the same length even if the drawing is slightly out. Matching arcs drawn inside two corners mean those two angles are equal in size. A small square tucked into a corner means that angle is exactly ninety degrees, so you never have to guess or measure. Matching arrowheads on two sides mean those sides are parallel, and if a second pair is also parallel it gets double arrowheads so you can tell the two pairs apart. Read the marks before you read the numbers, because the marks are part of the question.

Now put the letters and the marks together. To name a shape you walk round its corners in order, so this one is quadrilateral ABCD: A to B to C to D and back to A. Naming it ABDC would be wrong, because that path cuts across the middle instead of tracing the outline. Each side takes the two letters at its ends, so the sides here are AB, BC, CD and DA. The single arrowheads tell you AB is parallel to DC, and the double arrowheads tell you AD is parallel to BC. And angle ABC means the corner at B, sitting between the two sides BA and BC that meet there.