Chapter 4.1 — Introduction to Lines and Angles

Segments, Rays, and Points That Share a Line

A line segment has two fixed end points and a measurable length — segment AB is written AB, and its length is also written AB. A ray has only a starting point; it extends infinitely in one direction with no end point at all — a ray beginning at A and passing through B is written with an arrow over AB. A full line extends infinitely in both directions, with neither a start nor an end. The three objects form a natural progression by how much of them is fixed: a segment is fixed at both ends and has a length worth measuring, a ray is fixed at just one end and stretches away forever, and a line is fixed at neither end, unbounded in both directions at once.

Notation follows the same pattern throughout the chapter: a bare pair of letters like AB, with no arrow, usually means the segment or its length depending on context, while an arrow above the letters specifically marks a ray, and stating "line AB" in words removes any ambiguity when a segment's underlying infinite line is what's actually meant.

Collinear points are three or more points lying on one straight line — if A, X, P, M, and B all sit on the same line, all five are collinear, regardless of how they're spaced along it. Collinearity says nothing about distance or order, only that a single straight line can be drawn through every one of the points at once; three points that are almost, but not exactly, in a straight line are not collinear at all, no matter how small the deviation.

What Turns a Ray Into an Angle

An angle is the change in direction when a ray rotates about a fixed point, from an initial position to a final one. The two rays are the angle's arms; the fixed point they share is its vertex, the one point every measurement of the angle is taken relative to. If rays AB and AC share point A, the angle at A is written ∠BAC, with A as vertex and AB, AC as arms. The middle letter in that notation always names the vertex — ∠BAC and ∠CAB refer to the exact same angle, since both simply confirm that A is where the rotation happened, while B and C mark where each arm points — swap either of those outer letters for a different point on the same arm, and the notation still refers to the identical angle.

Five Angles, Sorted by Degree

These five categories cover every possible angle exactly once — any angle measure between 0° and 360° falls into precisely one row of this table, with the two boundary values (90° and 180°) each getting their own named category rather than being lumped in with the range on either side. It's worth noticing why the two named boundary cases matter so much in practice: a right angle marks the exact halfway point between "acute" and "obtuse," and a straight angle marks the exact halfway point between "obtuse" and "reflex" — both are the pivot values that later theorems (like the linear pair, which always sums to a straight angle) are built around, not arbitrary cutoffs.

Parallel, Intersecting, and Concurrent Lines

Parallel lines stay the same distance apart forever and never meet, however far extended — AB ∥ CD means AB and CD are parallel, with no point in common at all. Intersecting lines cross at exactly one point. When three or more lines all pass through that same single point, they're called concurrent lines, and the shared point is the point of concurrence.

These three categories aren't quite exhaustive in the way the five angle types were — they describe how lines behave, not an exhaustive partition of every possible pair. Two distinct straight lines in the same plane are always either parallel or intersecting, never both and never neither; there's no third option for two lines confined to a flat plane. Concurrency is a separate idea layered on top, describing what happens once a third (or fourth, or more) line joins an existing intersection at that exact same point rather than crossing somewhere else. Three lines drawn at random almost never manage this — concurrency at a single shared point is a special coincidence worth naming precisely because it doesn't happen by default; a small nudge to any one of the three lines is usually enough to break it, leaving three separate, ordinary intersection points instead of one shared one — which is exactly why concurrency, when it does show up deliberately in a construction, is often the whole point of the figure rather than an incidental detail.

It's easy to conflate "intersecting" with "concurrent," but the two aren't interchangeable: any two lines that cross are intersecting, full stop, regardless of how many other lines exist nearby. "Intersecting" is really just a description of a pair, while "concurrent" only becomes a meaningful word once at least three lines are on the table. Concurrency specifically requires three or more lines to share the exact same single point — two lines crossing at one point and a third line crossing either of them somewhere else entirely is not a concurrent arrangement, even though every pair among the three is still intersecting.

Where This Vocabulary Gets Used

Every term defined here reappears constantly across the rest of the chapter: Exercise 4.1 tests this vocabulary directly against real figures, and the angle-pair relationships in Exercise 4.2 and the parallel-line properties in Exercise 4.3 are both built directly on top of it. The algebraic habit of translating a diagram into an equation, needed throughout this chapter, draws on the same skills built in Polynomials and Factorisation — an unknown angle marked x° on a figure becomes, after one or two substitutions, exactly the kind of linear equation solved there. Nothing in the exercises ahead introduces a genuinely new object beyond what's defined here; every later result is really a statement about how these same segments, rays, angles, and line relationships behave once more of them are combined in a single figure.