On a wall plan, A and B mark the ends of a three-metre section, and C marks a socket 1.2 m from A. Elsewhere a direction branches from a line at 60°. Reading the plan means distinguishing an object, its notation and its measure.
A segment has two endpoints and a length; a ray has a starting point and a direction; a line extends without limit both ways. Two rays with a common starting point define an angle. We measure length in units such as metres and angles in degrees. The only prerequisite is models and proof.
Where does the line end?
Point A marks a position without having its own size. Line AB passes through distinct points A and B but does not end there. Ray AB starts at A, passes through B and continues. Ray BA has a different starting point and the opposite direction. Segments AB and BA, however, are the same figure.
In “AB=3 m”, AB denotes the segment’s length, rather than its set of points. Books may add bars or arrows to distinguish lines, rays and segments; accompanying words avoid ambiguity. Arrows in our diagram mean continuation, not physical motion.
Line, ray, segment
Bisector of a 60° angle
Two lines: 60° and 120° angles
Because C is between A and B, AB=AC+CB. The remaining distance from the socket to B is:
CB = AB − AC = 3 − 1.2 = 1.8 m.
All numbers are in the same unit: metres. If C lay beyond B on the extension, the same addition arrangement would no longer apply. A midpoint is an interior point dividing a segment into two equal parts; our C is not the midpoint because 1.2≠1.8.
Reading an angle
In ∠AOB, the middle letter O is the vertex; OA and OB are the side rays. The lengths of the drawn strokes OA and OB do not determine the angle. Extending a stroke without changing its direction leaves the angle unchanged.
A full turn is 360°, a straight angle between opposite rays is 180°, and a right angle is 90°. An acute angle is greater than 0° and less than 90°; an obtuse angle is greater than 90° and less than 180°. Here we use ordinary undirected angles up to 180°; an arc identifies the region intended. Turn direction and larger angles belong to trigonometry.
An angle bisector is a ray from the vertex inside the angle dividing it into two equal angles. In the diagram OC splits 60° into 30° and 30°. This does not mean halving an arbitrary side or distance. You need not know the compass construction yet: the meaning matters here. Euclid I.9.
Two lines provide more information
A linear pair consists of angles sharing one side whose other sides are opposite rays. Together they form a straight angle, so they sum to 180°. A shared side alone is insufficient: two adjacent 30° angles are not a linear pair. Euclid I.13.
Vertical angles lie opposite each other where two lines cross; each side of one is opposite a side of the other. They are equal. Each, together with the same third angle, makes 180°; subtract that third angle and equality follows. Euclid I.15.
If one of the four angles at an intersection is 60°, its linear-pair neighbour is 180°−60°=120°, its opposite is 60°, and the last is 120°. Check around the point:
60° + 120° + 60° + 120° = 360°.
Equal angles need not be vertical: they may occupy different places on the plan. Nor does “vertical” here mean upward: rotate the whole drawing and the property remains. When reading a plan, establish the given rays and points before applying length and angle rules.