A plan requires a second marking line parallel to the first and the shortest distance from an edge to a point. Looking “almost straight” establishes neither direction nor distance. The angles from the previous lesson will help.
Distinct coplanar lines are parallel if they never intersect; they are perpendicular if they meet at a right angle. The distance from a point to a line is the length of the perpendicular to it. A shared Euclidean plane is an essential assumption.
Not meeting on the sheet is not enough
Lines are unbounded. Two short segments may have no point in common even though their containing lines meet outside the drawing. The notation a∥b refers to parallel lines. In space there are also nonintersecting lines that are not coplanar; these plane rules do not apply to them.
Draw a third line c crossing a and b at distinct points. It is a transversal, creating four angles at each intersection. Pair names describe positions and make sense before parallelism has been established.
A transversal and eight angles
Two perpendiculars to c
Straight and broken paths
In our diagram, ∠1 and ∠5 are corresponding: they occupy the same positions at the two intersections. ∠3 and ∠5 are alternate interior angles: between a and b on opposite sides of the transversal. ∠4 and ∠5 are same-side interior angles: between the lines on the same side of the transversal.
A criterion and a property work in opposite directions
A criterion establishes parallelism. If corresponding angles are equal, the lines are parallel. Equal alternate interior angles are sufficient too, as is a sum of 180° for same-side interior angles. These are three ways to check the same geometric relationship. Euclid I.28.
For example, ∠1=60° and ∠5=60° are given. They correspond, so a∥b. Arbitrary equal angles will not do: vertical angles at one intersection are always equal and tell us nothing about the second line.
A property starts with known parallelism and gives angle conclusions. If a∥b and ∠1=60°, then ∠5=60°, ∠3=60° and ∠4=120°. Hence:
∠4 + ∠5 = 120° + 60° = 180°.
Reversing a criterion to obtain a property reverses the implication and needs justification too. In a Euclidean plane both directions hold. Euclid I.29.
A set square supplies the perpendicular
a⊥c means the lines make a right angle. All four angles at the intersection are then 90°: a linear-pair neighbour completes 90° to 180°, and a vertical angle equals it.
Two distinct coplanar lines perpendicular to the same line c are parallel: their corresponding angles with c are both 90°. Thus a set square placed twice against one guiding edge can establish parallel markings. This is a geometric principle; actual precision also depends on the instrument and the work.
Why an oblique segment overstates the distance
Let P be off a, PH⊥a, and H lie on a. H is the foot of the perpendicular; PH is the distance to the line. A point on the line has distance zero. Perpendicular from a point, Euclid I.12.
Take another point Q on a. Why is PQ longer? Imagine folding the sheet along a: the reflected point P′ lies opposite P and H is the midpoint of PP′. Matching the two sides preserves length, so PQ=P′Q. The straight path PP′ is shorter than the broken path P–Q–P′ when Q≠H. Thus 2PH is less than 2PQ, so PH is less than PQ. This reasoning uses the shorter-straight-path principle, rather than measuring the picture. Euclid I.20.
Distance to a line need not equal distance to a drawn segment: the perpendicular foot may fall beyond its endpoints. Before measuring, identify which object the distance concerns and then choose the direction.