Geometry

Given a line and a point not on the line, there is exactly one line through the point parallel to the line.

Two coplanar lines l₁ and l₂, cut by a transversal t are parallel if and only if

1. Alternate interior angles (e.g., d and e) are equal.
2. Corresponding angles (e.g., b and f) are equal.
3. Interior angles on the same side of the transversal are supplementary (see Figure 1).

These principles are used in a variety of ways. A draftsman uses 2) to rule a set of parallel lines. Number 1) is used to show that the sum of the angles of a triangle is equal to a straight angle.

If a set of parallel lines cuts off equal segments on one transversal, it cuts off equal segments on any other Figure 2. Illustration by Hans & Cassidy. Courtesy of Gale Group.
transversal (see Figure 2). A draftsman finds this useful when he or she needs to subdivide a segment into parts which are not readily measured, such as thirds. If transversal AC in Figure 2 is slanted so that AC is three units, then the parallel lines through the unit points will divide AB into thirds as well.

If a set of parallel planes is cut by a plane, the lines of intersection are parallel. This property and its converse are used when one builds a bookcase. The set of shelves are, one hopes, parallel, and they are supported by parallel grooves routed into the sides.