Parallel lines and transversals
When a line crosses two parallel lines, it creates eight angles in a predictable pattern. These relationships are among the most used facts in geometry.
Parallel and skew lines
Parallel lines are coplanar and never meet; they are marked with matching arrowheads. Skew lines are not coplanar and never meet, like a road on a bridge and a road passing underneath at a different angle. Parallel planes never intersect, like a floor and ceiling.
Transversals and angle pairs
A transversal crosses two or more lines, forming eight angles. Corresponding angles occupy the same position at each intersection. Alternate interior angles lie between the lines on opposite sides of the transversal. Alternate exterior angles lie outside the lines on opposite sides. Same-side interior angles lie between the lines on the same side.
The parallel postulate
Euclid's fifth postulate, in a common modern form, says that through a point not on a line there is exactly one line parallel to the given line. Mathematicians spent two thousand years trying to prove it from the other postulates; in the 1800s, Lobachevsky, Bolyai, and Gauss showed that consistent non-Euclidean geometries reject it.
Angle relationships
When the two lines are parallel: corresponding angles are congruent, alternate interior angles are congruent, alternate exterior angles are congruent, and same-side interior angles are supplementary. So if one angle at a parallel-line intersection measures 65 degrees, every one of the eight angles is either 65 or 115 degrees.
Proving lines parallel
The converses also hold: if corresponding angles are congruent, or alternate interior angles are congruent, or same-side interior angles are supplementary, then the lines are parallel. Builders use this to check that shelves or rails are parallel by measuring angles with a transversal.
Perpendicular transversals
If a transversal is perpendicular to one of two parallel lines, it is perpendicular to the other. Two lines perpendicular to the same line in a plane are parallel to each other, which is why the rungs of a ladder, all perpendicular to the rails, are parallel.
Algebra with parallel lines
If alternate interior angles measure 4x - 12 and 2x + 30 degrees and the lines are parallel, then 4x - 12 = 2x + 30, so x = 21 and each angle is 72 degrees. If instead same-side interior angles measure 3y and 2y, then 5y = 180, so y = 36 and the angles are 108 and 72 degrees.
Worked example
In a street grid, two parallel avenues are crossed by a diagonal street making a 50-degree angle with the first avenue. At the second avenue the corresponding angle is also 50 degrees, and the angle on the other side of the diagonal, forming a linear pair, is 130 degrees. A surveyor can find every angle at both intersections from one measurement.
Key takeaways
Parallel lines are coplanar and never meet; skew lines are non-coplanar and never meet. A transversal creates corresponding, alternate interior, alternate exterior, and same-side interior angles. With parallel lines, corresponding and alternate angles are congruent and same-side interior angles are supplementary, so all eight angles take just two values. The converses let you prove lines parallel. Euclid's parallel postulate cannot be proved from the others, which led to non-Euclidean geometry in the 1800s.
Flashcard
Coplanar, never meet
Check your understanding
Two parallel lines are cut by a transversal. One angle is 65 degrees. Which value can another of the eight angles take?