Transversal (geometry)

Transversals play a role in establishing whether two or more other lines in the Euclidean plane are parallel.

Proposition 1.27 of Euclid's Elements, a theorem of absolute geometry (hence valid in both hyperbolic and Euclidean Geometry), proves that if the angles of a pair of alternate angles of a transversal are congruent then the two lines are parallel (non-intersecting).

Proposition 1.28 of Euclid's Elements, a theorem of absolute geometry (hence valid in both hyperbolic and Euclidean Geometry), proves that if the angles of a pair of consecutive interior angles are supplementary then the two lines are parallel (non-intersecting).

If three lines in general position form a triangle are then cut by a transversal, the lengths of the six resulting segments satisfy Menelaus' theorem.

Euclid proves this by contradiction: If the lines are not parallel then they must intersect and a triangle is formed.

These follow from the previous proposition by applying the fact that opposite angles of intersecting lines are equal (Prop.

First, if a transversal intersects two parallel lines, then the alternate interior angles are congruent.

93-95 Euclid's proof makes essential use of the fifth postulate, however, modern treatments of geometry use Playfair's axiom instead.

To prove proposition 29 assuming Playfair's axiom, let a transversal cross two parallel lines and suppose that the alternate interior angles are not equal.

Unlike the two-dimensional (plane) case, transversals are not guaranteed to exist for sets of more than two lines.