Transversal (geometry)

In geometry, a transversal is a line that passes through two lines in the same plane at two distinct points. Transversals play a role in establishing whether two other lines in the Euclidean plane are parallel. The intersections of a transversal with two lines create various types of pairs of angles: consecutive interior angles, corresponding angles, and alternate angles. By Euclid's parallel postulate, if the two lines are parallel, consecutive interior angles are supplementary, corresponding angles are equal, and alternate angles are equal.

are always congruent.)

A transversal produces 8 angles, as shown in the graph at the above left:

A transversal that cuts two parallel lines at right angles is called a perpendicular transversal. In this case, all 8 angles are right angles

When the lines are parallel, a case that is often considered, a transversal produces several congruent and several supplementary angles. Some of these angle pairs have specific names and are discussed below:corresponding angles, alternate angles, and consecutive angles.

Corresponding angles are the four pairs of angles that:

Two lines are parallel if and only if the two angles of any pair of corresponding angles of any transversal are congruent (equal in measure).

Note: This follows directly from Euclid's parallel postulate. Further, if the angles of one pair are congruent, then the angles of each of the other pairs are also congruent. In our images with parallel lines, corresponding angle pairs are: α=α1, β=β1, γ=γ1 and δ=δ1.

Alternate angles are the four pairs of angles that:

If the two angles of one pair are congruent (equal in measure), then the angles of each of the other pairs are also congruent.

Proposition 1.27 of Euclid's elements, a theorem of absolute geometry (hence valid in both hyperbolic geometry 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) .

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