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 or more other lines in the Euclidean plane are parallel. The intersections of a transversal with two lines create various types of pairs of angles: vertical angles, consecutive interior angles, consecutive exterior angles, corresponding angles, alternate interior angles, alternate exterior angles, and linear pairs. As a consequence of Euclid's parallel postulate, if the two lines are parallel, consecutive angles and linear pairs are supplementary, while corresponding angles, alternate angles, and vertical angles are equal. Usually, transversals are parallel.
Angles of a transversal A transversal produces 8 angles, as shown in the graph at the above left:
4 with each of the two lines, namely α, β, γ and δ and then α1, β1, γ1 and δ1; and 4 of which are interior (between the two lines), namely α, β, γ1 and δ1 and 4 of which are exterior, namely α1, β1, γ and δ. 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 supplementary angles. Some of these angle pairs have specific names and are discussed below: corresponding angles, alternate angles, and consecutive angles.
Alternate angles
Alternate angles are the four pairs of angles that:
have distinct vertex points, lie on opposite sides of the transversal and both angles are interior or both angles are exterior. 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 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). It follows from Euclid's parallel postulate that if the two lines are parallel, then the angles of a pair of alternate angles of a transversal are congruent (Proposition 1.29 of Euclid's Elements).
Corresponding angles
Corresponding angles are the four pairs of angles that:
have distinct vertex points, lie on the same side of the transversal and one angle is interior and the other is exterior. 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). 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 corresponding angles of a transversal are congruent then the two lines are parallel (non-intersecting). It follows from Euclid's parallel postulate that if the two lines are parallel, then the angles of a pair of corresponding angles of a transversal are congruent (Proposition 1.29 of Euclid's Elements). If the angles of one sateur of corresponding angles are congruent, then the angles of each of the other pairs are also congruent. In the various images with parallel lines on this page, corresponding angle pairs are: α=α1, β=β1, γ=γ1 and δ=δ1.
Consecutive interior angles
Consecutive interior angles are the two pairs of angles that:
have distinct vertex points, lie on the same side of the transversal and are both interior. Two lines are parallel if and only if the two angles of any pair of consecutive interior angles of any transversal are supplementary (sum to 180°). 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). It follows from Euclid's parallel postulate that if the two lines are parallel, then the angles of a pair of consecutive interior angles of a transversal are supplementary (Proposition 1.29 of Euclid's Elements). If one pair of consecutive interior angles is supplementary, the other pair is also supplementary.
Other characteristics of transversals 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.
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