geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a transversal is a line that passes through two lines in the same plane at two distinct

points A point is a small dot or the sharp tip of something. Point or points may refer to: Mathematics * Point (geometry), an entity that has a location in space or on a plane, but has no extent; more generally, an element of some abstract topologica ...

. Transversals play a role in establishing whether two or more other lines in the

Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

are parallel. The intersections of a transversal with two lines create various types of pairs of

angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...

s: 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 In geometry, the parallel postulate is the fifth postulate in Euclid's ''Elements'' and a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: If a line segment intersects two straight lines forming two interior ...

, if the two lines are parallel, consecutive angles and linear pairs are supplementary, while corresponding angles, alternate angles, and vertical angles are equal.

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 angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...

s 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 Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...

supplementary angles In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight lines at a point. Formally, an angle is a figure lying in a plane formed by two rays, called the '' sides'' of the angle, sharing ...

. 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 Absolute geometry is a geometry based on an axiom system for Euclidean geometry without the parallel postulate or any of its alternatives. Traditionally, this has meant using only the first four of Euclid's postulates. The term was introduced by ...

(hence valid in both

hyperbolic Hyperbolic may refer to: * of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics ** Hyperbolic geometry, a non-Euclidean geometry ** Hyperbolic functions, analogues of ordinary trigonometric functions, defined u ...

and

Euclidean Geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

), 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

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

(hence valid in both

and

), 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

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 pair 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: (interactive) *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

(hence valid in both

and

), 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

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 In Euclidean geometry, Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle , and a Transversal (geometry), transversal line that crosses at points respectively, wi ...

Related theorems

Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...

's formulation of the

may be stated in terms of a transversal. Specifically, if the interior angles on the same side of the transversal are less than two right angles then lines must intersect. In fact, Euclid uses the same phrase in Greek that is usually translated as "transversal". Euclid's Proposition 27 states that if a transversal intersects two lines so that alternate interior angles are congruent, then the lines are parallel. Euclid proves this by contradiction: If the lines are not parallel then they must intersect and a triangle is formed. Then one of the alternate angles is an exterior angle equal to the other angle which is an opposite interior angle in the triangle. This contradicts Proposition 16 which states that an exterior angle of a triangle is always greater than the opposite interior angles. Euclid's Proposition 28 extends this result in two ways. First, if a transversal intersects two lines so that corresponding angles are congruent, then the lines are parallel. Second, if a transversal intersects two lines so that interior angles on the same side of the transversal are supplementary, then the lines are parallel. These follow from the previous proposition by applying the fact that opposite angles of intersecting lines are equal (Prop. 15) and that adjacent angles on a line are supplementary (Prop. 13). As noted by

Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor (, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers of late antiquity. He set forth one of th ...

, Euclid gives only three of a possible six such criteria for parallel lines. Euclid's Proposition 29 is a converse to the previous two. First, if a transversal intersects two parallel lines, then the alternate interior angles are congruent. If not, then one is greater than the other, which implies its supplement is less than the supplement of the other angle. This implies that there are interior angles on the same side of the transversal which are less than two right angles, contradicting the fifth postulate. The proposition continues by stating that on a transversal of two parallel lines, corresponding angles are congruent and the interior angles on the same side are equal to two right angles. These statements follow in the same way that Prop. 28 follows from Prop. 27. Euclid's proof makes essential use of the fifth postulate, however, modern treatments of geometry use

Playfair's axiom In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate In geometry, the parallel postulate is the fifth postulate in Euclid's ''Elements'' and a distinctive axiom in Euclidea ...

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. Draw a third line through the point where the transversal crosses the first line, but with an angle equal to the angle the transversal makes with the second line. This produces two different lines through a point, both parallel to another line, contradicting the axiom.A similar proof is given in Holgate 1901, Art. 93

In higher dimensions

In higher dimensional spaces, a line that intersects each of a set of lines in distinct points is a ''transversal'' of that set of lines. Unlike the two-dimensional (plane) case, transversals are not guaranteed to exist for sets of more than two lines. In Euclidean 3-space, a

regulus Regulus is the brightest object in the constellation Leo (constellation), Leo and one of the List of brightest stars, brightest stars in the night sky. It has the Bayer designation designated α Leonis, which is Latinisation of names, ...

is a set of

skew lines In three-dimensional geometry, skew lines are two Line (geometry), lines that do not Line-line intersection, intersect and are not Parallel (geometry), parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges ...

, , such that through each point on each line of , there passes a transversal of and through each point of a transversal of there passes a line of . The set of transversals of a regulus is also a regulus, called the ''opposite regulus'', . In this space, three mutually skew lines can always be extended to a regulus.

References

{{reflist, refs= {{cite book , title=Elementary Geometry, url=https://archive.org/details/elementarygeome00holggoog, first=Thomas Franklin, last=Holgate, publisher=Macmillan, year=1901 {{cite book , title=The thirteen books of Euclid's Elements, first=T.L., last=Heath, publisher=The University Press, year=1908, volume=1 Elementary geometry