geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

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

plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...

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

parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of ...

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

angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...

s: consecutive interior angles, consecutive exterior angles, corresponding angles, and alternate angles. As a consequence of Euclid's

parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ''If a line segmen ...

, if the two lines are parallel, consecutive interior angles are supplementary, corresponding angles are equal, and alternate 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 angles is called a perpendicular transversal. In this case, all 8 angles are right angles When the lines are

, 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 mod ...

several

supplementary angles In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...

. 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 Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet * Vertex (computer graphics), a data structure that describes the positio ...

points, *lie on opposite sides of the transversal and *both angles are interior or both angles are exterior. It is a very useful topic of mathematics 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, but since these are not suf ...

(hence valid in both

hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...

and

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

), 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 Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle ''ABC'', and a transversal line that crosses ''BC'', ''AC'', and ''AB'' at points ''D'', ''E'', and ''F'' respec ...

Related theorems

Euclid Euclid (; grc-gre, Εὐκλείδης; 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 ...

'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, 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 a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the ...

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 and one of the brightest stars in the night sky. It has the Bayer designation designated α Leonis, which is Latinized to Alpha Leonis, and abbreviated Alpha Leo or α Leo. Reg ...

is a set of

skew lines In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron. Two lines that both lie in the sa ...

, , 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