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Transversal (geometry)
Transversal (geometry)
Transversal (geometry)
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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.

   
Eight angles of a transversal.
(Vertical angles such as and

are always congruent.)

  Transversal between non-parallel lines.
Consecutive angles are not supplementary. 		Transversal between parallel lines.
Consecutive angles are supplementary.

Angles of a transversal

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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 [1]

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.[2][3]: Art. 87 

Alternate angles

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One pair of alternate angles. With parallel lines, they are congruent.

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

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For an alternate use see Corresponding angles (congruence and similarity)
One pair of corresponding angles. With parallel lines, they are congruent.

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 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

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One pair of consecutive angles. With parallel lines, they add up to two right angles

Consecutive interior angles are the two pairs of angles that:[4][2]

  • 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

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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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Euclid's formulation of the parallel postulate 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".[5]: 308, nfote 1 

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.[5]: 307 [3]: Art. 88 

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.[5]: 309–310 [3]: Art. 89-90 

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.[5]: 311–312 [3]: Art. 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. 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.[5]: 313 [6]

In higher dimensions

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

References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Transversal (geometry)

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In , a transversal is a line that intersects two or more other lines at distinct points, typically within the same plane. This creates various angles that are fundamental to understanding line relationships, particularly when the intersected lines are parallel. When a transversal crosses two , it forms eight angles at the points of intersection, which can be classified into specific pairs based on their positions relative to the transversal and the parallel lines. Corresponding angles, located on the same side of the transversal and in matching positions relative to the parallel lines, are congruent. Alternate interior angles, situated between the parallel lines on opposite sides of the transversal, are also congruent, as are alternate exterior angles, which lie outside the parallel lines on opposite sides. Consecutive interior angles, on the same side of the transversal between the parallel lines, are supplementary, summing to 180 degrees. These angle relationships form the basis of key theorems used to prove the parallelism of lines or to solve for unknown angles in geometric figures. Transversals extend beyond parallel lines to applications in non-parallel configurations, such as determining if lines are parallel by checking angle equality or in coordinate geometry for calculating slopes and intercepts. They are essential in for constructing proofs involving line properties and are also relevant in advanced topics like , where transversals help define harmonic divisions.

Definition and Configuration

Definition of a Transversal

In , a transversal is a line that intersects two or more other lines at distinct points, typically in the plane and most commonly applied to exactly two lines. This intersection occurs without the transversal coinciding with any of the lines it crosses, ensuring the points of intersection are separate. Formally, given two distinct lines l1l_1 and l2l_2 in a plane, a transversal tt is defined as a line that intersects l1l_1 at a point PP and l2l_2 at a point QQ, where P≠QP \neq Q. This setup establishes the basic configuration for analyzing line interactions, though transversals are often considered in relation to in subsequent geometric studies. The term "transversal" originates from the Latin "transversalis," meaning "crossing from side to side," reflecting its role in spanning multiple lines. A visual representation of a transversal depicts two straight lines in a plane, each crossed by a third line at different locations along their lengths, highlighting the points of without regard to the relative orientation of the original lines.

Basic Geometric Setup

In plane geometry, the basic setup involves two lines, denoted as l1l_1 and l2l_2, lying in the same plane and intersected by a third line called the transversal tt at distinct points. This configuration creates four angles at each intersection point—for a total of eight angles—arising from the transversal crossing each of the two lines. When l1l_1 and l2l_2 are non-parallel, they intersect at a single point elsewhere in the plane, and the transversal tt crosses both lines, producing eight angles with measures that generally vary based on the specific orientations and positions of the lines. In the parallel case, l1l_1 and l2l_2 do not intersect and maintain a constant apart, with the transversal tt intersecting both to form eight angles. These angles are conventionally labeled as ∠1\angle 1 through ∠8\angle 8, starting from one side of the transversal at the intersection with l1l_1 (e.g., ∠1\angle 1 and ∠2\angle 2 as adjacent angles), proceeding around that intersection, then to the intersection with l2l_2 (e.g., ∠5\angle 5 and ∠6\angle 6 adjacent), ensuring sequential numbering that highlights positional relationships.

Angle Relationships

Types of Angles Formed

When a transversal intersects two lines, it creates eight distinct angles at the two points of . These angles are typically labeled using a standard numbering system for clarity: at the upper , angles 1 and 2 lie above the upper line (adjacent along the transversal), angles 3 and 4 lie below it; at the lower , angles 5 and 6 lie above the lower line (between the two lines), and angles 7 and 8 lie below it. This configuration allows for systematic classification based on position relative to the two lines and the transversal. The angles are first categorized as interior or exterior. Interior angles are those positioned between the two intersected lines, specifically angles 3, 4, 5, and 6 in the standard diagram. Exterior angles, conversely, are those located outside the two lines, namely angles 1, 2, 7, and 8. Regarding their measures, each of these angles may be acute (less than 90°), right (exactly 90°), or obtuse (greater than 90°), depending on the angle at which the transversal crosses the lines; however, the primary focus in transversal geometry is their relative positions rather than individual measures. Among these, vertical angles form opposite pairs at each intersection point, such as ∠1 and ∠3 at the upper intersection, and ∠5 and ∠7 at the lower. Vertical angles are always equal in measure, regardless of whether the lines are parallel. This equality follows from the fact that adjacent angles at each intersection form a straight line, summing to 180° (a straight ); thus, each vertical angle equals 180° minus its adjacent angle, yielding congruence for the opposite pair. Adjacent angles, by contrast, share a common side along the transversal or one of the intersected lines and a common vertex at the . Examples include ∠1 and ∠2 (above the upper line) or ∠3 and ∠4 (below it), which together form a linear pair summing to 180°. This supplementary relationship holds because they compose a straight line at the .

Alternate Angles

Alternate interior angles are pairs of angles formed when a transversal intersects two lines, lying on opposite sides of the transversal and both within the region between the two lines. For standard labeling where the transversal crosses the first line forming angles 1 through 4 (with 3 and 4 interior) and the second line forming angles 5 through 8 (with 5 and 6 interior), one such pair is ∠4 and ∠5, while the other is ∠3 and ∠6. These angles are characterized by their positions interior to the two lines but alternating relative to the transversal's direction. Alternate exterior angles, similarly, are pairs located on opposite sides of the transversal but outside the region between the two lines. Using the same labeling, examples include ∠1 and ∠8, as well as ∠2 and ∠7, where ∠1 and ∠2 are exterior to the first line, and ∠7 and ∠8 to the second. The alternate interior angles theorem states that if two are cut by a transversal, then each pair of alternate interior angles is congruent. This result, originally established in Euclid's Elements as Proposition I.29, follows from the parallel postulate and properties of straight lines. Likewise, the alternate exterior angles theorem asserts that under the same conditions, each pair of alternate exterior angles is congruent, also derivable from the same Euclidean proposition. A simple proof of the alternate interior angles theorem uses the consecutive interior angles theorem, which states that if two parallel lines are cut by a transversal, then consecutive interior angles are supplementary (summing to 180°). Consider parallel lines l1 and l2 cut by transversal t, with ∠4 and ∠6 as consecutive interior angles on the same side of t, so ∠4 + ∠6 = 180°. At the intersection with l2, ∠5 and ∠6 form a linear pair along the straight line l2, so ∠5 + ∠6 = 180°. Subtracting these equations yields ∠4 = ∠5, establishing congruence for this alternate interior pair; the other pair follows analogously. For alternate exterior angles, a parallel argument applies: consecutive exterior angles ∠1 and ∠7 are supplementary (∠1 + ∠7 = 180°), and ∠7 + ∠8 = 180° along l2, so ∠1 = ∠8. For illustration, suppose two are intersected by a transversal at a 30° , forming ∠4 = 30° as an interior adjacent to the transversal. Then, the alternate interior ∠5 must also measure 30°, confirming the theorem's application in determining parallelism or measures in diagrams.

Corresponding Angles

Corresponding angles are pairs of angles formed by a transversal intersecting two lines, positioned in identical relative locations with respect to both lines and the transversal—for instance, both angles above their respective lines and on the same side of the transversal, such as ∠1 (formed above the first line) and ∠5 (formed above the second line)./01%3A_Lines_Angles_and_Triangles/1.04%3A_Parallel_Lines) These angles occupy matching orientations, either both interior (between the two lines) or both exterior (outside the two lines), and on the same side of the transversal. A transversal crossing two lines creates four such pairs of corresponding angles: two interior pairs (one on each side of the transversal between the lines) and two exterior pairs (one on each side outside the lines). In standard diagrams, these are typically labeled as ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, and ∠4 and ∠8, with corresponding pairs sharing the same relative position, such as the upper-left angles at each . This configuration highlights their , as visualized in diagrams where shaded or numbered angles emphasize the matching pairs across the lines. The corresponding angles postulate asserts that if a transversal intersects two parallel lines, then each pair of corresponding angles is congruent. Conversely, if a transversal intersects two lines such that a pair of corresponding angles is congruent, then the two lines are parallel. This bidirectional relationship serves as a key criterion for determining parallelism without direct measurement. The notion of corresponding angles plays a foundational role in Euclid's , the fifth postulate of his Elements, which underpins by guaranteeing that through a point not on a line, exactly one parallel line exists and implies the equality of these angles in parallel configurations.

Consecutive Interior Angles

Consecutive interior angles, also known as same-side interior angles or co-interior angles, are a pair of angles formed when a transversal intersects two lines, where both angles lie between the two lines and on the same side of the transversal. For standard labeling in a with two horizontal lines intersected by a slanting transversal, these angles are typically ∠3 and ∠6 (or ∠4 and ∠5 on the opposite side). The consecutive interior angles theorem states that if a transversal intersects two , then each pair of consecutive interior angles is supplementary, meaning their measures sum to 180°. This theorem can be proved using the corresponding angles postulate and the linear pair postulate. Consider two l1l_1 and l2l_2 intersected by transversal tt, with consecutive interior angles ∠c (adjacent to corresponding ∠a on l1l_1) and ∠e (corresponding to ∠a on l2l_2). By the corresponding angles postulate, m∠a=m∠em\angle a = m\angle e. By the linear pair postulate, m∠c+m∠a=180∘m\angle c + m\angle a = 180^\circ. Substituting gives m∠c+m∠e=180∘m\angle c + m\angle e = 180^\circ, so ∠c and ∠e are supplementary. The converse of the provides a test for parallelism: if a transversal intersects two lines such that a pair of consecutive interior angles is supplementary, then the lines are parallel. Conversely, if the sum of a pair of consecutive interior angles is not 180°, the lines are not parallel. For example, suppose two are intersected by a transversal that forms a 45° angle with one of the ; the adjacent consecutive interior on the other will then measure 135°, since 45∘+135∘=180∘45^\circ + 135^\circ = 180^\circ.

Properties and Theorems

Characteristics of Transversals with Parallel Lines

When a transversal intersects two , it forms equal angles with each of the , a property arising from the equality of alternate interior angles. This uniform inclination ensures that the direction of the transversal relative to the parallels remains consistent, independent of the specific points. The between two remains constant regardless of the transversal used to approach them, as this distance is an intrinsic property of the lines in . Consequently, the segment of the transversal lying between the intersection points with the parallel lines has a length determined by this fixed and the angle of inclination; for instance, if the parallel lines are horizontal, the length equals the perpendicular distance divided by the sine of the angle the transversal makes with the parallels. In coordinate , parallel lines share the same , and the of the transversal dictates the angles it forms with both lines identically, given by the tan⁡θ=∣mp−mt1+mpmt∣\tan \theta = \left| \frac{m_p - m_t}{1 + m_p m_t} \right|
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