Transversal
A transversal is a line that intersects two other lines at two distinct points, resulting in the creation of eight angles. This geometrical concept is particularly significant when the intersected lines are parallel, as the angles formed exhibit specific relationships that can indicate whether the lines are indeed parallel. For example, when a transversal crosses two parallel lines, it produces pairs of angles that are either congruent or supplementary, with a total angle sum of 360 degrees.
The angles generated can be categorized into four types: internal angles (between the lines) and external angles (outside the lines), alongside three specific types of angle pairs: consecutive angles, corresponding angles, and alternate angles. Consecutive angles, which are both interior and located on the same side of the transversal, sum to 180 degrees. Corresponding angles, consisting of one interior and one exterior angle from the same side, are always congruent. Finally, alternate angles, found on opposite sides of the transversal, are also congruent if the lines are parallel. These properties were historically studied by the ancient Greek mathematician Euclid, contributing to our understanding of geometry.
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Subject Terms
Transversal
Within the study of plane geometry, a transversal is a line that intersects two other lines at two separate points. Because each intersection of the transversal with a line creates four angles, in total the transversal produces eight angles. When a transversal intersects two lines that are parallel to one another, the eight angles that are produced have special relationships with one another. For this reason, the angles of a transversal are often studied as a means of determining whether two lines are parallel, because if they exhibit the same properties that they would if the lines were parallel, then it can be concluded that the lines are in fact parallel. When the lines crossed by the transversal are parallel, for example, several pairs of angles are created, some of which are congruent and others of which are supplementary, meaning that the sum of their values is equal to 180 degrees. The sum of all four angles created by the intersection of a transversal with a line is thus 360 degrees. Of the eight angles created by the transversal’s intersection of the two lines, four are called internal angles because they are found between the two lines that the transversal crosses, and the other four are called external, because they lie outside the area between the lines.
Overview
If a transversal crosses two parallel lines at right angles, then each of the eight angles created measures 90 degrees; this is called a perpendicular transversal. If the transversal is not perpendicular but the lines it crosses are parallel, then three types of angles will be created. The first type of angles created are called consecutive angles. Consecutive angles are identifiable because they are both interior angles (meaning that they occur between the two parallel lines rather than outside of them), they appear on the same side of the transversal, and they have different vertex points. The most significant characteristic of consecutive angles is that their sum is 180 degrees.
The second type of angle created when a non-perpendicular transversal intersects two parallel lines is known as a corresponding angle. Corresponding angles, like consecutive angles, have different vertex points and lie on the same side of the transversal, but they differ from consecutive angles in the fact that with corresponding angles, one of the pair is interior and the other is exterior. Corresponding angles will always be congruent with each other.
The final type of angle pair created are called alternate angles. Alternate angles also have different vertex points, like consecutive and corresponding angles, but the similarity ends there. Alternate angles are found on opposite sides of the transversal (hence the use of the term "alternate") and are either both internal angles or both external angles. Alternate angles will always be congruent with one another if the lines crossed by the transversal truly are parallel to one another. These properties of the angles created by a transversal were studied and proven by the Greek mathematician Euclid, who was active at about the year 300 BC.
Bibliography
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