Transversal Angles

Introduction to Transversal Angles?

Parallel lines are lines that always stay the same distance apart and never meet.  A transversal is a line that crosses two or more other lines. When a transversal crosses two parallel lines, like in the diagram below, it creates 8 angles. Corresponding angles are found on matching corners. They are on the same side of the transversal. One is in between two parallel lines, and other is on the outside.

Corresponding angles

Two lines, line segments, or rays (or any combination of those) are parallel if they never meet and are always the same distance apart. Both lines have to be in the same plane (be coplanar).

You encounter parallel lines in geometry, of course, but also in everyday life. The lines of notebook paper are parallel. Sides of doors, edges of cereal boxes, and the floorboards of home are parallel.

Alternate Interior Angles

When two parallel lines are intersected by the transversal, eight angles are formed. The eight angles include corresponding angles, alternate interior and exterior angles, vertically opposite angles, and co-interior angles. When two parallel lines crossed by a transversal, they formed same-side interior angles and their sum is equal to 180 degrees.

Alternate Exterior Angles

Transversals are straight lines that intersect two or more lines at different points. A transversal line meets the other line at one point which forms four angles around the point of intersection. A transversal line, in geometry, passes through two lines in the same plane at two distinct points. Transversals play a role in establishing the parallelism of two or more other straight lines in the Euclidean plane. It intersects two lines at distinct points. Intersection caused by transversal forms several angles. These are corresponding angles, alternate interior angles, alternate exterior angles, and co-interior angles.

Co-Interior Angles

Consecutive interior angles are formed on the inner sides of the transversal and are also known as co-interior angles or same-side interior angles. When a transversal crosses any two parallel lines, it forms many angles like alternate interior angles, corresponding angles, alternate exterior angles, consecutive interior angles.

Pairs of Angles

Consecutive interior angles are defined as the pair of non-adjacent interior angles that lie on the same side of the transversal. The word 'consecutive' refers to things that appear next to each other. Consecutive interior angles are located next to each other on the internal side of a transversal. Observe the following figure and the properties of consecutive interior angles to identify them.

Transversal and Transversal Lines

When two lines are cut by a transversal, the pair of angles on one side of the transversal and inside the two lines are called the consecutive interior angles. If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles formed are supplementary. The consecutive interior angles theorem states that when the two lines are parallel, then consecutive interior angles are supplementary to each other. Supplementary means that the two angles add up to 180 degrees. As a consequence of Euclid's parallel postulate, if the two lines are parallel, consecutive interior angles are supplementary, corresponding angles are equal, and alternate angles are equal.

Constructing a Transversal on Parallel Lines

There is only one situation in which two consecutive angles are congruent. In this situation, the two lines must be parallel and the transversal must be perpendicular to those two lines. Recall, perpendicular means the two lines create a 90 angle.

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Most Important Frequently Asked Questions Searched By Students

Q: What are the angles formed by a transversal? 

Answer: 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).

Q: Is transversal equal to 180?

Answer: Corresponding angles, alternate angles, and consecutive angles.

If the two angles of one pair are congruent (equal in measure), then the angles of each of the other pairs are also congruent.

Q: What does transversal mean in angles?

Answer: 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: consecutive interior angles, consecutive exterior angles, corresponding angles, and alternate angles. As a consequence of Euclid's parallel postulate, if the two lines are parallel, consecutive interior angles are supplementary, corresponding angles are equal, and alternate angles are equal.

Q: How do you find transversal angles?

Answer: 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 the contradiction.

Q: What angles are supplementary in a transversal?

Answer: If we draw to parallel lines and then draw a line transversal through them, we will get eight different angles. The eight angles will together form four pairs of corresponding angles. The eight angles include corresponding angles, alternate interior and exterior angles, vertically opposite angles, and co-interior angles.

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 and several supplementary angles. Some of these angle pairs have specific names and are discussed below.

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).

A transversal line, in geometry, passes through two lines in the same plane at two distinct points. Transversals play a role in establishing the parallelism of two or more other straight lines in the Euclidean plane. It intersects two lines at distinct points. Intersection caused by transversal forms several angles. These are corresponding angles, alternate interior angles, alternate exterior angles, and co-interior angles.