Parallel lines transversals angle

According to geometry, lines can be divided into several categories such as parallel, perpendicular, intersecting, and non-intersecting lines, among others. If we have two lines that do not intersect, we can draw a special line called a transversal that crosses both lines at different points.

Parallel lines

A pair of lines is said to be parallel if they do not cross each other at any point. Parallel lines are also defined as two lines that run parallel to one another and meet at an infinity point.

Transversal

When a line crosses two other lines at two different points, the intersection is referred to as a transversal. In the figure below, line XY intersects lines PQ and RS at two distinct points, denoted by the letters X and Y. As a result, line l is known as the transversal line.

Angle relationship between the parallel lines and transversal

When a transversal intersects two or more parallel lines, a number of different angle pairs are formed. Let us quickly review the angle relationships for the parallel lines that are cut by a transversal in order to avoid confusion. Assume that a and d are two parallel lines that are intersected by the transversal l at the points P and Q, as shown in the illustration below.

After that, we can write different pairs of angles and the relationships that exist between them.

Corresponding angles

Corresponding angles are the angles formed in matching corners or corresponding corners with the transversal when two parallel lines and the transversal are intersected by a transversal line.

∠1 and ∠6

∠4 and ∠8

∠2 and ∠5 

∠3 and ∠7 

A transversal that intersects two parallel lines has the property that each pair of corresponding angles is the same as the other. Thus,

∠1 = ∠6

∠4 = ∠8

∠2 = ∠5  

∠3 = ∠7 

Alternate Interior angles

An alternate interior angle is formed when a transversal intersects two coplanar lines at an angle other than 90 degrees.

∠4 and ∠5  

∠3 and ∠6 

If a transversal intersects two parallel lines, on the other hand, each pair of alternate interior angles is the same as the other.

∠4 = ∠5

∠3 = ∠6

Alternate exterior angles

It is the pair of angles formed on the outer side of two lines but on the opposite side of the transversal that is known as alternate exterior angles.

∠1 and ∠7 

∠2 and ∠8

When a transversal is used to cut two parallel lines, the alternate exterior angles that result are congruent with one another. 

∠1 = ∠7  

∠2 = ∠8 

Interior angles on the same side of the transversal

Those interior angles that are located on the same side of the transversal are known as consecutive interior angles, allied interior angles, or co-interior angles, among other terms.

∠3 and ∠5 

∠4 and ∠6 

A transversal intersects two parallel lines, and each pair of interior angles on the same side of the transversal is supplementary, meaning that they add up to 180° when they are added together.

∠3 + ∠5 = 180° 

∠4 + ∠6 = 180° 

Vertically opposite angles

When two lines intersect each other, the opposing angles that are formed as a result of the intersection are referred to as vertical angles or vertically opposite angles, respectively. In addition, a pair of vertical angles are always congruent with itself. 

So, 

∠1 = ∠3 

∠2 = ∠4

∠7 = ∠6 

∠8 = ∠5

Conclusion

When a transversal is used to cut two parallel lines, the pairs of consecutive interior angles that are formed are known as supplementary interior angles. When two lines are intersected by a transversal, the pairs of angles on either side of the transversal and within the two lines are referred to as the alternate interior angles. 

Parallel lines are also defined as two lines that run parallel to one another and meet at an infinity point.

When a line crosses two other lines at two different points, the intersection is referred to as a transversal. When a transversal intersects two or more parallel lines, a number of different angle pairs are formed.

Corresponding angles are the angles formed in matching corners or corresponding corners with the transversal when two parallel lines and the transversal are intersected by a transversal line.