Transversal

In transversal geometry, we talk about a transversal vertical that shows how a transversal intersects two lines at a time lying in the same plane, but at two different points.

A transversal vertical helps us in deciding whether two or more other lines in the Euclidean plane are parallel or not.

Also, the intersections of a transversal with two lines create different types of pairs of angles, like consecutive interior angles, consecutive exterior angles, corresponding angles, and alternate angles. 

Following Euclid's parallel postulate, a transversal line crossing the two parallel lines, we see that the consecutive interior angles are supplementary, corresponding angles are equal, and alternate angles are equal.

Transversal math discusses the transversal meaning, which we will understand on this page along with the transversal view. In transversal math, we will also go through the triangle transversal.

Transversal View

(Image will be uploaded soon)

Corresponding Angles

The pairs of angles are corresponding angles are as follows:

∠1 and ∠5

∠2 and ∠6

∠3 and ∠7

∠4 and ∠8

Alternate Interior Angles

The pairs of alternate interior angles are as follows:

∠3 and ∠6

∠4 and ∠5

Similarly, a pair of alternate Exterior Angles are as follows:

∠1 and ∠8

∠2 and ∠7

Another, we have are Co-interior Angles

The following pairs of co-interior angles are as follows:

∠3 and ∠5

∠4 and ∠6

Transversal Math

From the above text, we understand that a transversal line is a line that passes through two lines lying in the same plane at two different points in the geometry. 

Also, we got to know that transversals play a role in setting up the parallelism of two or more other straight lines in the plane, i.e., a Euclidean plane.

A transversal intersects two lines at distinct points from several angles that we can see in the image below. The types of angles formed are:

• Corresponding angles, 

• Alternate interior angles

• Alternate exterior angles, and 

• Co-interior angles.

(Image will be uploaded soon)

When this line cuts the two parallel lines:      

Angles of A Transversal Line

A transversal makes eight angles, as shown in the above figure.1.

Four  with each of the two lines are α, β, γ and δ and then α1, β1, γ1 and δ1; and

The interior angles, i.e., between the two lines are α, β, γ1 and δ1 and four of these are exterior, namely α1, β1, γ and t.

When a transversal line cuts two parallel lines at right angles, we call it a perpendicular transversal. From Fig.1, we see that all 8 angles are right-angled. 

In the case of the following figure, when the lines are parallel, we often see that a transversal produces several congruent and supplementary angles. Some of these angle pairs have particular names discussed are corresponding angles, alternate angles, and consecutive angles.

(Image will be uploaded soon)

Alternate Angles Formed By A Transversal Line

Alternate angles are the four pairs of angles that have the following:

• Distinct vertex points,

• Lie on opposite sides of the transversal, as we can see in the above diagram, and 

• Either both the angles are interior or both of them are exterior.

However, when the two angles of one pair are congruent, i.e., equal in measure, the angles of each of the other pairs also become congruent.

Proposition 1.27 under Euclid's Elements explains a theorem of absolute geometry that remains valid in both hyperbolic and Euclidean Geometry), this theorem proves that if the angles of a pair of alternate angles of a transversal are congruent, then the two lines are parallel (non-intersecting) or meet each other at infinity.

We see that a transversal line follows Euclid's parallel postulate, meaning that if the two lines are parallel, the angles of a pair of alternate angles of a transversal are also congruent, as per the Proposition 1.29 of Euclid's Elements.

Corresponding Angles Formed By A Transversal Line

(Image will be uploaded soon)

Corresponding angles are the four pairs of angles that have the following attributes:

• Distinct vertex points,

• Lie on the same side of the transversal, as we can see in the above figure, and

• One angle is interior and another is exterior (Fig. 2).

The two lines are parallel if and just if the two angles of any pair of progressive interior angles of any cross-over are linear (total to 180°). 

Recommendation 1.28 of Euclid's Elements: a hypothesis of supreme calculation (consequently legitimate in both exaggerated and Euclidean Geometry), demonstrates that if the points of a couple of relating points of a cross-over are consistent then the two lines are equal (non-meeting). 

It follows from Euclid's equal hypothesis that assuming the two lines are equal, the points of a couple of relating points of a cross-over are harmonious (Proposition 1.29 of Euclid's Elements). 

Assuming the points of one set of corresponding angles are congruent, the points of every one of different sets are likewise consistent. In the different images with parallel lines on this page, comparing point sets are: α = α1, β = β1, γ = γ1 and δ = δ1.

Consecutive Interior Angles

Consecutive interior angles are the two sets of angles that carries the following attributes:

• have particular vertex points, 

• lie on a similar plane as the transversal and 

• are both inside. 

The two lines are parallel if and just if the two angles of any pair of successive interior angles of any transversal are supplementary (aggregate to 180°). 

Proposition (Suggestion) 1.28 of Euclid's Elements, a hypothesis of total calculation (subsequently legitimate in both hyperbolic and Euclidean Geometry), demonstrates that assuming the pairs of consecutive angles are supplementary, the two lines are parallel (non-meeting). 

It follows from Euclid's equal hypothesis that assuming the two lines are parallel, the points of a couple of consecutive interior angles of a transversal are supplementary (Proposition 1.29 of Euclid's Elements). 

In the event that one pair of consecutive interior angles is supplementary, the other pair is also supplementary.

Triangle Proportionality Theorem

The Triangle Proportionality Theorem discusses the transversal triangle. 

This theorem states that a transversal lying parallel to one of the sides in a triangle divides the other two sides proportionally as we study in a BPT (Basic Proportionality Theorem).

(Image will be uploaded soon)

Proof: We all need to know the parameters to prove the theorem and that is the area of a triangle, given by:

Area of the above triangle = ½ * base * height (half of the product of base and height)

Using a BPT, we get a/b = c/d because Δ ABC ≃ Δ DEC

Now, let us understand the corollary of the Triangle Proportionality Theorem:

If a transversal line lies parallel to one side in a triangle, we get a new smaller triangle that looks similar to that of the original one.

We get the following relation:

(a  +  b)/a   is equal to   (c  +  d)/c

(Image will be uploaded soon)

From the above diagram, we see that:

(a  +  b)/a   =   (c  +  d)/c   =  AB/DE

In conclusion, a transversal is a straight line that intersects two parallel or a non-parallel line at distinct points.

Intersection caused by transversals composes several angles. These are :

a) Corresponding Angles

b) Alternate Interior Angles

c) Alternate Exterior Angles

d) Co-interior Angles