What Does Transversal Mean in Math

Introduction to Transversal in Math:

Definition:

A line that cuts (passes through) across two or more (usually parallel) lines then it is said to be a transversal.
It can also be defined as a line that intersects two or more co-planar lines each at a different point is also termed as a Transversal.

Transversal Postulates:
If two lines are parallel to each other and intersected by a transversal then the corresponding angles are congruent.

Some important points:

• Supplementary angles are the angles whose sum is equal to 180⁰.
• Vertical angles are the angles that are not adjacent angles and are formed by two intersecting lines and are always equal to each other.

Properties of transversal of parallel lines:

• If two parallel lines are cut by a straight line (transversal), then the corresponding angles around each intersection are equal in measure or we can say mathematically as these angles are congruent.
• If two parallel lines are cut by a straight line (transversal) then the alternate interior angles are congruent.
• If two parallel lines are cut by a straight line (transversal) then the interior angles on the same side of the transversal are supplementary.

Proofs for transversal by solved examples

Solved Problems to prove the above given points:
Ex 1: In the given figure, If the angles 2 and 3 are congruent then prove that r and s are parallel.

parallel lines 

Sol: Step 1: Given in the problem that angle 2 = angle 3
       Step 2: From the data given above angle 1 and angle 2 are vertically opposite angles and are congruent.
                               therefore, angle 1 = angle 2
       Step 3:The transitive property says that if a=b and b=c then a=c
 Using this property we have,
          angle 2 = angle 3    (given in the problem)
          angle 1 = angle 2    (vertically opposite angles)
so ,    angle 1 = angle 3   
As the angles made by the transversal with the two lines are equal the two lines are parallel.
Ex 2: Show that the transversal shown in the figure cuts the parallel lines.

Sol: In the above figure the angles 4 and 3 are known as alternate interior angles.Accoding to above discusssed points the alternate interior angles are congruent.
           `=>`    angle 4 = angle 3
So, the two lines are parallel lines.
Ex 3: Prove that the lines a and b shown in the figure below are parallel.

Sol: Clearly it is shown that the angles 1 and 2 are equal to` 90@` . That means that they are supplementary angles based upon the above discussion.
As the  angles made by the transversal with the two straigth are supplementary(equal) the two lines are parallel.Hence, proved.

Practice Problem on traversal line

Pro: Find the measures of the unknown angles in the following figure.Given r is parallel to s and angle 1 = `60@.`

Ans:  angle 2 =`120@.`
angle 3 = `60@` .
angle 4 = `120@`
angle 5 = angle 1 = 60⁰
angle 6 = angle 2 = 120⁰
angle 7 = angle 3 = 60⁰
angle 8 = angle 4 = 120⁰