Important Axioms and Theorems: Transversal and Parallel lines

Axiom 1: If a transversal intersects two parallel lines, then each pair of corresponding angles is equal.

In the picture transversal AD intersects two parallel PQ and RS at points B and C respectively.

According the given axiom

Angle ABQ (angle 1) = Angle BCS (angle 2) because they are corresponding angles

Angle QBC (angle 3) = Angle SCD (angle 4) because they are corresponding angles

There are two more pairs of corresponding angles. Can you find them?

Angle ABP = Angle BCR

Angle PBC = Angle RCD

Converse of the axiom: If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel to each other.

Given that XY is a transversal and it intersecting two lines PQ and RS at M and N respectively such that 

Angle XMQ ( angle 1) = Angle MNS (angle 2)

Then according to the axiom

PQ II RS ( PQ is parallel to RS)

Theorem 1 : If a transversal intersects two parallel lines then each pair of alternate interior angles is equal.

In the above picture AB and CD are two parallel lines. Transversal PS is intersecting AB and CD at points Q and R respectively.

Then according the theorem:

Angle CRQ = Angle RQB (Alternate interior angles)

There is one more pair of alternate interior angles. Can you find?

YES! Angle AQR = Angle QRD

Converse of the theorem: Just remember that if a transversal intersect two lines such that pair of alternate interior angles is equal, then the lines are said to be parallel.

Theorem 2 : If a transversal intersects two parallel lines ,then the each pair the interior angles on the same side of the transversal is supplementary ( In other words their sum is 180 degree)

In the above picture MN and XY are two parallel lines.

Transversal PS is intersecting MN and XY at points Q and R respectively.Then according the theorem:

Angle XRQ ( angle 1)+Angle MQR(angle 2) = 180 degree

In other words Angle XRQ and Angle MQR are supplementary.

Converse of the theorem: Just remember that if a transversal intersects two lines such that a pair of interior angles on the same side are supplementary, then the lines are said to be parallel to each other.

Now let us do a question

• In the figure given below AB ‖ CD and CD ‖ EF. Also EA is perpendicular to AB . If angle BEF = 55 degree. Find the values of x, y and z.

Solution:

Y+ 55⁰ =180⁰  (  Interior angles on the same side of the transversal ED are supplementary)

Y = 180⁰- 55⁰

    = 125⁰

x = y = 125⁰ ( corresponding angles because AB ‖ CD )

Now since AB ‖ CD and CD ‖ EF

Therefore AB ‖ EF

Angle EAB + Angle FEA = 180⁰  ( Interior angles on the same side of the transversal EA)

90⁰ + z +55⁰ =180⁰

 Z= 180⁰- (90⁰ + 55⁰)

     = 180⁰-145⁰

     = 35⁰