Suppose we are given,

two lines & a transversal

70.jpg

We know that

For parallel lines

  • Corresponding angles are equal
  • Alternate interior angles are equal
  • Interior angles on same side of transversal is supplementary
  • Alternate exterior angles are equal

Proving Lines Parallel (using Angles made by transversal property) - Part 2

But the opposite is true as well

  • If corresponding angles are equal,
    Line are parallel

  • If alternate interior angles are equal,
    lines are parallel

  • If sum of interior angles on same side of transversal is 180°,
    lines are parallel.

  • If alternate exterior angles are equal,
    lines are equal.

Let’s do some questions

 

Is l m ?

Proving Lines Parallel (using Angles made by transversal property) - Part 3

-a-

Here,

Proving Lines Parallel (using Angles made by transversal property) - Part 4

Here,

  ∠1 = ∠2 = 50°  

 

For lines l & m,

With transversal p

∠1 & ∠2 are alternate interior angles.

 

And they are equal.

So, lines l & m are parallel

-ea-

 

Is l m ?

Proving Lines Parallel (using Angles made by transversal property) - Part 5

-a-

Here,

Proving Lines Parallel (using Angles made by transversal property) - Part 6

Here,

  ∠1 = ∠2 = 120° 

 

For lines l & m,

With transversal p

∠1 & ∠2 are alternate interior angles.

 

And they are equal.

So, lines l & m are parallel

-ea-

 

Is l ∥ m ?

Proving Lines Parallel (using Angles made by transversal property) - Part 7

-a-

Here,

Proving Lines Parallel (using Angles made by transversal property) - Part 8

Here,

  ∠1 = ∠2 = 45°  

 

For lines l & m,

With transversal p

∠1 & ∠2 are corresponding angles.

 

And they are equal.

So, lines l & m are parallel

-ea-

 

Is l m ?

Proving Lines Parallel (using Angles made by transversal property) - Part 9

-a-

Proving Lines Parallel (using Angles made by transversal property) - Part 10

Here,

  ∠1 = ∠2 = 100°  

 

For lines l & m,

With transversal p

∠1 & ∠2 are corresponding angles.

 

And they are equal.

So, lines l & m are parallel

-ea-

 

Is l m ?

Proving Lines Parallel (using Angles made by transversal property) - Part 11

-a-

Here,

Proving Lines Parallel (using Angles made by transversal property) - Part 12

Here,

  ∠1 = ∠2 = 105°  

 

For lines l & m,

With transversal p

∠1 & ∠2 are corresponding angles.

 

And they are equal.

So, lines l & m are parallel

-ea-

 

Is l m ?

Proving Lines Parallel (using Angles made by transversal property) - Part 13

-a-

Here

Proving Lines Parallel (using Angles made by transversal property) - Part 14

Here,

  ∠1 = ∠2 = 60°  

 

For lines l & m,

With transversal p

∠1 & ∠2 are corresponding angles.

 

And they are equal.

So, lines l & m are parallel

-ea-

 

Is l m ?

Proving Lines Parallel (using Angles made by transversal property) - Part 15

-a-

Here

Proving Lines Parallel (using Angles made by transversal property) - Part 16

Here,

  ∠1 + ∠2 = 45° + 135°

= 180°    

 

For lines l & m,

With transversal p

∠1 & ∠2 are interior angles on the same side of transversal

And they are supplementary

So, lines l & m are parallel

-ea-

 

Is l ∥ m ?

Proving Lines Parallel (using Angles made by transversal property) - Part 17

-a-

Here,

Proving Lines Parallel (using Angles made by transversal property) - Part 18

Here,

  ∠1 + ∠2 = 110° + 70°

= 180°    

For lines l & m,

With transversal p

∠1 & ∠2 are interior angles on the same side of transversal

 

And their sum is 180°.

So, they are supplementary

So, lines l & m are parallel

-ea-

 

Is l m ?

Proving Lines Parallel (using Angles made by transversal property) - Part 19

-a-

Here,

Proving Lines Parallel (using Angles made by transversal property) - Part 20

Here,

  ∠3 = ∠2                    (Vertically opposite angles)

  ∠3 = 135°    

 

Now, ∠1 = ∠3 = 135°

 

For lines l & m,

With transversal p

∠1 & ∠3 are corresponding angles.

 

And they are equal.

So, lines l & m are parallel

-ea-

 

Is l m ?

Proving Lines Parallel (using Angles made by transversal property) - Part 21

-a-

Here,

Proving Lines Parallel (using Angles made by transversal property) - Part 22

Here,

  ∠3 = ∠2                     (Vertically opposite angles)

         ∠3 = 115°   

 

Now, ∠1 = ∠3 = 115°

 

For lines l & m,

With transversal p

∠1 & ∠3 are corresponding angles.

 

And they are equal.

So, lines l & m are parallel

-ea-

 

Is l m ?

Proving Lines Parallel (using Angles made by transversal property) - Part 23

-a-

Here,

Proving Lines Parallel (using Angles made by transversal property) - Part 24

Here,

  ∠1 ≠ ∠2

 

For lines l & m,

With transversal p

∠1 & ∠2 are alternate interior angles.

 

And they are not equal.

So, lines l & m are not parallel

-ea-

 

Is l m ?

Proving Lines Parallel (using Angles made by transversal property) - Part 25

-a-

Here,

Proving Lines Parallel (using Angles made by transversal property) - Part 26

Here,

  ∠1 ≠ ∠2

 

For lines l & m,

With transversal p

∠1 & ∠2 are corresponding angles.

 

But they are not equal.

So, lines l & m are not parallel

-ea-

 

Is l m ?

Proving Lines Parallel (using Angles made by transversal property) - Part 27

-a-

Here,

Proving Lines Parallel (using Angles made by transversal property) - Part 28

Here,

  ∠3 = ∠2             (Vertically opposite angles)

  ∠3 = 80°

≠ 100° 

 

∴ ∠1 ≠ ∠3

For lines l & m,

With transversal p

∠1 & ∠3 are corresponding angles.

But they are not equal.

So, lines l & m are not parallel

-ea-

  1. Chapter 5 Class 7 Lines and Angles
  2. Concept wise

About the Author

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.