Proving Lines Parallel (using Angles made by transversal property)

Suppose we are given,

two lines & a transversal

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

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 ?

-a-

Here,

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 ?

-a-

Here,

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 ?

-a-

Here,

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 ?

-a-

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 ?

-a-

Here,

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 ?

-a-

Here

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 ?

-a-

Here

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 ?

-a-

Here,

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 ?

-a-

Here,

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 ?

-a-

Here,

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 ?

-a-

Here,

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 ?

-a-

Here,

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 ?

-a-

Here,

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-