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-