Properties Of Parallel Lines

When a transversal intersects two parallel lines:

1. The corresponding angles are equal.
2. The vertically opposite angles are equal.
3. The alternate interior angles are equal.
4. The alternate exterior angles are equal.
5. The pair of interior angles on the same side of the transversal is supplementary.

Examples:

1. If the lines m and n are parallel to each other, then determine the angles ∠5 and ∠7.

Solution:

Determining one pair can make it possible to find all the other angles. The following is one of the many ways to solve this question.

∠2 = 125°

∠2 = ∠4 since they are vertically opposite angles.

Therefore, ∠4 = 125°

∠4 is one of the interior angles on the same side of the transversal.

Therefore, ∠4 + ∠5 = 180°

125 + ∠5 = 180 → ∠5 = 180 – 125 = 55°

∠5 = ∠7 since vertically opposite angles.

Therefore, ∠5 = ∠7 = 55°

2.If ∠A = 120° and ∠H = 60°. Determine if the lines are parallel.

Solution:

Given ∠A = 120° and ∠H = 60°.

Since adjacent angles are supplementary, ∠A + ∠B = 180°

120 + ∠B = 180 → ∠B = 60°.

It is given that ∠H = 60°. We can see that ∠B and ∠H are exterior alternate angles.

When exterior alternate angles are equal, the lines are parallel.

Hence the lines p and q are parallel.

We can verify this using other angles.

If ∠H = 60°, ∠E = 120° since those two are on a straight line, they are supplementary.

Now, ∠A = ∠E = 120°. ∠A and ∠E are corresponding angles.

When corresponding angles are equal, the lines are parallel.

Likewise, we can prove using other angles too.