0When a transversal intersects two parallel lines:
- The corresponding angles are equal.
- The vertically opposite angles are equal.
- The alternate interior angles are equal.
- The alternate exterior angles are equal.
- 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.
