Question Video: Identifying and Proving Lines Parallel Using Angle Relationships Mathematics • First Year of Preparatory School

Video Transcript

Are lines 𝐿 one and 𝐿 five parallel?

𝐿 five is the blue line. And 𝐿 one is the red line. To prove these lines are parallel, we need to look for transversals. A transversal is a line that cuts across two other lines. We need a line that cuts through both the red and the blue line. 𝐿 two, the pink line, is a transversal. 𝐿 four, the green line, is a transversal. And the purple line 𝐿 three is also a transversal.

However, we only need one of these to help us prove that 𝐿 one and 𝐿 five are parallel. So let’s consider the pink line, 𝐿 two. In order for 𝐿 one and 𝐿 five to be parallel, the transversals have to create corresponding congruent angles.

Is there a way we can find out what this angle is? Well, we know that these three angles together make up a straight line. And that means if we subtract 45 and 80 from 180 degrees, we can find the missing angle. 180 degrees minus 45 degrees minus 80 degrees equals 55 degrees. The angle across from that 55 degrees is a vertical angle and also must measure 55 degrees.

If we look at the angles created with 𝐿 five and 𝐿 two in the blue line, we also see a pair of vertical angles that measure 55 degrees, these two corresponding angles. These two pair of corresponding congruent angles prove that 𝐿 one and 𝐿 five are parallel. 𝐿 one and 𝐿 five are parallel by corresponding congruent angles. They’re angles that are at the same corner at each intersection.

It’s a little bit hard to see. So I’m gonna remove the other two transversals since we only need one to prove these lines are parallel. Okay, it looks like this. When two lines intersect, they form four angles.

If we labelled them like this at the intersection of 𝐿 one and 𝐿 two, we would label them in the same way at the intersection of 𝐿 two and 𝐿 five. We’re seeing that for top left angle, here the one labelled two, is 55 degrees for both 𝐿 five intersection and 𝐿 one intersection. That’s what we mean by corresponding. And the fact that they’re both equal to 55 degrees makes them congruent angles.

Yes, by corresponding congruent angles, 𝐿 one and 𝐿 five are parallel.