The diagram shows three straight lines: 𝐴𝐵, 𝐶𝐷, and 𝐸𝐹. The line 𝐸𝐹 intersects 𝐴𝐵 and 𝐶𝐷. Determine whether the line 𝐴𝐵 is parallel to 𝐶𝐷. You must give reasons for your answer.
There are a number of rules for parallel lines that we should recall. The first is corresponding angles are equal. When we have a single straight line that intersects two parallel lines, these are the ones that look like they’re enclosed in the letter F. We’re not allowed to say F angles, but it’s a good way to remember them.
Alternate angles are also equal. These are the ones that look like they’re enclosed in the letter Z. Again, we’re not allowed to say Z angles; it’s just a good way to remember it.
The third rule relates to the angles that look like they’re enclosed in a letter C. These are called supplementary or co-interior and they add to 180 degrees.
Let’s compare these three rules to our diagram. We can see that we have a single straight line that intersects the lines 𝐴𝐵 and 𝐶𝐷. These angles look like they’re enclosed in the letter C. If angles 𝐵𝐸𝐹 and 𝐷𝐹𝐸 do indeed add to 180 degrees, then this will mean that they are supplementary and the two lines 𝐴𝐵 and 𝐶𝐷 must be parallel.
Let’s check this. If necessary, we can add 129 and 51 using the column method. Nine plus one is 10. So we put a zero in the units column and carry the one. Two plus five is seven. And then, when we add that carried one, we get eight, so looking good so far. And then, one add nothing is just one. 129 plus 51 is indeed 180.
We can say that the angles 𝐵𝐸𝐹 and 𝐸𝐹𝐷 were formed by the intersection of a single straight line with the lines 𝐴𝐵 and 𝐶𝐷. Since these two angles add to 180 degrees, we can say that they’re supplementary and the lines 𝐴𝐵 and 𝐶𝐷 are indeed parallel.