When two parallel lines both intersect a third line, corresponding angles (angles in the same relative positions, like angles 1 and 7 or angles 3 and 5 in the picture below) have the same measure.
One way to understand this is to imagine sliding a copy of the picture above along line j until line k sits on top of line l. Note 3
Now 1 sits exactly where 7 used to be, 3 sits exactly where 5 used to be, and so on.
To prove that corresponding angles are congruent, we could add another line segment, , parallel to line j. By doing so we have created a parallelogram, and thus we know that the adjacent angles of a parallelogram (in this case 2 and 7) equal 180°.
So, to prove that 1 and 7 are congruent, we write the following:
1 + 2 = 180° (because they form a straight line)
2 + 7 = 180° (because they are adjacent angles of a parallelogram)
It follows that 1 + 2 = 2 + 7 = 180°.
And thus, 1 = 7