Two lines are either parallel or they intersect at some point. At the point of intersection four angles are formed. These angles are called "vertical" -- which is somewhat confusing. The more familiar meaning of "vertical" is the opposite of "horizontal," but another meaning of "vertical" is sharing the same vertex. From the point of view of one of these angles, there are two adjacent angles and one opposite angle, and all of these angles are vertical -- they all share the same vertex. The adjacent angels are supplemental -- they add up to 180 degrees, because any two adjacent angles make up a straight line.
With the four vertical angles formed by intersecting lines, opposite angles are congruent -- they have the same number of degrees. This is not obvious, but it is easy to prove. Let the angles be named A, B, C and D when labeled in clockwise order. To prove that opposite angles are congruent, it is sufficient to prove that A is congruent to C. Because adjacent angles are supplemental. A + B = 180 degrees and A + D = 180 degrees. This means that A + B + A + D = 360. But it is obvious that A + B + C + D = 360 so A = C.
When a single line cuts across a couple of parallel lines, the relationships between each of the sets of four vertical angles are the same. If the vertical angles made with one of the parallel lines are A, B, C and D, then the vertical angles made with the other parallel line will also be A, B, C and D, and the angles in similar locations will have similar measurements. This probably seems fairly obvious and not very interesting. It turns out that this is a very valuable theorem in the sense that it is useful in proving other less obvious relationships.
Many proofs use the theorem about a line cutting across a set of parallel lines, but one of the simplest is the fact that the interior angles of a triangle always add up to 180 degrees. Place any triangle between two parallel lines so that the base of the triangle is on one line and the top of the triangle is touching the other line. By the parallel theorem, the angle between the top of the triangle and the line that the top of the triangle touches is equal to one of the interior angles of the base of the triangle. This picture makes it clear that the interior angles of any triangle add up to 180 degrees.