It is unclear what you mean by each of the terms you have used in this question. Here's what I'm assuming you mean, and my answers based on my assumption.
This terminology is usually used when talking about various angles made by lines (usually parallel, although they don't have to be) that are both intersected by another line (called a transversal). It is not so much that the lines are alternate or adjacent, but the angles might be categorized one of these ways or the other.
Two angles, each of which is formed by one of the parallel lines and the transversal are called alternate angles if they are on opposite sides of the transversal. They can be alternate interior, or alternate exterior angles depending on whether they are between the parallel lines(interior) or outside the parallel lines (exterior).
Two angles that are formed by one of the parallel lines and the transversal are adjacent angles if they share a common side. If they do not share a common side then they are called vertical angles.
If the two lines are parallel then a pair of alternate interior or a pair of alternate exterior angles have equal measure.
Any pair of adjacent angles are supplementary - their measures add up to 180 degrees. And any pair of vertical angles have equal measures.
As far as whether the "lines" have to "match" or "end the same" - recall that a line extends infinitely in both directions. A line segment would be any portion of the line with a beginning and a terminal point. For the above statements to hold, it is not necessary that the line segments be the same length.