Exercise 49 Page 386

Let's begin by drawing two arbitrary lines that intersect.

Since these are not parallel, or coincidental, they will intersect each other once. We can continue by drawing a third line in our diagram. Let's draw this additional line so that it intersects each of the existing lines once.

Notice that the three coplanar lines have a total of three intersection points. Now, we can add a fourth line and make it intersect each of the other lines once.

Looking at our four coplanar lines, we can see that they intersect $6$ times. As each line intersects all other lines once, six is the greatest number of intersection points we can have.