Select the end point of each line with the direct selection tool (white arrow) and go through Object>Path>Join (Keyboard shortcut: Command (Mac)/Control (PC)-J). If you use the option (Mac) or the alt (PC) along with the keyboard command you'll get a dialog with options as to how you want the points to join (see screen shot).
I does work, but not in the way I want it to. I want each line to stay in the same position and angle. I just want them to grow until they touch eachother. Your technique makes each line meet at an average position, which is not what I want.
Or use the following tool (In German called: Zusammenfügen-Werkzeug)
This works perfectly! Thanks!
Join Tool in English, 😉