1. Use the CUT Tool. [di]
2. Cut the lines needed in parts with the scissor tool. Use [si] intersection points to match intersection lines. Then left click do the cut.
3. Then use DELETE to delete.
OR better and faster:
3. Use the TRIMM tool [d2].
Yes, [d2] is your friend here!
Check also http://www.flagsoft.ch/qcadtutorial/qcadtutorial.html
for a few video online tutorials.