See image 1. Circles A,B,C and D contain the anchor points of the blue shape. Let's assume you want to continue straight line segments AB and DC to form a sharp corner.

[![enter image description here][1]][1]

2. Draw lines AB and DC. To do it exactly have smart guides and snap to points ON, no other snaps!

3. Holding Shift at the same time drag the line new objects with the normal selection tool to bigger sizes to make a crossing

[![enter image description here][2]][2]

4. With the anchor removal tool remove anchor B and with the anchor type conversion tool make C a corner point

5. Drag C to the crossing of the lines with the direct selection tool


  [1]: https://i.sstatic.net/ekzop.jpg
  [2]: https://i.sstatic.net/ZhLEb.jpg