froodster,
To keep the left segment unchanged, you may:
1) Cut the path by Clicking the Anchor Point with the Scissors Tool;
2) With the straight path (still) selected Click the Anchor Point with the Convert Anchor Point Tool, then ClickDrag it to drag out the new Handle;
3) Select both paths (you may ClickDrag across the Anchor Point with the Direct Selection Tool) and Ctrl/Cmd to Join, ticking Corner (or all will be in vain).