portfelio,
Smart Guides are your friends.
As a general solution, you may:
1) With the Line Segment Tool ClickDrag between the two Anchor Points (Smart Guides say anchor);
2) With the appropriate Reference Point selected in the Transform palette, increase the W or H value and Ctrl/Cmd+Enter;
3) Lock the new auxiliary line;
4) ClickDrag the relevant Anchor Point of the original line along the extended auxiliary line (Smart Guides say path);
5) Unlock and delete the auxiliary line.
After 1), you may also click the Line Segment Tool to see the angle.
Or, in a case with a straight path (segment), you may use the Scale Tool:
1) (Direct) Select the straight path (segment);
2) Switch to the Scale Tool and Click the Anchor Point that is to stay, then ClickDrag the one to move (Smart Guides say uniform when you are dragging in the right direction).