The commands in the Path menu, such as Union can help with those finishing touches. There's a good chance that your drawing is more complex than this and you'll need to integrate this with other objects. This operation isn't too difficult if you have 2 simple straight lines. If you're working with lines that are right angles, it might be easy to draw a single line between two end points, add a node in the middle, and then manually assign coordinates to that node instead of drawing a new line. (If Inkscape doesn't automatically snap to these, check what snapping options you have set and try again). Draw a new line Shift + F6, using the end points and intersection of your two lines as snapping points.Make sure that 'Snap to Intersections' is enabled. Repeat with the other line until they intersect. Hold Ctrl + Alt and drag the node to extend it.Using the Node Tool F2, select the nodes to be extended.

The closest work around would be to extend the two lines, and using snapping to draw a new path over them. This is not easily accomplished in Inkscape.

Inkscape crop cancas