- Click on the line
- Switch to Direct Selection tool and click on the top anchor-point of the line
- Drag it over the other lines and it will snap to the other line's end
It should look like this:
From your update id looks like the "line" is not a line. It looks more like you used the rectangle tool to create the line or something like it.
Please try to recreate the line and snap it like I suggested above or better yet, select the line tool and create the line directly from one anchor point to another! This has to work. There's nothing that can stand in it's way!
Line vs Line like Rectangle view: