Intersect Off-Grid Wires in Eagle PCB
There are actually a couple of ways to do this if we assume that both lines orthogonal. If they are not, neither of these methods will work, and you have to instead do a series of trigonometric calculations to work out the intersection point.

The first method involves opening the properties dialog for both of the lines in turn to find the intersection point. For the horizontal line, you would find the Y coordinate of the line and then set this as the end Y coordinate for the vertical line. Then for the vertical line you find the X coordinate, and set this as the end X for the horizontal line.

Personally I find this somewhat of a pain as you have to open each dialog box twice and copy numbers between them. It also means you have to make sure you have the correct end of the line and it's easy to get mixed up.

I have recently come across a simpler method which doesn't require opening the properties boxes, so I figured I'd share it.


With the wire tool, CTRL+LeftClick on the horizontal line (could do vertical first if you want). This will automatically start extending the line, automatically selecting the same layer and width of the existing line. route the line until you are near to where the intersection will be, but on a nice grid dot. Make sure the grid point you choose is inside the corner.


Now repeat with the vertical line, ending on the same grid point as in the first step. This will join the lines.


Next select the miter tool, selecting a very large straight miter. You can do this from the command buttons, or by simply typing miter -10 where the value 10 represents an amount significantly larger than your current grid, and the minus sign indicates straight.

Click on both outer corners that were created during routing to miter them:

 
The final step is to set the size of this miter to however large you need. If you want a right angle rather than miter, that is the same as setting a miter of 0. Run the miter command again, this time with your desired size, e.g. miter -0 for right angle. Click on the middle of the mitered corner a couple of times, and the miter updates to the correct size:




Once you get used to the process, this approach is far faster than trying to merge them using the properties dialog.