The Paths you are joining must all be on the same Layer.

Go to View - Outline (Command-Y) and you will see only the Outlines of everything on your artboard – or – Command-Click a Layer eyeball and you will see only the Outlines of that Layer on your artboard.

Click the Layer triangle so you can see the Paths you are cutting (see below):

Use the Scissors Tool (C) and line up the crosshairs right over where the lines intersect and then Click. New endpoints will appear on your new cut section (see below):

 

You will see a new Path appear in the Layers Panel. Toggle the visibility of the Paths in the Layers Panel to see what you did (see below):

Delete the Path you don’t want by selecting it in the Layers Panel and then clicking the Trash. Now my Outlines on the artboard look like this (see below):

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If you try and cut the other line at the same spot you will get a warning and it won’t cut (see below):

To solve this:

• go to the Layers Panel
• grab the Path layer you want to cut
• and move it the top of its Group

– now you should be able to cut that Path.

You will again see a new Path appear in the Layers Panel. Toggle the visibility of the Paths in the Layers Panel to see what you did.

Delete the Path you don’t want by selecting it in the Layers Panel and then clicking the Trash. Now my Outlines on the artboard look like this (see below):

Now use the Direct Selection Tool (A) to make a marquis around the two points of the two Paths that you want to join (see below):

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Go to Object - Path - Join (Command-J) and now you will see in the Layers Panel that your other two Paths have disappeared and you’ve created a new joined Path (see below):

The endpoints are now joined and the new single Path will move as one object (see below in Outline Mode):

The same new object in Preview Mode with the stroke thickness and color I chose (see below):

Is the joined corner not quite right? Funky corner? If so, un-do your previous join. Re-select your corners again using the Direct Selection Tool (A) to make a marquis around the two points of the two Paths that you want to join and press:
Shift-Option-Command-J
This averages and joins your corners in one keyboard shortcut.

If the keyboard shortcut doesn't work for some reason, the long way to accomplish the above correction of a funky joining of Paths is the following:

• Un-do your previous join. (You can check what you’ve done by going to the Edit Menu and seeing what it will now let you un-do.)

• Then go to Object - Path - Average (Option-Command-J) and check Both in the dialog box (see below):

• And then go to Object - Path - Join (Command-J) and you’ll get this dialog box (see below):

• If it’s a corner – click OK. If you want a straighter line – use smooth.

Your Paths should now be properly joined and made into one object.

Any questions?

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