Have already dried off pipes & even wrapped pipes with paper towels & have found no sign of a leak; no moisture points on the copper pipes. I have turned off the MAIN water in to the house; I have turned off the HOT & COLD water pipes under the sink; The picture shows the hot water line , the connection that would normally be connected to the faucets' Hot input . With Main & sinks valves closed, I am still getting a steady trickle of water from the hot line . The Washer in the Basement, which runs from the same feed off the Hot Water Heater seems to be OK; no leak yet found . NOTE: I have replaced the Hot & cold supply lines between the copped pipes & the faucet; the dripping was occurred before AND after. Here is a picture of the Faucet supply line, where the dripping is; whether attached or not. Devan Kirk's Photos | Facebook (http://www.facebook.com/photo.php?pid=1294308&l=24481de491&id=1487328934)

And thank you...

It appears that the washers in angle valve is no good. Also the valve on your main is not holding. Need to change washers.

Chuck

Everything from the copper supply pipes up(Braided hoses, Hot & cold, Faucet,but not any copper itself) ; are all brand new; washers included. Is the "Angle Valve" the shut off itself? I have been suspecting a bad shutoff valve...

the dripping was occurred before AND after Before and after what?
The angle stop? Another connection? Follow Chucks advice and replace the washer in the angle stop,(see image). Good luck, Tom

@Speedball1... "Before & After" refers to the preceding line "replaced the Hot & cold supply lines between the copped pipes & the faucet;" MEANING... the dripping has occurred before & after the replacement of the supply lines. Looking at my valves, they appear to be soldered to the copper piping, so may still have to cut off & replace with a compression fitting valve.
Thankx all

Se the image below I have updated you picture with some notes, Thanks.

John

Thank you John.. I really don't mean to pick, but.. would this ALSO cause a chronic leak when the valve is OPEN & connected to the faucet? A leaking Faucet is the original issue I began with; it has been isolated to the leak from the supply hose/valve. If all is connected & the valve is open, WOULD the bad valve/washer result in a leak up the faucet.
Now, I have removed the knob, Packing Nut & washer,and the trickle stopped. If I replace the washer & Packing nut, it begins to trickle again (very weird) . The Valve stem is not removable , so this is all that comes out. Again, I imagine at this point, it means replacing the whole SO valve. Se la vie

I think you have two issues that are not related other than they are on the same supply. The faucet leaks because of a bad washer. The supply line is not fully closing because of a different valve, the stop, not closing all of the way. Valve stems do come out, not sure how you replaced a washer on the stem if you could not remove it. Frequently the seat inside the valve needs to be refaced with a stem resurfacing tool. General Purpose Parts (http://faucet-fix.com/html/general_purpose_parts.html)

I thin bb is right. No washer in the hose and possibly two bad stops. The one below the sink and the main.

However, before I would suspect the main. Turn off the ho water supply and drain some of the hot water from a fixture below this one or some from the tank. Sometimes the hot water heater burps.

If you need to replace the whole house stop, you may be able to borrow the tools to access the curb stop to replace the one inside. You may not be allowed to operate the stop outside because of laws. e.g only a licensed plumber or the city can operate the stop.