how to draw a tangent line from a straight line without disturbing the straight line???

I appreciate all your help! I will take a look at the script and see if that helps. As for the other suggestions, I suspect that my question was not very clear as I could find a solution. The 15° was just an arbitrary number. It could be any degree or better yet, imagine I'm tracing something.

I've been using AI for almost 20 years, It still baffles me that this isn't possible in AI.

If you're saying "I understand this can be easily done if the straight line was vertical or horizontal as I can just hold down shift and drag the handle," simply draw them horizontally. Then with both objects selected, double-click on the Rotate tool (R) and rotate 15°.

vajratir,

View>Smart Guides are your friends.

Just ClickDrag from a point on A-B and follow the line; Smart Guides will say path when you are within snapping distance from A-B with the Handle.

If you need to start on A and avoid a continuation of the existing path, you may start a bit away, then press the Spacebar and Drag into the end point, where Smart Guides will say anchor; this is the same as if you are going to ShiftClickDrag along a vertical/horizontal line.

I suggest the Subscribe plugin from AstuteGraphics - it is free. SubScribe - Astute Graphics