How do you determine the angle of a path?

You may also use Illustrator's highly acclaimed Measure tool together with the Info palette to display angles.

The Measure tool is snap-sensitive and it can be precise. You may just not notice it at first glance and some dexterity is certainly required.

JLG Lines always measures from the horizontal line

Yes, unfortunately this is still not possible with Adobe Illustrator after many years. You must use a tool. FOR EXAMPLE. JLG Dimensions or JLG Lines

Thank you everyone! I was to vague with my question, to clarify:

Is the another Palette or Menu that will display the angle value.

My concern with drawing lines for measurements, there's a chance of deleting the original path. Yes Lock/Unlock is an option - but each work around takes me further into a rabbit hole.

If you want to constrain the angle in which you are drawing a path to something other than the default of 0, 45 and 90 degrees then change the constrain angle in the General section of your Illustrator preferences from 0° to whatever angle you wish. If, for example you change the constrain angle to 15° then by holding down the shift key while using the pen tool all line segments will be drawn at that angle.

you have to draw the segment with the Line tool to get the angle in the Transform panel.

If you make a multi-segment shape with the pen tool, the angle shown would be the whole shape's angle.

so, draw a Line on top the segment you want to measure then delete it

Draw a line on top of the line you want to measure with snap to point on, hit return.

Does it change if you use the Direct Select tool and click on the path segment?