Lay a straightedge across a minimum of 3 stair nosings. Put your universal protractor on the straightedge and measure the angle. Note that the more stair nosings you have under the straightedge, the more accurate the angle will be. In the case that the run of stairs is very long, measure the stair angle at 2 or 3 places along the run to check that the angle remains constant along the full run.
I am proposing that Kha the architect at Deir El-Medina, Egypt, supervisor of the projects of three kings of the 18th Dynasty used his protractor to check the inclination of stairs for instance. You can see it at the Egyptian Museum of Torino. Read more at http://stretchingtheboundaries.blogspot.com/2011/08/khas-protractor.html
A modern universal protractor for building is that shown in the following figure.