You could use trigonometry.
Use a set square against the newel post, draw a pencil line until it hits the end of the stair string and measure the length of the line. In the image, measure the lengths of the red and the green lines.
To calculate angle number one, use a website such as the following
Effortlessly find trigonometric function values (sin, cos, tan, cot) or solve for missing sides or angles in a right triangle using our remarkable tool crafted by experts.
www.omnicalculator.com
In the website, enter the length of red as "b", enter green as "a", then hit return- all you need is those two dimensions. In my image, the angle you seek will be listed as "β"
By using a set square, you have created a right angle triangle. That means that the angle between the red and green lines is 90 degrees. The total of all three angles in a triangle equates to 180 degrees. We know that one by definition is 90 degrees, that means that the other two combined equal 90 degrees.
The great thing about trigonometry is that so long as one angle is 90 degrees, you only need two further dimensions to calculate all of the other dimensions.
Sorry, the above is not intended to be condescending. I have no idea if you remember trig from school. Hopefully it will help others that have forgotten it.