I'm glad that people find this useful. Many of the regulars on this forum are familiar with the principles behind cutting compound angles but for the sake of folks new to this:
Before we begin a layout with a framing square or start punching a formula from a book or the Internet into a calculator, it’s important to understand what we are trying to accomplish. An equation or calculation without reference to reality can be misleading - and expensive!
In all cases when measuring any dihedral angle, Backing angle or otherwise, the key idea is: cross section. Note the locations of the right angles in the triangles forming the tetrahedra … the geometry of
any compound angle and a Hip roof is exactly the same. Study the diagrams in the links below, try cutting wooden models of the compound angles, and make 3D cardboard models by developing the angles.
If we look at a Hip roof, we see that the rise as fixed while the common run is stretched through the Plan angle or Deck angle (divided by sin
Plan Angle).
In the case of the compound angle the rise shared by the
Dihedral Angle (complementary to the
Saw Blade Bevel) and the
Angle on the Adjoining Face remains fixed. The run of the
Dihedral Angle is drawn out through the
Angle of Saw Travel angle (divided by sin
Angle of Saw Travel), creating the run of the
Angle on the Adjoining Face. This operation can be reversed; the run of the
Angle on the Adjoining Face turned through the
Angle of Saw Travel (multiplied by sin
Angle of Saw Travel) thus forms the run of the
Dihedral Angle created by the
Saw Blade Bevel.
Here’s another thought: compound angle may be cut from either face of the stick. So who is to say which of the angles of the compound angle is the
Angle of Saw Travel, and which is the
Angle on the Adjoining Face? We can create an equally valid geometric model of the compound angle composed of a different arrangement of angles.
Combining this elementary concept with a bit of basic trigonometry we can easily deduce hundreds of formulas relating every angle in a complex Hip-Valley roof.
Development of Hip Roof Development of Compound Angle Compound Angle: Jack Rafter intersects Valley Rafter Construction of the Backing Angle Model of Backing Angle extracted from Regular Octogonal Gazebo 
