Jack Purlin meets Hip or Valley Rafter
Study of Chamfer Cut or Square Cut along the Line of Intersection

P1 ... Saw Miter Angle on the face of the jack purlin perpendicular to the surface of the roof, at the intersection of this face of the purlin with the side (plumb) face of the Hip-Valley rafter. C1 ... Saw Blade Bevel along miter line created by angle 90° – P1. 90° – Q1 ...The Backing Angle, C5, as viewed in section on a backed Hip-Valley rafter if the cutting plane is revolved through angle R2. P4 ... Tenon Layout Angle on the jack purlin shoulder set in the surface of the roof to produce a tenon face perpendicular to the side (plumb) face of the Hip-Valley rafter. Also the angle created on the jack purlin by the housing chamfer cut (square cut) to remove the feather edge along miter angle P1. Cutting the housing to accommodate the jack purlin along the line of angle 90° ± R2 on the Hip-Valley rafter creates angle 90° ± A8 on the backed shoulder of the Hip-Valley. Angles P4 and A8 are produced by a section plane or cutting plane intersecting the backed shoulder of a Hip-Valley rafter. This section plane follows the line of angle 90° ± R2 on the side (plumb) face of the Hip-valley rafter (or angle Q3 measured from the plumb line), and also passes through the line at right angles to the Hip-Valley run in plan view. P4BV ... Saw Blade Bevel to cut the tenon from the upper shoulder of the jack purlin, along the miter line produced by angle P4.

Development of General Tetrahedron composed of Right Triangles
Overview of Tetrahedra and Triangles defining a Hip or Valley Roof