Journal for Geometry and Graphics 29 (2025), No. 2, 127--137
Copyright by the authors licensed under CC BY SA 4.0
Angles of Skew Quadrilaterals and Tetrahedra
Hidefumi Katsuura
San Jose State University, San Jose, U.S.A.
hidefumi.katsuura@sjsu.edu
In Theorem 1 we prove angle sum conditions for a skew quadrilateral to be planar. Theorem 2 is about the angle sum of a non-planar skew quadrilateral. Theorem 3 proves that a tetrahedron must have a vertex with all three angles acute.
A skew quadrilateral with pairwise equal opposite edges is called reversible. A tetrahedron that contains a reversible skew quadrilateral is reversible. An equal-angled skew quadrilateral may not be reversible. However, Theorem 4 states that if a tetrahedron contains an equal-angled skew quadrilateral, then the tetrahedron must be reversible. Our last Theorem 5 is on an angle condition of an isosceles tetrahedron.
Keywords: Skew quadrilateral, quadrilateral, law of cosines, spherical law of cosines, reversible skew quadrilateral, reversible tetrahedron, isosceles tetrahedron.
MSC: 51M04
[ Fulltext-pdf (454 KB)]