In school we have learned about objects in $2$-space and in $3$-space, with heavy emphasizes on the properties in $2$-space. My question can be formulated as follows:
What would we have learned in school, if the geomtry lessons didn't essentially restrict to the geometry of simplices in the euclidean plane, but fully generalized these inspections for $d$-dimensional simplices?
Whereas knowledge on the geometry of a triangle is taken for granted, I admit I don't know about corresponding results for a tetrahedron, (e.g.: What is the sum of the angles between faces?) not to speak of the higher dimensional analogues.
Whereas this is elementary, I don't think this is completely trivial. Do you know articles or other resources which develop the geometric theory of $d$-simplices in the elementary way I described?