## Proposition 21

In proposition XI.23 the condition stated here and the condition in XI.20 (the sum of any two plane angles is less than the third) together are shown to be sufficient to construct a solid angle.

The proof only shows that the sum of the plane angles in all cases is less than four right angles when there are three plane angles, not when there are more than three. When there are four ore more plane angles, the proof is analogous, but it is necessary to invoke Proclus' first corollary to I.32, which states that "the sum of the interior angles of a convex rectilinear figure equals twice as many angles as the figure has sides, less four."

#### Use of this proposition

This proposition is used in the proof of remark after proposition XIII.18 to show that the five regular polyhedra constructed in Book XIII are the only five possible.

Next proposition: XI.22

Previous: XI.20

 Select from Book XI Book XI intro XI.Def.1-2 XI.Def.3-5 XI.Def.6-8 XI.Def.9-10 XI.Def.11 XI.Def.12-13 XI.Def.14-17 XI.Def.18-20 XI.Def.21-23 XI.Def.24 XI.Def.25-28 XI.1 XI.2 XI.3 XI.4 XI.5 XI.6 XI.7 XI.8 XI.9 XI.10 XI.11 XI.12 XI.13 XI.14 XI.15 XI.16 XI.17 XI.18 XI.19 XI.20 XI.21 XI.22 XI.23 XI.24 XI.25 XI.26 XI.27 XI.28 XI.29 XI.30 XI.31 XI.32 XI.33 XI.34 XI.35 XI.36 XI.37 XI.38 XI.39 Select book Book I Book II Book III Book IV Book V Book VI Book VII Book VI Book IX Book X Book XI Book XII Book XI Select topic Introduction Table of Contents Geometry applet About the text Euclid Web references A quick trip