G.526 solid angle contained

XI.21

Any solid angle is contained by plane angles less than four right angles.

solid_angle_contained_xi_21(A, < B A C ), < C A D ), < D A B ))

given the angle at A be a solid angle contained by the plane angles < B A C ), < C A D ), < D A B )

prove < B A C ), < C A D ), < D A B ) are less than four right angles

given the angle at A be a solid angle contained by the plane angles < B A C ), < C A D ), < D A B )

construct segment() → B, segment() → C, segment() → D, segment(A, B) → / A B /, segment(A, C) → / A C /, segment(A, D) → / A D /, segment(B, C) → / B C /, segment(C, D) → / C D /, segment(D, B) → / D B /

assert the solid angle at B is contained by the three plane angles < C B A ), < A B D ), < C B D ),

∵ solid_angle_containedassert Any two are greater than the remaining one; therefore < C B A ), < A B D ) are greater than < C B D )

assert For the same reason < B C A ), < A C D ) are also greater than < B C D ), and < C D A ), < A D B ) are greater than < C D B ); therefore the six angles < C B A ), < A B D ), < B C A ), < A C D ), < C D A ), < A D B ) are greater than the three angles < C B D ), < B C D ), < C D B )

∵ triangle_angle_sumassert The three angles < C B D ), < B D C ), < B C D ) are equal to two right angles

assert the six angles < C B A ), < A B D ), < B C A ), < A C D ), < C D A ), < A D B ) are greater than two right angles

assert since the three angles of each of < A B C >, < A C D >, < A D B > are equal to two right angles,

assert the nine angles of the three triangles, < C B A ), < A C B ), < B A C ), < A C D ), < C D A ), < C A D ), < A D B ), < D B A ), < B A D ) are equal to six right angles

assert of them the six angles < A B C ), < B C A ), < A C D ), < C D A ), < A D B ), < D B A ) are greater than two right angles

∴ the remaining three angles < B A C ), < C A D ), < D A B ) containing the solid angle are less than four right angles

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