G.585 angles equilateral pentagon

XIII.7

If three angles of an equilateral pentagon, taken either in order or not in order, be equal, the pentagon is equiangular.

angles_equilateral_pentagon()

prove the pentagon < A B C D E > is equiangular; in this case too the pentagon < A B C D E > is equiangular

assert In the equilateral pentagon < A B C D E > let, first, three angles taken in order, those at A, B, C, =

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

assert / C B /, / B A / = / B A /, / A E /, and < C B A ) = < B A E ),

∵ sas_congruenceassert / A C / = / B E /, < A B C > is equal to < A B E >, and the remaining angles = the remaining angles, namely those which the equal sides subtend, that is, < B C A ) to < B E A ), and < A B E ) to < C A B )

∵ equal_angles_isoscelesassert / A F / = / B F /

assert / A C / = / B E /

assert / F C / = / F E /

assert / C D / = / D E /. Therefore / F C /, / C D / = / F E /, / E D /; and / F D / is common to them

∵ sss_congruenceassert < F C D ) = < F E D )

assert < B C A ) = < A E B )

assert < B C D ) = < A E D )

assert But, by hypothesis, < B C D ) = the angles at A, B

assert < A E D ) = the angles at A, B

assert Similarly we can prove that < C D E ) = the angles at A, B, C; therefore the pentagon < A B C D E > is equiangular

assert Next, let the given equal angles not be angles taken in order, but let the angles at A, C, D be =

construct segment(B, D) → / B D /

assert / B A /, / A E / = / B C /, / C D /, and they contain = angles,

∵ sas_congruenceassert / B E / = / B D /, < A B E > is equal to < B C D >, and the remaining angles will be equal to the remaining angles, namely those which the equal sides subtend; therefore < A E B ) = < C D B )

∵ isosceles_base_anglesassert < B E D ) = < B D E ), since / B E / = / B D /

assert < A E D ) = < C D E )

assert < C D E ) is, by hypothesis, = to the angles at A, C

assert < A E D ) = the angles at A, C

∴ For the same reason < A B C ) = the angles at A, C, D. Therefore the pentagon < A B C D E > is equiangular

Dependency Graph

Required for

G.600, G.601