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

First, let three angles *A, B,* and *C* taken in order in the equilateral pentagon *ABCDE* be equal to one another.

I say that the pentagon *ABCDE* is equiangular.

Join *AC, BE,* and *FD.*

Now, since the two sides *CB* and *BA* equal the two sides *BA* and *AE* respectively, and the angle *CBA* equals the angle *BAE,* therefore the base *AC* equals the base *BE,* the triangle *ABC* equals the triangle *ABE,*and the remaining angles equal the remaining angles, namely those opposite the equal sides, that is, the angle *BCA* equals the angle *BEA,* and the angle *ABE* equals the angle *CAB.*

Hence the side *AF* also equals the side *BF.*

But the whole *AC* equals the whole *BE,* therefore the remainder *FC* equals the remainder *FE.* But *CD* also equals *DE.* Therefore the two sides *FC* and *CD* equal the two sides *FE* and *ED,* and the base *FD* is common to them, therefore the angle *FCD* equals the angle *FED.*

But the angle *BCA* was also proved equal to the angle *AEB,* therefore the whole angle *BCD* equals the whole angle *AED.* And, by hypothesis, the angle *BCD* equals the angles at *A* and *B,* therefore the angle *AED* also equals the angles at *A* and *B.* Similarly we can prove that the angle *CDE* also equals the angles at *A, B,* and *C.* Therefore the pentagon *ABCDE* is equiangular.

Next, let the given equal angles not be angles taken in order, but let the angles at the points *A, C,* and *D* be equal.

I say that in this case too the pentagon *ABCDE* is equiangular.

Join *BD.*

Then, since the two sides *BA* and *AE* equal the two sides *BC* and *CD,* and they contain equal angles, therefore the base *BE* equals the base *BD,* the triangle *ABE* equals the triangle *BCD,* and the remaining angles equal the remaining angles, namely those opposite the equal sides. Therefore the angle *AEB* equals the angle *CDB.*

But the angle *BED* also equals the angle *BDE,* since the side *BE* equals the side *BD.*

Therefore the whole angle *AED* equals the whole angle *CDE.*

But the angle *CDE* is, by hypothesis, equal to the angles at *A* and *C,* therefore the angle *AED* also equals the angles at *A* and *C.*

For the same reason the angle *ABC* also equals the angles at *A, C,* and *D.* Therefore the pentagon *ABCDE* is equiangular.

Therefore, *if three angles of an equilateral pentagon, taken either in order or not in order, are equal, then the pentagon is equiangular.*

Q.E.D.