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Euclid Book 13 Proposition 7b

Euclid Book 13 Proposition 7b

Statement

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

Computational Explanation

GeometricSceneA.,B.,C.,D.,E.,{},GeometricAssertionPolygonA.,B.,C.,D.,E.,"Equilateral","Convex",PlanarAngle[B.,A.,E.]PlanarAngle[A.,E.,D.]PlanarAngleD.,C.,B.,GeometricAssertionPolygonA.,B.,C.,D.,E.,"Equiangular"

GeometricAssertionPolygonA.,B.,C.,D.,E.,Equiangular

Explanations

Let the given equal angles not be angles taken in order, but let the angles at the points

A

,

C

,

D

be equal; I say that in this case too the pentagon

ABC

DE

is equiangular.

For let

BD

be joined.

Then, since the two sides

BA

,

AE

are equal to the two sides

BC

,

CD

, and they contain equal angles, therefore the base

BE

is equal to the base

BD

, the triangle

ABE

is equal to the triangle

BCD

, and the remaining angles will be equal to the remaining angles, namely those which the equal sides subtend;

therefore the angle

AEB

is equal to the angle

CDB

.

But the angle

BED

is also equal to the angle

BDE

, since the side

BE

is also equal to the side

BD

.

Therefore the whole angle

AED

is equal to the whole angle

CDE

.

But the angle

CDE

is, by hypothesis, equal to the angles at

A

,

C

; therefore the angle

AED

is also equal to the angles at

A

,

C

. For the same reason the angle

ABC

is also equal to the angles at

A

,

C

,

D

. Therefore the pentagon

ABC

DE

is equiangular.

Q. E. D.

Classes

Euclid's Elements
MathWorld
Theorems
Polygons
EuclidBook13

MathWorld

RegularPentagon

Related Theorems

EuclidBook13Proposition7a
EuclidBook13Proposition8