Euclid’s Elements — Book XIII, Proposition 7

When Three Equal Angles Make a Pentagon Equiangular

In Book XIII of Euclid’s Elements, Proposition 7 explores a fascinating property of the regular pentagon.

It shows that if an equilateral pentagon has three equal interior angles, then the pentagon must necessarily be equiangular — meaning all five angles are equal.

This result reveals the deep symmetry of the pentagon and the inherent geometric constraints imposed by equal sides and angles.

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Proposition Statement

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

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Geometric Setup

• Consider an equilateral pentagon $ABCDE$.

• Two cases arise:

  1. The three equal angles are consecutive (e.g., $\angle A = \angle B = \angle C$).

  2. The three equal angles are not consecutive (e.g., $\angle A = \angle C = \angle D$).

In both cases, we want to prove that all five interior angles are equal, making the pentagon equiangular.

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Case 1 — Three Equal Angles Taken in Order

1. Suppose $\angle A = \angle B = \angle C$.

2. Join diagonals AC, BE, and FD.

3. Using triangle congruence properties, Euclid shows:

   • The diagonals AC and BE are equal.

   • The sides FC and FE are also equal.

4. This symmetry implies that:

   • $\angle BCD = \angle AED$.

   • Since $\angle BCD$ was assumed equal to $\angle A$ and $\angle B$, it follows that $\angle AED$ equals them as well.

5. Extending the same reasoning shows:

   $$\mathrm{\angle }A=\mathrm{\angle }B=\mathrm{\angle }C=\mathrm{\angle }D=\mathrm{\angle }E\mathrm{.}$$

6. Therefore, the pentagon is equiangular.

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Case 2 — Three Equal Angles Taken Out of Order

1. Suppose $\angle A = \angle C = \angle D$.

2. Join diagonal BD.

3. By applying triangle congruence again, Euclid shows:

   • $\angle AED = \angle CDE$.

   • Since $\angle CDE$ was assumed equal to $\angle A$ and $\angle C$, we deduce that $\angle AED$ is also equal to them.

4. Similarly, we prove that $\angle B$ and $\angle E$ are also equal to the rest.

5. Therefore, the pentagon is equiangular.

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Key Insight

This proposition relies on two geometric facts:

• In an equilateral pentagon, equal angles force equal diagonals.

• Equal diagonals then enforce equality of the remaining angles.

Thus, in an equilateral pentagon, the symmetry of the sides propagates through the entire figure, making equal angles “spread” to the whole shape.

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Summary

In an equilateral pentagon, if any three interior angles are equal — whether they are consecutive or not —
then all five angles are equal, and the pentagon is equiangular.

This is a beautiful result showing the rigidity of regular pentagons:
once three angles match, symmetry forces the rest.

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Why This Matters

This proposition is crucial in the construction of regular pentagons and decagons, both of which are deeply connected to the golden ratio explored throughout Book XIII.

It highlights the elegant interplay between:

• Equal sides,

• Equal diagonals, and

• Equal angles.