Solid Angle contained by Plane Angles is Less than Four Right Angles

Theorem

In the words of Euclid:

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

(The Elements: Book $\text{XI}$: Proposition $21$)

Proof

Let $A$ be a solid angle which is contained by the three plane angles $\angle BAC$, $\angle CAD$ and $\angle DAB$.

It is to be demonstrated that $\angle BAC + \angle CAD + \angle DAB$ is less than $4$ right angles.

Let $B$, $C$ and $D$ be arbitrary points on the straight lines $AB$, $AC$ and $AD$.

Let $BC$, $CD$ and $DB$ be joined.

We now have that the solid angle at $B$ is contained by the three plane angles $\angle CBA$, $\angle ABD$ and $\angle CBD$.

From Proposition $20$ of Book $\text{XI} $: Sum of Two Angles of Three containing Solid Angle is Greater than Other Angle:

any two of these is greater than the other one.

Therefore $\angle CBA + \angle ABD > \angle CBD$.

For the same reason:

$\angle BCA + \angle ACD > \angle BCD$

$\angle CDA + \angle ADB > \angle CDB$

Therefore the sum of the six plane angles:

$\angle CBA + \angle ABD + \angle BCA + \angle ACD + \angle CDA + \angle ADB$

are greater than the three plane angles:

$\angle CBD + \angle BCD + \angle CDB$

But from Proposition $32$ of Book $\text{I} $: Sum of Angles of Triangle equals Two Right Angles:

$\angle CBD + \angle BCD + \angle CDB$ equals $2$ right angles.

Therefore $\angle CBA + \angle ABD + \angle BCA + \angle ACD + \angle CDA + \angle ADB$ is greater than two right angles.

Also from Proposition $32$ of Book $\text{I} $: Sum of Angles of Triangle equals Two Right Angles:

$\angle ABC + \angle ACD + \angle ADB$ equals $2$ right angles.

Therefore from Proposition $32$ of Book $\text{I} $: Sum of Angles of Triangle equals Two Right Angles:

$\angle CBA + \angle ACB + \angle BAC + \angle ACD + \angle CDA + \angle CAD + \angle ADB + \angle DBA + \angle BAD$ equals $6$ right angles.

Of them, $\angle ABC + \angle BCA + \angle ACD + \angle CDA + \angle ADB + \angle DBA$ are greater than $2$ right angles.

Therefore the remaining three angles:

$\angle BAC + \angle CAD + \angle DAB$

are less than $4$ right angles.

$\blacksquare$

Historical Note

This proof is Proposition $21$ of Book $\text{XI}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{XI}$. Propositions