Sum of Two Angles of Three containing Solid Angle is Greater than Other Angle
Theorem
In the words of Euclid:
- If a solid angle be contained by three plane angles, any two, taken together in any manner, are greater than the remaining one.
(The Elements: Book $\text{XI}$: Proposition $20$)
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 any two of these plane angles together are greater than the third.
Suppose $\angle BAC$, $\angle CAD$ and $\angle DAB$ are all equal.
Then any two together are greater than the remaining one and the result follows.
Suppose otherwise.
Without loss of generality, suppose $\angle BAC$ is greater than $\angle BAD$.
Let $E$ be constructed on $AB$ such that $\angle BAE$, in the plane through $BA$ and $AC$, is equal to $\angle DAB$.
Let $AE = AD$.
Let $BEC$ be drawn across through $E$ such that the straight lines $AB$ and $AC$ are cut at $B$ and $C$.
Let $DB$ and $DC$ be joined.
We have that:
- $DA = AE$
and $AB$ is common.
So we have two sides equal to two sides, while $\angle DAB = \angle DAE$.
Therefore from Proposition $4$ of Book $\text{I} $: Triangle Side-Angle-Side Congruence:
- $DB = BE$
We have from Proposition $20$ of Book $\text{I} $: Sum of Two Sides of Triangle Greater than Third Side:
- $BD + DC > BC$
But we have that $BD = BE$.
Therefore $DC > EC$.
We have that:
- $DA = AE$
and $AC$ is common.
We also have that $DC > DE$.
Therefore from Proposition $25$ of Book $\text{I} $: Converse Hinge Theorem:
- $\angle DAC > \angle EAC$
But:
- $\angle DAB = \angle BAE$
and so:
- $\angle DAB + \angle DAC > \angle BAC$
Similarly it can be shown that the remaining plane angles, taken two together, are greater than the remaining one.
$\blacksquare$
Historical Note
This proof is Proposition $20$ 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