Side of Spherical Triangle is Less than 2 Right Angles

Theorem

Let $ABC$ be a spherical triangle on a sphere $S$.

Let $AB$ be a side of $ABC$.

The length of $AB$ is less than $2$ right angles.

Proof

$A$ and $B$ are two points on a great circle $E$ of $S$ which are not both on the same diameter.

So $AB$ is not equal to $2$ right angles.

Then it is noted that both $A$ and $B$ are in the same hemisphere, from Three Points on Sphere in Same Hemisphere.

That means the distance along $E$ is less than one semicircle of $E$.

The result follows by definition of spherical angle and length of side of $AB$.

$\blacksquare$

Sources

• 1976: W.M. Smart: Textbook on Spherical Astronomy (6th ed.) ... (previous) ... (next): Chapter $\text I$. Spherical Trigonometry: $2$. The spherical triangle.