Proposition: Prop. 11.22: Extremities of Line Segments containing three Plane Angles any Two of which are Greater than Other form Triangle

Euclid's Formulation

If there are three rectilinear angles, of which (the sum of any) two is greater than the remaining (one), (the angles) being taken up in any (possible way), and if equal straight lines contain them, then it is possible to construct a triangle from (the straight lines created by) joining the (ends of the) equal straight lines.

Let $ABC$, $DEF$, and $GHK$ be three rectilinear angles, of which the sum of any) two is greater than the remaining (one), (the angles) being taken up in any (possible way) - (that is), $ABC$ and $DEF$ (greater) than $GHK$, $DEF$ and $GHK$ (greater) than $ABC$, and, further, $GHK$ and $ABC$ (greater) than $DEF$.
And let $AB$, $BC$, $DE$, $EF$, $GH$, and $HK$ be equal straight lines.
And let $AC$, $DF$, and $GK$ have been joined.
I say that that it is possible to construct a triangle out of (straight lines) equal to $AC$, $DF$, and $GK$ - that is to say, that (the sum of) any two of $AC$, $DF$, and $GK$ is greater than the remaining (one).

fig22e