Proof: By Euclid
(related to Proposition: 11.02: Two Intersecting Straight Lines are in One Plane)
∎
^{1}
 For let the random points $F$ and $G$ have been taken on $EC$ and $EB$ (respectively).
 And let $CB$ and $FG$ have been joined, and let $FH$ and $GK$ have been drawn across.
 I say, first of all, that triangle $ECB$ is in one (reference) plane.
 For if part of triangle $ECB$, either $FHC$ or $GBK$, is in the reference [plane], and the remainder in a different (plane) then a part of one the straight lines $EC$ and $EB$ will also be in the reference plane, and (a part) in a different (plane).
 And if the part $FCBG$ of triangle $ECB$ is in the reference plane, and the remainder in a different (plane) then parts of both of the straight lines $EC$ and $EB$ will also be in the reference plane, and (parts) in a different (plane).
 The very thing was shown to be absurb [Prop. 11.1].
 Thus, triangle $ECB$ is in one plane.
 And in whichever (plane) triangle $ECB$ is (found), in that (plane) $EC$ and $EB$ (will) each also (be found).
 And in whichever (plane) $EC$ and $EB$ (are) each (found), in that (plane) $AB$ and $CD$ (will) also (be found) [Prop. 11.1].
 Thus, the straight lines $AB$ and $CD$ are in one plane, and every triangle (formed using segments of both lines) is in one plane.
 (Which is) the very thing it was required to show.
Thank you to the contributors under CC BYSA 4.0!
 Github:

 nonGithub:
 @Fitzpatrick
References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Footnotes