If with any straight line, and at a point on it, two straight lines not lying on the same side make the adjacent angles equal to two right angles, the two straight lines will be in a straight line with one another.

For with any straight line $AB$, and at the point $B$ on it, let the two straight lines $BC$, $BD$ not lying on the same side make the adjagent angles $ABC$, $ABD$ equal to two right angles;I say that $BD$ is in a straight line with $CB$.For if, $BD$ is not in a straight line with $BC$, let $BE$ be in a straight line with $CB$.Then, since the straight line $AB$ stands on the straight line $CBE$,the angles $ABC$, $ABE$ are equal to two right angles. [Prop. 1.13]But the angles $ABC$, $ABD$ are also equal to two right angles;therefore the angles $CBA$, $ABE$ are equal to the angles $CBA$, $ABD$. [Post. 4] and [C.N. 1]Let the angle $CBA$ be subtracted from each;therefore the remaining angle $ABE$ is equal to the remaining angle $ABD$, [C.N. 3]the less to the greater: which is impossible.Therefore $BE$ is not in a straight line with $CB$.Similarly we can prove that neither is any other straight line except $BD$.Therefore $CB$ is in a straight line with $BD$.Therefore etc.Q.E.D.