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

With any straight line *AB,* and at the point *B* on it, let the two straight lines *BC* and *BD* not lying on the same side make the sum of the adjacent angles *ABC* and *ABD* equal to two right angles.

I say that *BD* is in a straight line with *CB.*

If *BD* is not in a straight line with *BC,* then produce *BE* in a straight line with *CB.*

Since the straight line *AB* stands on the straight line *CBE,* therefore the sum of the angles *ABC* and *ABE* equals two right angles. But the sum of the angles *ABC* and *ABD* also equals two right angles, therefore the sum of the angles *CBA* and *ABE* equals the sum of the angles *CBA* and *ABD.*

Subtract the angle *CBA* from each. Then the remaining angle *ABE* equals the remaining angle *ABD,* the less equals 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 *if with any straight line, and at a point on it, two straight lines not lying on the same side make the sum of the adjacent angles equal to two right angles, then the two straight lines are in a straight line with one another.*

Q.E.D.

This is a proposition in plane geometry. If *A, B, C,* and *D* do not lie in a plane, then *CBD* cannot be a straight line. An ambient plane is necessary to talk about the sides of the line *AB*

The qualifying sentence, “Similarly we can prove that neither is any other straight line except *BD,*” is meant to take care of the cases when *E* does not lie inside the angle *ABD.*

This is the first use of Postulate 4 that all right angles are equal.