If a straight line stands on a straight line, then it makes either two right angles or angles whose sum equals two right angles.

Let any straight line *AB* standing on the straight line *CD* make the angles *CBA* and *ABD.*

I say that either the angles *CBA* and *ABD* are two right angles or their sum equals two right angles.

Now, if the angle *CBA* equals the angle *ABD,* then they are two right angles.

But, if not, draw *BE* from the point *B* at right angles to *CD.* Therefore the angles *CBE* and *EBD* are two right angles.

Since the angle *CBE* equals the sum of the two angles *CBA* and *ABE,* add the angle *EBD* to each, therefore the sum of the angles *CBE* and *EBD* equals the sum of the three angles *CBA, ABE,* and *EBD.*

Again, since the angle *DBA* equals the sum of the two angles *DBE* and *EBA,* add the angle *ABC* to each, therefore the sum of the angles *DBA* and *ABC* equals the sum of the three angles *DBE, EBA,* and *ABC.*

But the sum of the angles *CBE* and *EBD* was also proved equal to the sum of the same three angles, and things which equal the same thing also equal one another, therefore the sum of the angles *CBE* and *EBD* also equals the sum of the angles *DBA* and *ABC.* But the angles *CBE* and *EBD* are two right angles, therefore the sum of the angles *DBA* and *ABC* also equals two right angles.

Therefore *if a straight line stands on a straight line, then it makes either two right angles or angles whose sum equals two right angles.*

Q.E.D.

But that’s not the only addition that occurs here. Euclid also says that the sum of the angles *CBE* and *EBD* equals the sum of the three angles *CBA, ABE,* and *EBD.* That sum being mentioned is a straight angle, which is not to be considered as an angle according to Euclid. It is a formal sum equal to two right angles. In other propositions formal sums of four right angles occur. These and larger formal sums are not angles themselves, merely sums of angles. Only if an angle sum is less than two right angles can it be identified with a single angle.

Clark University