If two straight lines cut one another, then they make the vertical angles equal to one another.

Let the straight lines *AB* and *CD* cut one another at the point *E.*

I say that the angle *CEA* equals the angle *DEB,* and the angle *BEC* equals the angle *AED.*

Since the straight line *AE* stands on the straight line *CD* making the angles *CEA* and *AED,* therefore the sum of the angles *CEA* and *AED* equals two right angles.

Again, since the straight line *DE* stands on the straight line *AB* making the angles *AED* and *DEB,* therefore the sum of the angles *AED* and *DEB* equals two right angles.

But the sum of the angles *CEA* and *AED* was also proved equal to two right angles, therefore the sum of the angles *CEA* and *AED* equals the sum of the angles *AED* and *DEB.* Subtract the angle *AED* from each. Then the remaining angle *CEA* equals the remaining angle *DEB.*

Similarly it can be proved that the angles *BEC* and *AED* are also equal.

Therefore *if two straight lines cut one another, then they make the vertical angles equal to one another.*

Q.E.D.

A *corollary* that follows a proposition is a statement that immediately follows from the proposition or the proof in the proposition. It is possible that this and the other corollaries in the *Elements* are interpolations inserted after Euclid wrote the *Elements.* During the writing, he could have either bundled the corollary into the proposition or made it a separate proposition.