The Equality of Vertically Opposite Angles

When two straight lines intersect, the vertically opposite angles are equal.

For example, in the figure above, the sum of ∠A and ∠C is a straight angle of 180°, and the sum of ∠B and ∠C is also 180°. Subtracting the common ∠C from both sums shows that ∠A and ∠B are equal. This is a well-known basic property in geometry, and the same statement appears in Euclid’s Elements (Book I, Proposition 15).

The attribution of this property to the ancient Greek philosopher Thales comes from Proclus, who reported (quoting Eudemus) that Thales discovered it. However, since none of Thales’ writings survive, it cannot be confirmed with certainty that he proved it himself.