Opposite Angle Congruence

Opposite angle congruence is a property of triangles.^[2] If two angles of a triangle are equal, the sides opposite the equal angles are also equal. This property is sometimes called base angle congruence. In his book Elements, book 1 proposition 6, Euclid described this property.

Let ΔABC be a triangle having the angle ∠ABC equal to the angle ∠ACB. The claim is that side AB is equal to side BC.

Initial conditions and statement of claim.

If side AB is not equal to side AC, then one of them must be larger.

Euclid, Elements, Book 1, Common Notions, Translated and annotated by D. Joyce.

Let segment AC be the larger. Put point D on AC such that DC is equal to AB.

Draw a line segment between points D and B.

Euclid. Elements, Book 1 Postulate 1, translated by D. Joyce: A line can be drawn using any two points.

Since DB = AC, BC is in common and ∠ABC = ∠ACB by definition, triangles ΔABC = ΔDCB.

Euclid. Elements Book 1 Proposition 4: SAS Congruence.

But, if ΔABC = ΔDCA, then line segment AC = DB which contradicts the assumption that AC > DB. So AC can not be greater than AB, and can not be less than AB, so it must be equal to AB.

Euclid. Elements, Book 1, Common Notions 5, translated by D. Joyce.