Triangle Side-Angle-Angle Congruence
Theorem
If two triangles have:
then the remaining angles are equal, and the remaining sides equal the respective sides.
That is to say, if two pairs of angles and a pair of opposite sides are equal, then the triangles are congruent.
In the words of Euclid:
- If two triangles have two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that subtending one of the equal angles, they will also have the remaining sides equal to the remaining sides and the remaining angle equal to the remaining angle.
(The Elements: Book $\text{I}$: Proposition $26$)
Proof
Let:
- $\angle ABC = \angle DEF$
- $\angle BCA = \angle EFD$
- $AB = DE$
Aiming for a contradiction, suppose that $BC \ne EF$.
If this is the case, one of the two must be greater.
Without loss of generality, let $BC > EF$.
We construct a point $H$ on $BC$ such that $BH = EF$, and then we construct the segment $AH$.
Now, since we have:
- $BH = EF$
- $\angle ABH = \angle DEF$
- $AB = DE$
from Triangle Side-Angle-Side Congruence we have:
- $\angle BHA = \angle EFD$
But from External Angle of Triangle is Greater than Internal Opposite, we have:
- $\angle BHA > \angle HCA = \angle EFD$
which is a contradiction.
Therefore $BC = EF$.
So from Triangle Side-Angle-Side Congruence:
- $\triangle ABC = \triangle DEF$
$\blacksquare$
Also known as
Triangle Side-Angle-Angle Congruence is also known as:
- Triangle Angle-Angle-Side Congruence
- SAA or the SAA Condition
- AAS or the AAS Condition
Also see
Historical Note
This proof is the second part of Proposition $26$ of Book $\text{I}$ of Euclid's The Elements.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 1 (2nd ed.) ... (previous) ... (next): Book $\text{I}$. Propositions
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): congruent: 1. $(3)$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): congruent: 1. $(3)$