Triangle Side-Angle-Angle Congruence

From ProofWiki
(Redirected from AAS)
Jump to navigation Jump to search

Theorem

If two triangles have:

two angles equal to two angles, respectively
the sides opposite one pair of equal angles equal

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

Euclid-I-26-2.png

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

Retrieved from "https://proofwiki.org/w/index.php?title=Triangle_Side-Angle-Angle_Congruence&oldid=660987"