Triangle Angle-Side-Angle and Side-Angle-Angle Equality

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Theorem

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 $26$)


Triangle Angle-Side-Angle Equality

If two triangles have:

two angles equal to two angles, respectively
the sides between the two 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 the included sides are equal, then the triangles are congruent.


Triangle Side-Angle-Angle Equality

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.


Proof

Both parts of this proposition follow trivially from the other part and Sum of Angles of Triangle equals Two Right Angles.

However, it is important to note that both of these are provable without the parallel postulate, which the proof of that theorem requires.


Historical Note

This theorem is Proposition $26$ of Book $\text{I}$ of Euclid's The Elements.


Sources