Triangle Congruence
Theorem
Triangle Side-Angle-Side Congruence
If $2$ triangles have:
- $2$ sides equal to $2$ sides respectively
- the angles contained by the equal straight lines equal
they will also have:
- their third sides equal
- the remaining two angles equal to their respective remaining angles, namely, those which the equal sides subtend.
Triangle Side-Side-Side Congruence
Let two triangles have all $3$ sides equal.
Then they also have all $3$ angles equal.
Thus two triangles whose sides are all equal are themselves congruent.
In the words of Euclid:
- If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they will also have the angles equal which are contained by the equal straight lines.
(The Elements: Book $\text{I}$: Proposition $8$)
Triangle Angle-Side-Angle Congruence
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 the included sides are equal, then the triangles are congruent.
Triangle Side-Angle-Angle Congruence
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.
Triangle Right-Angle-Hypotenuse-Side Congruence
If two right triangles have:
- their hypotenuses equal
- another of their respective sides equal
they will also have:
Ambiguous Case for Triangle Side-Side-Angle Congruence
Let $\triangle ABC$ be a triangle.
Let the sides $a, b, c$ of $\triangle ABC$ be opposite $A, B, C$ respectively.
Let the sides $a$ and $b$ be known.
Let the angle $\angle B$ also be known.
Then it may not be possible to know the value of $\angle A$.
This is known as the ambiguous case.