Triangular Gaps

There is an unfortunate gap in the triangle congruency theorems. It would be nice to be able to say that we can declare that two triangles are congruent based on a pair of sides and exactly two other bits of information, but we cannot.

If we can match up all three pairs of sides as congruent, the triangles are congruent.

If we can match up two pairs of angles and one pair of sides, the triangles are congruent.

If we can match up two pairs of sides and the angle between them, the triangles are congruent.

If we can match up two pairs of sides and a non-acute angle, the triangles are congruent.

But if we can match up two pairs of sides and an acute angle not between them, then we could be describing either of two triangles.

This gap is painful to the mathematician who prefers clear order.

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