Congruence

In geometry, two objects are congruent if one of them is placed on the other, and they exactly coincide. Congruent objects are equal in all respects. They have same shape, same size, and same area.

1. Same area
2. Same shape
3. Same size
4. It preserves five invariants introduced by Flex Klein

The idea of congruent was first mention in Euclid’s Elements. In the Hilbert and SMSG model, the same idea was replicated.

1. Euclid, Book 1, Proposition 4
2. Hilbert Model, Postulate III-13
3. SMSG Model, Postulate VI-15

According to Euclidean geometry, triangle congruence theorem is mentioned in Book 1, proposition 4, that is given as below.

If two triangles have two sides and the angles contained are equal, then the triangle equals the triangle, the remaining angles and sides are respectively equal (SAS).

In Hilbert’s model, it is written in postulate 13 (congruence group III).

If, in the two triangles ABC and A′B′C′ the congruence AB≅A′B′, AC≅A′C′, ∠BAC≅∠B′A′C′ hold, then the congruence ∠ABC≅∠A′B′C′ holds (and, by a change of notation, it follows that ∠ACB≅∠A′C′B′ also holds) (SAS).

In SMSG model, it is written in postulate 15 (Congruence group VI).

Given a one-to-one correspondence between two triangles (or between a triangle and itself), if two sides and the included angle of the first triangle are congruent to the corresponding parts of the second triangle, then the correspondence is a congruence (SAS).

Other congruence theorems are

Theorem

Show that, ASA is triangle congruence theorem.

Show that, AAS is triangle congruence theorem.

Show that, SSS is triangle congruence theorem.

Show that, RHS is triangle congruence theorem.

Theorem 1: AAS




Theorem 2: ASA