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

Parallelism

In Euclidean Geometry, parallelism is a geometrical reasoning based on Euclid’s fifth postulate. It is

1. One of the invariant introduced by Flex Klein (the five invariants are: distance, angle, parallelism, straight-ness, continuity)
2. Origin of the birth of non-Euclidean Geometry
3. Demarcation of Euclidean and Non- Euclidean Geometry
4. Extends Euclidean to Projective Geometry

The idea of parallelism are mention in following works

1. Euclid: Fifth postulate
2. Playfair axiom
3. Hilbert: postulate 14
4. Brikhoff: postulate 4
5. SMSG: postulate 16

The details are given below.

Euclid’s Fifth postulate

If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Play-fair

Let l is a line and P be a point not on l, then there is at most one line on P parallel to l.

Hilbert-IV-14

Let l be any line and P be a point not on it. Then there is at most one line in the plane, determined by l and P, that passes through P and does not intersect l.

Brikhoff’s 4

Given two triangles ABC and A'B'C' and some constant k > 0, 

if

d(A', B' ) = kd(A, B), 

d(A', C' ) = kd(A, C and 

<B'A'C'  = <BAC, 

then

d(B', C' ) = kd(B, C), 

<C'B'A'  = <CBA, nd 

<A'C'B'  = <ACB

SMSG Group VI, 16

Through a given external point there is at most one-line parallel to a given line.