## Tuesday, July 21, 2015

### Unizor - Geometry3D - Symmetry - Congruence

Unizor - Creative Minds through Art of Mathematics - Math4Teens

The purpose of this lecture is to prove that symmetrical objects in three-dimensional space are congruent.

Mini-theorem 1
A centrally symmetrical counterpart to a segment is a segment of the same length.

Mini-theorem 2
A counterpart to a segment reflected relatively to a plane is a segment of the same length.

Mini-theorem 3
A centrally symmetrical counterpart to a triangle is a triangle congruent to the original.

Mini-theorem 4
A counterpart to a triangle reflected relatively to a plane is a triangle congruent to the original.

Mini-theorem 5
A centrally symmetrical counterpart to an angle is an angle congruent to the original.

Mini-theorem 6
A counterpart to an angle reflected relatively to a plane is an angle congruent to the original.

Mini-theorem 7
A centrally symmetrical counterpart to a dihedral angle is a dihedral angle congruent to the original.

Mini-theorem 8
A counterpart to a dihedral angle reflected relatively to a plane is a dihedral angle congruent to the original.

Let me add more philosophical statement that is based on the above theorems.
Since
most of complex three-dimensional objects contain elements discussed
above (like pentagon consists of five segments and five angles,
parallelepiped consists of six parallelograms and eight dihedral angles
etc.), these complex objects are transformed into congruent ones by a
transformation of symmetry (central or relative to a reflection plane).