Geometry for Elementary School/The Side-Side-Side congruence theorem

Geometry for Elementary School
Congruence The Side-Side-Side congruence theorem The Side-Angle-Side congruence theorem

The first congruence theorem we will discuss is the Side-Side-Side theorem.

The Side-Side-Side congruence theorem edit

Given two triangles   and   such that their sides are equal, hence:

  1. The side   equals  .

  2. The side   equals  .

  3. The side   equals  .

Then the triangles are congruent and their angles are equal too.

Method of Proof edit

In order to prove the theorem we need a new postulate. The postulate is that one can move or flip any shape in the plane without changing it. In particular, one can move a triangle without changing its sides or angles. Note that this postulate is true in plane geometry but not in general. If one considers geometry over a ball, the postulate is no longer true.

Given the postulate, we will show how can we move one triangle to the other triangle location and show that they coincide. Due to that, the triangles are equal.

The construction edit

  1. Copy The line Segment side   to the point D.
  2. Draw the circle  .
  3. The circle   and the segment   intersect at the point E hence we have a copy of   such that it coincides with  .
  4. Construct a triangle with   as its base,  ,   as the sides and the vertex at the side of the vertex F. Call this triangle triangles  

The claim edit

The triangles   and   congruent.

The proof edit

  1. The points A and D coincide.
  2. The points B and E coincide.
  3. The vertex F is an intersection point of   and  .
  4. The vertex G is an intersection point of   and  .
  5. It is given that   equals  .
  6. It is given that   equals  .
  7. Therefore,   equals  and   equals  .
  8. However, circles of different centers have at most one intersection point in one side of the segment that joins their centers.
  9. Hence, the points G and F coincide.
  10. There is only a single straight line between two points, therefore   coincides with   and   coincides with  .
  11. Therefore, the   coincides with   and the two are congruent.
  12. Due to the postulate   and   are equal and therefore congruent.
  13. Hence,   and   are congruent.
  14. Hence,   equals  ,   equals   and   equals  .

Note edit

The Side-Side-Side congruence theorem appears as Book I, prop 8 at the Elements. The proof here is in the spirit of the original proof. In the original proof Euclid claims that the vertices F and G must coincide but doesn’t show why. We used the assumption that “circles of different centers have at most one intersection point in one side of a segment that joins their centers”. This assumption is true in plane geometry but doesn’t follows from Euclid’s original postulates. Since Euclid himself had to use such an assumption, we preferred to give a more detailed proof, though the extra assumption.