Geometry for elementary school/The Side-Side-Side congruence theorem

Introduction

In this chapter, we will start the discussion of congruence and congruence theorems. We say the two triangles are congruent if they have the same shape. The triangles $\mathrm{△}ABC$ and $\mathrm{△}DEF$ congruence if and only if all the following conditions hold:

1. The side $\overline{AB}$ equals $\overline{DE}$.
   
   
   
   
2. The side $\overline{BC}$ equals $\overline{EF}$.
   
   
   
   
3. The side $\overline{AC}$ equals $\overline{DF}$.
   
   
   
   
4. The angle $\mathrm{\angle }ABC$ equals $\mathrm{\angle }DEF$.
   
   
   
   
5. The angle $\mathrm{\angle }BCA$ equals $\mathrm{\angle }EFD$.
   
   
   
   
6. The angle $\mathrm{\angle }CAB$ equals $\mathrm{\angle }FDE$.
   
   
   
   

Note that the order of vertices is important. It is possible that $\mathrm{△}ABC$ and $\mathrm{△}ACB$ won’t congruence though it is the same triangle.

Congruence theorems give a set of less conditions that are sufficient in order to show that two triangles congruence.

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

The Side-Side-Side congruence theorem

Given two triangles $\mathrm{△}ABC$ and $\mathrm{△}DEF$ such that their sides are equal, hence:

1. The side $\overline{AB}$ equals $\overline{DE}$.
   
   
   
   
2. The side $\overline{BC}$ equals $\overline{EF}$.
   
   
   
   
3. The side $\overline{AC}$ equals $\overline{DF}$.
   
   
   
   

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

Method of Proof

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

1. Copy The line Segment side $\overline{AB}$ to the point D.
2. Draw the circle $\circ D,\overline{AB}$.
3. The circle $\circ D,\overline{AB}$ and the segment $\overline{DE}$ intersect at the point E hence we have a copy of $\overline{AB}$ such that it coincides with $\overline{DE}$.
4. Construct a triangle with $\overline{DE}$ as its base, $\overline{BC}$, $\overline{AC}$ as the sides and the vertex at the side of the vertex F. Call this triangle triangles $\mathrm{△}DEG$

The claim

The triangles $\mathrm{△}DEF$ and $\mathrm{△}ABC$ congruent.

The proof

1. The points A and D coincide.
2. The points B and E coincide.
3. The vertex F is an intersection point of $\circ D,\overline{DF}$ and $\circ E,\overline{EF}$.
4. The vertex G is an intersection point of $\circ D,\overline{AC}$ and $\circ E,\overline{BC}$.
5. It is given that $\overline{DF}$ equals $\overline{AC}$.
6. It is given that $\overline{EF}$ equals $\overline{BC}$.
7. Therefore, $\circ D,\overline{DF}$ equals $\circ D,\overline{AC}$and $\circ E,\overline{EF}$ equals $\circ E,\overline{BC}$.
8. However, circles of different centers has at most one intersection point in one side of the segment the joins their centers.
9. Hence, the points G and F coincide.
10. Thorough two points only an straight line passes, Therefore $\overline{EG}$ coincides with $\overline{EF}$ and $\overline{GD}$ coincides with $\overline{DF}$.
11. Therefore, the $\mathrm{△}DEG$ coincides with $\mathrm{△}DEF$ and therefore congruent.
12. Due to the postulate $\mathrm{△}DEG$ and $\mathrm{△}ABC$ are equal and therefore congruent.
13. Hence, $\mathrm{△}DEF$ and $\mathrm{△}ABC$ congruent.
14. Hence, $\mathrm{\angle }ABC$ equals $\mathrm{\angle }DEF$, $\mathrm{\angle }BCA$ equals $\mathrm{\angle }EFD$ and $\mathrm{\angle }CAB$ equals $\mathrm{\angle }FDE$.

Note

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 has at most one intersection point in one side of the segment the 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 assumption, we preferred to give a more detaild proof, though the extra assumption. it:Geometria per scuola elementare/Il teorema di congruenza lato-lato-lato