SSS Congruence Theorem and Its Proof

Many high textbooks consider the congruence theorems (SSS Congruence Theorem, SAS Congruence Theorem, ASA Congruence Theorem) as postulates. This is because their proofs are complicated for high school students.  However, let us note that strictly speaking, in Euclidean Geomtery (the Geometry that we learn in high school), there are only five postulates and no others. All of other postulates mentioned in textbooks aside from these five are really theorems without proofs.

In this post, we are going to prove the SSS Congruence Theorem. Recall that the theorem states that if three corresponding sides of a triangle are congruent, then the two triangles are congruent.

Before proving the SSS Congruence theorem, we need to understand several concepts that are pre-requisite to its proof. These concepts are isometries particulary reflection and translation, properties of kites, and the transitive property of congruence. If you are familiar with these concepts, you can skip them and go directly to the proof.

Isometry

In the figure below, \triangle MNP is slid to the right forming \triangle M^\prime N^\prime P^\prime. Clearly, when you side a figure, the size and shape are preserved, so clearly, the two triangles are congruent.

translation

Sliding or translation is a form of isometry, a type of mapping that preserves distance.  In the isometry above, the preimage \triangle MNP is mapped onto  the image \triangle M^\prime N^\prime P^\prime. Notice that there is a 1-1 mapping between the objects in the preimage and the objects in the image. Each object in the preimage has exactly one image.  Also, each object in the image has exactly one preimage.

Properties of Kites

A kite is a polygon with two distinct pairs of congruent sides. In the figure below, ABCD is a kite with \overline{AB} \cong \overline{AD} and \overline{BC} \cong \overline{CD}.

kite

The diagonal \overline{AC} is a line of symmetry of the kite. This means that \triangle ABC mirrors \triangle ADC. Mirroring an image or reflection preserves distance. This means that \triangle ABC and \triangle ADC congruent. Thus, we say that a kite is reflection-symmetric.

The Transitive Properties of Congruence

Let us recall the transitive property of equality of real numbers. It says that for any real numbers a, b, and c, if a = b and b = c, then a = c. This is also true in congruence. For any figure P, Q and R,

If P \cong Q and Q \cong R, then P \cong R.

Now that we finished the prerequisite, we now prove the theorem.

The SSS Congruence Theorem

If in two triangles, three sides of one are congruent to three sides of the other, then the two triangles are congruent.

Proof

Given triangles ABC and DEF with AB \cong DE, AC \cong DF and AC \cong DF.

In proving the theorem, we will use the transitive property of congruence. We show that if a third triangle exists, and \triangle ABC is congruent to it, then \triangle DEF is also congruent to it. Let the third triangle be \triangle A^\prime B^\prime C^\prime, an image of \triangle ABC under an isometry.

To begin, since \overline{AB} \cong \overline{DE}, there is an isometry that maps \overline{AB} to \overline{DE}. So, there is a triangle A^\prime B^\prime C^\prime which is an image of \triangle ABC that has a common side with \triangle DEF.

sss congruence

Now, \overline{AC} \cong \overline{A^\prime C^\prime} and \overline{BC} \cong \overline{B^\prime C^\prime}. By the transitive property of congruence, \overline {A^\prime C^\prime} \cong \overline{DF} and \overline{B^\prime C^\prime} \cong \overline{EF}. Therefore, \triangle A^\prime B^\prime C^\prime and \triangle DEF form a kite. Since this kite is reflection-symmetric over line DE, \triangle DEF is a reflection of \triangle A^\prime B^\prime C^\prime which means that \triangle A^\prime B^\prime C^\prime \cong \triangle DEF.

So, if the three pairs of sides of \triangle ABC can be mapped onto \triangle DEF by an isometry, by the definition of congruence, \triangle ABC \cong \triangle DEF. This prove the SSS Congruence Theorem.

Reference: An old edition of Geometry (University of Chicago School Mathematics Project)

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