Congruence in Triangles

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Introduction:

Two geometrical figures are said to be congruent if they are identical in every respects.

For example, two squares of the same side-length are congruent, as shown below:

Congruent squares

Note: If a figure \(A\) is congruent to a figure \(B\), we will write this fact as follows: \(A \cong B\). Notice the following obvious facts about congruence:

  1. If \(A \cong B\), then \(B \cong A\). This simply means that saying that \(A\) is congruent to \(B\) is the same as saying that \(B\) is congruent to \(A\).

  2. If \(A \cong B\) and \(B \cong C\), then \(A \cong C\). This should once again be easy to understand. If \(A\) is congruent to \(B\), it can be superimposed exactly on \(B\). Similarly, \(B\) can be superimposed exactly on \(C\). Thus, \(A\) can be superimposed exactly on \(C\), and hence \(A \cong C\).


Congruence in Triangles:

Two triangles will be congruent if they can be superimposed upon each other exactly. This means that if two triangles are congruent, then:

  1. the three sides of one triangle will be (respectively) equal to the three sides of the other.

  2. the three angles of one triangle will be (respectively) equal to the three angles of the other.

Consider the following figure which shows two congruent triangles:

Two congruent triangles

We will write this congruence relation as follows: \(\Delta ABC \cong \Delta DEF\). Note this carefully. In the congruence relation, the vertex \(A\) corresponds to the vertex \(D\), the vertex \(B\) corresponds to the vertex \(E\), and the vertex \(C\) corresponds to the vertex \(F\).

Note: The order in which you write the vertices in a congruence relation is extremely important. For example, you cannot write the above congruence relation as \(\Delta ABC \cong \Delta DFE\), even though the triangles on both sides are still the same as earlier.

As another example, consider the following two congruent triangles:

Example two congruent triangles

yesChallenge: How will you write the congruence relation for above figure?

Tip: The order of the vertices should be corresponding. The vertex \(S\) corresponds to the vertex \(L\), the vertex \(G\) corresponds to the vertex \(J\).


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