Congruence of triangles

Two triangles are congruent if both their corresponding sides and angles are equal.

In a simpler way, two triangles are congruent if they have the same shape and size, even if their position and orientation are different. The corresponding parts of congruent triangles are congruent.

Look at the following figure:

Figure 1. Congruent triangles.

In this figure we can see the triangles $\Delta ABC$ and $\Delta DEF$

If we translate on of the triangles and put it over the other, we can see they match perfectly, that's what congruence is about, each pair of sides and each pair of angles are equal in measure, we don't say that triangles $\Delta ABC$ and $\Delta DEF$ are equal, we say the are congruent and we use the following special notation $\Delta ABC \cong \Delta DEF$, which tells us that triangles $\Delta ABC$ \ and $\Delta DEF$ \ are congruent.

When two triangles are congruent, we can also say that each of their corresponding parts are congruent.

What we mean is that the 3 sides and 3 angles (figure 1) of both triangles are congruent, so there are $6$ congruences in two congruent triangles, if $\Delta ABC \cong \Delta DEF$ then the following is also true:

$\left\{ \begin{array}{c} AB\cong DE \\ BC\cong EF \\ CA\cong FD% \end{array}% \right\}$ Their corresponding sides are congruent.

$\left\{ \begin{array}{c} \measuredangle \alpha \cong \measuredangle \alpha \\ \measuredangle \beta \cong \measuredangle \beta \\ \measuredangle \gamma \cong \measuredangle \gamma% \end{array}% \right\} $ Their corresponding angles are congruent.
(See figure 1)

The Congruence Postulates of triangles tell us that it is not necessary to verify the congruence of all 6 pairs of elements (3 pairs of sides and 3 pairs of angles). Under certain conditions we can verify the congruence of three pairs of elements.

First Congruence Postulate of triangles (SAS)

Two triangles that have two sides and the angle between them equal are congruent. This Congruence Postulate is called Side-Angle-Side (SAS).

Figure 2. Congruence postulate SAS

In these two triangles we can see there is congruence between two sides and the angle between them, so:

$\left\{ \begin{array}{c} AB\cong DE \\ \measuredangle B\cong \measuredangle E \\ CA\cong FD \end{array} \right\}$ we can see the SAS postulate is met.

So, applying the SAS postulate, we can conclude that $\Delta ABC \cong $ $\Delta DEF$

Second Congruence Postulate of triangles (SSS)

Two triangles with three equal sides are congruent. This is known as the Side-Side-Side postulate (SSS).

Figure 3. Congruence Postulate SSS

Here we can see that

$\left\{ \begin{array}{c} AB\cong DE \\ BC\cong EF \\ CA\cong FD \end{array} \right\}$ All corresponding sides of the triangles are congruent.

So, applying the SSS postulate, we conclude that $\Delta ABC \cong \Delta DEF$

Third Congruence Postulate of triangles (ASA)

Two triangles with one equal side and two equal adjacent angles are congruent. This is known as the Angle-Side-Angle postulate (ASA).

Figure 4. Congruence Postulate ASA

In figure 4 we can see that the following congruences are met:
$\left\{ \begin{array}{c} \measuredangle B\cong \measuredangle E \\ BC\cong EF \\ \measuredangle C\cong \measuredangle F% \end{array}% \right\} $ one side and two adjacent angles are congruent.

So, through the ASA postulate, we can conclude that $\Delta ABC \cong \Delta DEF$

Fourth Congruence Postulate of triangles (SSA)

Two triangles are congruent if they have two congruent sides and the opposite angle to the longest side is also congruent. This is known as the Side-Side-Angle postulate (SSA).

Figure 5. Congruence Postulate SSA

In figure 5 we can see that the following congruences are met:
$\left\{ \begin{array}{c} AC\cong A\prime C\prime \\ AB\cong A\prime B\prime \\ \measuredangle C\cong \measuredangle C\prime \end{array} \right\} $ notice that there are congruences between two pairs of sides, so the angle that must be congruent is the opposite angle to the longest side that is congruent.

This way, through the congruence postulate SSA we can conclude that $\Delta ABC \cong \Delta A\prime B\prime C\prime$

Problems and solutions involving congruence of triangles