Congruent Triangles

Notations for congruent triangles

We know that two triangles can be congruent, but now the question is: "How can we tell which side or angle from each triangle corresponds with which?" So, let's see how equal sides and angles are marked and identified.

Typically, we mark equal sides of congruent triangles with dash-like lines, while the angles have curved markings over them. We can see these notations in the figure of congruent triangles below. Notice that different corresponding sides and angles have their own matching notations to show which one matches with which. To avoid mixing up the separate sides and angles, different pairs have differing numbers of marks (i.e., one line, two lines, three lines).

Congruent triangles, StudySmarter Originals

In the above figure, sides AB and DE are equal, so it is marked by a single dash. Similarly, sides BC and EF are equal, and sides AC and DF are equal. Also, $\angle A=\angle D,$ which is shown by a single curved mark. Double and triple curved marks are used as a notation to show that $\angle B=\angle E$ and $\angle C=\angle F$, respectively. Note that the $\cong$ sign is used to show that the triangles are congruent. For the above figure, we can say that $△ABC\cong △DEF$; that is, triangle ABC is congruent to triangle DEF.

Congruent triangles versus non-congruent triangles

It's important to note that for congruent triangles to stay congruent, we can only perform transformations of translation (location change) or rotation on any one of them. If we need to transform the shape or size (or some angles or lengths) of a triangle to make them exactly overlap one another, then the triangles are non-congruent. Let's define non-congruent triangles.

Let's look at an example.

Rules for determining congruent triangles

Now that we know the concept of congruent triangles, how can we determine if triangles are congruent? In the first two examples, we dragged and turned the triangles to see if they were congruent. You may be wondering if we can tell if triangles are congruent just by looking at them closely. However, relying on vision alone can be faulty and inaccurate. Rotating and dragging the triangles is also not the best method! There are more accurate methods for finding out if triangles are congruent.

As you might have guessed, we can determine congruency by measuring the triangles. Those with equal corresponding sides and angles are congruent. Let's see an example of this method.

It would take quite some time to measure every side and every angle whenever we wanted to determine triangle congruency. For this reason, we rely on triangle congruency theorems which help us reduce the number of measurements needed to evaluate congruency.

Congruent triangles types and theorems

Suppose you only know some information about two triangles' measurements, such as the measurements of two of their angles and the side length between those angles. If these specific measurements are equal between the two triangles, this limited amount of information is enough to prove that they are congruent. Why don't we need all of the triangles' angle measurements and side lengths to confirm it? This is because we can refer to the known triangle congruency theorem, Angle-Side-Angle (ASA), which states that our equal measurements (two angles and their shared side) are sufficient.

Depending on the type of information that we have about the triangles' measurements, we can select from the five triangle congruency theorems to help us evaluate congruency. The known theorems for triangle congruence are shown in the table below. We can also think of these theorems as shortcuts of measurement, each with their own specific rules and conditions.