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Congruent Triangles

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How is being "equal" different from being "congruent" in Geometry? When talking about shapes, we can say that one side is equal to another or the respective angles are equal: this term works well when talking about lengths or another numeric value. If you say, 'These two triangles are equal,' you're not providing the full story. The next questions would be, 'What is equal? The area? The lengths of the sides?' When two or more triangles have all equal areas, angles, and side lengths, we can more precisely describe them as **congruent**.

Congruent triangles have the same shape and size, with equal sides and angles, but they can be positioned differently from each other in space. When talking about congruent triangles, there must be two or more triangles in order to compare them with each other. You can't evaluate congruence on one triangle because it will always be congruent to itself! Let's look at an example that compares two triangles.

Imagine you have a right triangle, and your friend is sitting on the opposite side of a table from you with a copy of your triangle. You both put your triangles on the table with the right angle on the left side like this:

These two triangles are congruent: they are the same size and shape. With a turn and a drag, the triangles can precisely overlap each other, like so:

When two or more triangles can overlap each other exactly, we know that they are congruent.

Given the example above, can you define congruent triangles? Try comparing your definition with the following one:

**Congruent triangles** are triangles of the same shape and size. However, they can be positioned differently in space.

Let's see a different example.

Three triangles are positioned differently from each other. One of them is also oriented differently; that is, it is rotated relative to the others. Just by looking at them, do you think these triangles are congruent? Take a look at the given triangles in the picture below.

The first two triangles on top look congruent, right? They are the same shape and size as each other. The third triangle on the bottom may look a bit different from the first two because of the way it's oriented. If you were to rotate the third triangle 70° clockwise, you could more easily see that it is in fact congruent to the other two: it's the same shape and size. See the picture below.

From this example, we see that triangles can still be congruent even if they are rotated or oriented differently in space. The same goes for congruent triangles that are flipped (reflected) or slid over (translated).

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).

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, which is shown by a single curved mark. Double and triple curved marks are used as a notation to show that and , respectively. Note that the sign is used to show that the triangles are congruent. For the above figure, we can say that ; that is, triangle ABC is congruent to triangle DEF.

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.

Non-congruent triangles are triangles differing in shape and/or size relative to each other.

Let's look at an example.

Two triangles are given in the picture below. Do they look congruent?

**Solution:** This case is pretty evident: the given triangles are different in shape and size, no matter how you move or rotate them. This means the given triangles are non-congruent.

Check whether the given triangles are congruent or not.

**Solution:** Based on the triangles' angle notations, we can see that By looking at the figure, it may seem as though the triangles are congruent. However, the triangle ABC is bigger than the triangle DEF. Although the triangles have the same shape, they are not congruent. When we drag triangle DEF onto triangle ABC, they don't fit each other exactly. This is because their sides' lengths are not equal.

Triangle DEF fits inside triangle ABC. Hence, the triangles have the same shape but different sizes, making them **non-congruent triangles**. They are, however, **similar**!

**Similar triangles** are triangles that have the same shape (due to their equal corresponding angles) but are different in size.

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.

Two triangles are positioned right next to each other. The second one is rotated. We need to find out whether these two are congruent.

First, let's rotate the second triangle, so it's positioned in the same orientation as the first triangle. This step is not mandatory, but it helps to see the shapes more clearly and compare the right sides and angles.

Next, let's measure the angles and sides of both triangles. To understand our results more clearly, let's name the triangles ABC and DEF.

The measurements for the given triangles are:

By comparing the triangles' measurements, we can see that they are all equal. This means the triangles are congruent. We denote this with the following symbol:

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.

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.

Theorem Name | Statement | Figure |

Side-Side-Side(SSS) | The triangles are congruent if... all three sides of one triangle are equal to all three sides of another triangle. | |

Side-Angle-Side(SAS) | The triangles are congruent if... two sides and the included angle of these sides of one triangle are equal to the two sides and included angle of another triangle. | |

Hypothenuse-Leg(HL) | The triangles are congruent if... the hypothenuse side and any one side of one right triangle are equal to the hypotenuse and another side of another right triangle. | |

Angle-Side-Angle(ASA) | The triangles are congruent if... two angles and the included side of one triangle are equal to the two angles and included side of another triangle. | |

Angle-Angle-Side(AAS) | The triangles are congruent if... two angles and a non-included side of one triangle are equal to the two angles and a non-included side of another triangle. |

- Triangles are congruent if their respective sides and angles are equal (they have the same shape and size).
- Congruent triangles can be placed in different locations and rotated relative to each other.
- Non-congruent triangles differ in shape and/or size.
- There are rules or theorems that help determine whether triangles are congruent or not.

__ congruence__, it usually involves

__ equal respective sides and angles__. You can also use one of the

__ equal respective sides and angles__ are

__ equal__ respective

** example** of congruent triangles is

More about Congruent Triangles

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