Congruent Shapes Worksheet, Examples, and Definition

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Key Points about Congruent Shapes

Congruent figures are identical in every way, except for their position and orientation.
Two shapes must be able to be superimposed on each other so that they match up exactly to be considered congruent.
Congruent shapes are important in geometry, architecture, art, and fashion.

What does Congruent mean in Math?

Congruent Shapes are figures that have the same shape and the same size. Congruent Shapes also have equal side lengths and equal angle measures. You can determine if two shapes are Congruent by following a series of transformations that will prove that they are congruent. Transformations can include translations, reflections, and rotations. You can always double check to make sure that the figures are Congruent Shapes by ensuring that they are identical once they ar transformed.

Congruent shapes are an important concept in geometry that help to identify figures that have the same size and shape. In other words, congruent figures are identical in every way, except for their position and orientation. This means that if two figures are congruent, they will have the same angles, the same side lengths, and the same area.

To be considered congruent, two shapes must be able to be superimposed on each other so that they match up exactly. This is often accomplished through a combination of translations, reflections, and rotations. Congruent figures can be found in a variety of geometric shapes, including triangles, quadrilaterals, and circles.

Understanding the concept of congruent shapes is important not only in geometry but also in real life. For example, architects and engineers use congruent shapes to design buildings and structures that are symmetrical and balanced. Additionally, congruent shapes can be used to create patterns and designs in art and fashion.

Common Core Standard: 8.G.C
Related Topics: Similar Figures, Translation on a Coordinate Grid, Rotation on a Coordinate Grid, Reflection on a Coordinate Grid, Dilation on a Coordinate Grid
Return To: Home, 8th Grade

What are Congruent Figures?

Congruent figures are identical in shape and size. In geometry, two figures are said to be congruent if they have the same shape and size. This means that all corresponding angles and sides of the figures are equal. Congruent figures can be obtained by rigid motions, such as translations, rotations, and reflections.

Another definition of congruence is that if one of the figures can be obtained after a series of rigid motions of the other, the figures are said to be congruent. This also means that the sides and the angles of both these figures are exactly the same. Angles of these congruent figures with the same measure are called congruent angles.

Congruence is an important concept in geometry, and it is used to prove various theorems about triangles and parallelograms. It is also used to determine whether two figures are congruent or not.

In order to determine whether two figures are congruent, one can use the following methods:

Side-Side-Side (SSS) Congruence: If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent.
Side-Angle-Side (SAS) Congruence: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
Angle-Side-Angle (ASA) Congruence: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
Angle-Angle-Side (AAS) Congruence: If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the two triangles are congruent.

Overall, congruent figures are important in geometry, and they play a key role in proving theorems and solving problems.

Congruent Figures Definition

Congruent figures are two or more shapes that have the same size and shape. This means that if one figure is placed on top of the other, they will match up exactly. In other words, congruent figures are identical in every way.

The term “congruent” comes from the Latin word “congruere,” which means “to come together.” In geometry, congruent figures come together perfectly because they have the same size and shape.

Congruent figures have several important properties. First, all of their corresponding sides are congruent. This means that if you pair up the sides of one figure with the sides of another figure, each pair will have the same length. Second, all of their corresponding angles are congruent. This means that if you pair up the angles of one figure with the angles of another figure, each pair will have the same measure.

Congruent figures can be two-dimensional or three-dimensional. In two-dimensional geometry, congruent figures are often used to prove theorems about triangles and other shapes. For example, if two triangles are congruent, then all of their corresponding sides and angles are congruent, and the two triangles are therefore equal in every way.

In three-dimensional geometry, congruent figures are used to describe shapes such as cubes, spheres, and cylinders. For example, if two cubes are congruent, then they have the same length, width, and height, and they are therefore equal in every way.

Overall, congruent figures are an important concept in geometry. They allow mathematicians to prove theorems about shapes and to describe the properties of three-dimensional objects.

Similar and Congruent Shapes

When it comes to shapes, two of the most important concepts to understand are similarity and congruence. While these terms are often used interchangeably, they actually have distinct meanings and applications.

Similarity

Similarity refers to shapes that have the same shape but different sizes. In other words, they have the same angles and proportions but are not necessarily the same size. For example, a small triangle and a large triangle with the same angles are similar shapes.

When comparing similar shapes, you can use ratios to determine how they are related. For example, if two triangles are similar, you can compare the length of their corresponding sides to find the ratio of their sizes. This ratio is known as the scale factor.

Congruence

Congruence, on the other hand, refers to shapes that are exactly the same size and shape. If two shapes are congruent, you can superimpose one on top of the other and they will match up perfectly.

To determine if two shapes are congruent, you need to compare all of their corresponding angles and sides. If they are exactly the same, then the shapes are congruent.

Differences between Similarity and Congruence

While similarity and congruence both refer to relationships between shapes, they have some key differences. Here are a few important distinctions:

Similarity applies to shapes that have the same shape but different sizes, while congruence applies to shapes that are exactly the same size and shape.
Similarity is determined by comparing the ratios of corresponding sides, while congruence is determined by comparing all corresponding angles and sides.
Similarity can be used to resize shapes while maintaining their proportions, while congruence is useful for determining when two shapes are exactly the same.

In summary, similarity and congruence are important concepts in geometry that help us understand relationships between shapes. While they are related, they have distinct meanings and applications that are important to understand.

Congruent Shapes Solution

4 Simple Congruent Shapes Examples

You have to match the original shape up to the shape that has been transformed.
To prove that they are the same shape you can use either translations, reflections, and rotations.
You must check each shape to make sure it is lined up once you are done.

When it comes to congruent shapes, there are several examples that can be used to illustrate this geometric concept. These examples include triangles, rectangles, quadrilaterals, and circles.

Triangles

In geometry, congruent triangles are those that have the same size and shape. This means that all corresponding sides and angles of the triangles are equal. For example, if two triangles have sides of length 3, 4, and 5, they are congruent because they have the same size and shape.

Rectangles

Rectangles are another example of congruent shapes. A rectangle is a four-sided figure with four right angles. If two rectangles have the same length and width, they are congruent. This means that all corresponding sides and angles of the rectangles are equal.

Quadrilaterals

Quadrilaterals are four-sided figures that can have different shapes, such as a square, rectangle, parallelogram, or rhombus. If two quadrilaterals have the same shape and size, they are congruent. This means that all corresponding sides and angles of the quadrilaterals are equal.

Circles

Circles are another example of congruent shapes. A circle is a geometric figure that is defined as the set of all points in a plane that are equidistant from a given point called the center. If two circles have the same radius, they are congruent. This means that they have the same size and shape.

In conclusion, congruent shapes are an important concept in geometry. By understanding these examples of congruent shapes, individuals can better understand how to identify and work with geometric figures.

5 Quick Congruent Figures Practice Problems

Real Life Examples of Congruent Shapes

Congruent shapes are identical in shape and size. They may appear different because one is shifted or rotated a certain way, but they are still the same shape, and all the sides of one are the same length as the corresponding sides of the other. Here are some real-life examples of congruent shapes:

1. Pages of a Book

Imagine a book that has pages with uneven measurements. It is sure to look messy to you and will not be easy to store. Every page of the book fits inside the cover perfectly only because these are conforming to a set specification or are congruent.

2. Wheels of a Vehicle

The wheels of a vehicle are congruent to each other. They are identical in shape and size, and each wheel has the same number of spokes. This is important for the smooth functioning of the vehicle, as any difference in size or shape can cause imbalance and lead to accidents.

3. Tiles on a Floor

Tiles on a floor are congruent to each other. They are identical in shape and size, and each tile has the same number of sides and angles. This is important for the aesthetics of the floor, as any difference in size or shape can make the floor look uneven and unattractive.

4. Windows on a Building

Windows on a building are congruent to each other. They are identical in shape and size, and each window has the same number of sides and angles. This is important for the symmetry of the building, as any difference in size or shape can make the building look asymmetrical and unbalanced.

5. Mirrors

Mirrors are congruent to each other. They are identical in shape and size, and each mirror has the same number of sides and angles. This is important for the reflection of images, as any difference in size or shape can distort the reflection and make it look inaccurate.

In conclusion, congruent shapes are important in our daily lives, as they ensure symmetry, balance, and accuracy in various objects and structures.

FAQ about Congruent Shapes

What are the rules for congruent shapes?

Congruent shapes are identical in shape and size. The rules for congruent shapes are that they must have the same number of sides, same angles, and same lengths. If two shapes satisfy these conditions, they are considered congruent.

How do you identify congruent shapes?

To identify congruent shapes, you need to compare their corresponding sides and angles. If all the corresponding sides and angles of two shapes are equal, then they are congruent.

What shape has congruent sides?

A square is an example of a shape that has congruent sides. All four sides of a square are equal in length, which makes them congruent.

Can different shapes be congruent?

No, different shapes cannot be congruent. Congruent shapes must have the same number of sides, same angles, and same lengths.

What is an example of a congruent shape?

An example of a congruent shape is an equilateral triangle. All three sides of an equilateral triangle are equal in length, and all three angles are equal, which makes them congruent.

How can you tell if two shapes are congruent?

To tell if two shapes are congruent, you need to compare their corresponding sides and angles. If all the corresponding sides and angles of two shapes are equal, then they are congruent.

How many shapes are congruent?

There are an infinite number of shapes that can be congruent. As long as two shapes have the same number of sides, same angles, and same lengths, they are congruent.

What shapes are not congruent?

Shapes that do not have the same number of sides, same angles, and same lengths are not congruent. For example, a rectangle and a square are not congruent because they do not have the same angles.

Congruent Shapes Worksheet Video Explanation

Watch our free video on how to solve Congruent Shapes. This video shows how to solve problems that are on our free Congruent Shapes worksheets that you can get by submitting your email above.

Watch the free Congruent Figures video on YouTube here: Congruent Shapes Video

Video Transcript:
This video is about the congruent shapes definition. You can get the similar and congruent figures worksheet used in this video for free by clicking on the link in the description below.

In talking about congruent shapes you have to understand that congruent shapes mean that the shapes have to be identical. This means that they have the same size and shape.

Problem one gives us two examples of congruent shapes with triangles. These triangles look like they are the same size and shape even though the orientation is different. Even though the triangles are rotated, it looks like this has been spun or spun over, they look like they are the same exact size and the same exact shape. Because of this they are congruent.

Our second problem gives us two rectangles the rectangles are the same shape but they are different sizes. This one is obviously bigger than this one so this would not be congruent. Number 3, same thing. We have two arrows pointing straight up they are the same shape but they are different sizes. They are also not congruent.

The second part of our congruent shape worksheet asks us to describe the sequence of transformations that result in the transformation from figure a to figure a prime. Now when they say a sequence of transformations what they’re referring to is either translation, rotations, or reflections. When it says describe the sequence of transformations, what they want you to do is to look at figure a and explain in terms of translations, rotations, and reflections, how you move from shape a to a prime.

Looking at the first problem in the second part of our congruent shapes worksheet, we have figure a here, which I’m going to outline in red, and you have to get it to figure a prime, which is over here which I’m going to use blue for. In order to go from one to the other you have to use either rotation, reflection, or a translation. We can eliminate a couple of those just by looking at this problem. A translation is a slide which means the whole shape would slide either left or right or up or down. And you can tell if you slide this it’s not going to line up. We can go ahead and eliminate translation.

The easiest way to transform this from this shape into this shape is to reflect across the Y axis. If we take our figure and we reflect across the Y axis all the points will line up. In order to double check this we can count how many spaces away from the y axis each point is. This is three so if you go in this direction that is also three.

And then down bottom we’re going to count three over and then if you go this way I count three over and it’s here. This is eight away and then this will also be eight away and then this will also line up with this over here. We can use a reflection across the y-axis to translate our figure a into a prime. The solution for this first problem is that we have to reflect over the y-axis and is a congruent shapes examples.

Problem three gives us two triangles. Now we have to ask ourselves what are congruent shapes before transforming figure a into figure a Prime. We can go ahead and outline a in red. You can see it better if we were to translate this straight up. This would not work because we have to flip the shape around has to be spun.

What we’re going to do is we’re going to reflect across the x-axis this time so that we change the orientation of the shape. We’ll reflect and when we reflect we’re going to count the spaces away from the x-axis each point is or each vertices. This will be one two three four five six so you go now this time when we draw our triangle. It does not line up perfectly with our new figure so we still have to move it one more time.

What we’re going to do is we’re going to take our shape and we’re going to move it to the right two and then down one. Each vertices will go right to and then down one and then right two and then down one. Our new and final shape will be here. Our solution will be we first have to reflect across the x-axis and then translate x + 2 y -1 because we went right 2 and then down 1 and that’s our solution. Hopefully this help you understand what is a congruent figure and you can try these practice problems on the free congruent figures worksheets above.