Congruent Figures — Definition, Shapes & Examples
What are congruent figures?
Congruent figures in geometry are identical in shape and size. Congruent figures have congruency. That seems simple enough, but congruent figures need not be turned the same way or face the same direction to be congruent.
| Key Concept | Description |
|---|---|
| Congruent Figures | Two or more figures that are identical in both shape and size |
| Orientation | Congruent figures do not need to face the same direction |
| Congruency Preserved By | Rotation, translation, reflection |
| Congruency Broken By | Dilation (enlarging or shrinking) |
Suppose you have two playing cards from two different decks, both Queens of Spades. Everything about these cards is the same:
Size
Shape
Color
Details
If we returned them to the decks, would we know which card was from which deck? No. The two cards are congruent, meaning they are identical in size and shape. The cards are rectangles with congruent sides and congruent interior angles.
In geometry, we do not worry about color; we pay attention only to size and shape. Suppose we turn one card on its side:
They are still congruent, even though one is rotated. Rotation does not interfere with congruency.
Congruent definition in geometry
The word "congruent" is an adjective that describes two objects whose corresponding parts are identical. Congruent figures, like two equal squares, have congruency. Whether you have just two figures or a whole chessboard of congruent squares, they are all congruent.
| Term | Meaning |
|---|---|
| Congruent (adjective) | Describes objects with identical corresponding parts |
| Congruency (noun) | The state of being congruent in shape and size |
| Congruence Symbol | ≅ (used to denote two congruent figures) |
| What Must Match | All corresponding side lengths and all corresponding angles |
Transformations in geometry
In transformation geometry, we can manipulate shapes in four ways:
| Transformation | Action | Preserves Congruency? |
|---|---|---|
| Rotation | Turn the figure around a point | Yes |
| Translation | Slide the figure without turning it | Yes |
| Reflection | Flip the figure to create a mirror image | Yes |
| Dilation | Enlarge or shrink the figure | No |
Rotate (turn)
Translate (slide)
Reflect (flip)
Dilate (enlarge or shrink)
For the first three transformations, congruent figures stay congruent. Rotate the Queen of Spades, and she is still the Queen. Slide the card around the table, and our Queen is still congruent with the other playing card.
Reflect a congruent figure (turn it to a mirror image), and it is still congruent.
Learn more about the different transformations in geometry.
Similar vs. congruent
If we enlarge or shrink the Queen, it is still the same shape, but the figures are now different sizes. The shapes still have congruent angles, but the line segments that make up the card are now different lengths, so the two shapes are no longer congruent. Dilating one of two congruent shapes creates similar figures, but it prevents congruency.
Figures are similar if they are the same shape and the ratios of their corresponding side lengths are equal. So, are congruent figures similar? Technically, yes, all congruent figures are also similar shapes. But not all similar shapes have congruency.
In geometry, similar triangles are important, and three theorems help mathematicians prove if triangles are similar or congruent.
| Theorem | What It Compares |
|---|---|
| Side-Angle-Side (SAS) | Two sides and the included angle |
| Side-Side-Side (SSS) | All three corresponding sides |
| Angle-Side-Angle (ASA) | Two angles and the included side |
Congruent shapes examples
Usually, we reserve congruence for two-dimensional figures, but three-dimensional figures, like chess pieces, can be congruent, too. Think of all the pawns on a chessboard. They are all congruent.
| Figure Type | Example of Congruency |
|---|---|
| 2D Shapes | Two triangles with equal side lengths and angles |
| 2D Shapes | Squares on a chessboard |
| 3D Shapes | Identical chess pawns |
| Asymmetrical Shapes | Irregular polygons with matching sides and angles |
The geometric figures themselves do not matter. You could be working with congruent triangles, quadrilaterals, or even asymmetrical shapes. Here are two congruent figures with an example of one rotated, one translated up, and one reflected (flipped):
To summarize, congruent figures are identical in size and shape; the side lengths and angles are the same. They can be rotated, reflected, or translated and still be congruent. They cannot be dilated (enlarged or shrunk) and remain congruent.