Congruence facts for kids
In geometry, two shapes or objects are congruent if they are exactly the same size and shape. Think of it like this: if you can pick up one shape and move it, turn it, or flip it over so it perfectly matches the other shape, then they are congruent.
You can't make a congruent shape bigger or smaller. If you have to change its size to make it fit, then the shapes are not congruent; they are just similar.
For example, if you draw two identical squares on a piece of paper, you could cut one out and place it exactly on top of the other. This means they are congruent!
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What Makes Shapes Congruent?
When we say shapes are congruent, it means they have the same measurements. For example:
- All squares with sides of the same length are congruent.
- All equilateral triangles with sides of the same length are congruent.
How to Check if Triangles are Congruent
There are special rules to check if two triangles are congruent without measuring every single side and angle:
- ASA (Angle-Side-Angle): If two angles and the side between them are the same in both triangles.
- AAS (Angle-Angle-Side): If two angles and a side not between them are the same in both triangles.
- SSS (Side-Side-Side): If all three sides of both triangles are the same length.
- SAS (Side-Angle-Side): If two sides and the angle between them are the same in both triangles.
How to Create Congruent Shapes
You can make a new shape that is congruent to an original one using these moves:
- Sliding (Translation): If you slide a shape across a flat surface, its size and shape don't change. The new shape is congruent to the old one.
- Turning (Rotation): If you spin a shape around a point, it stays the same size and shape. The new shape is congruent.
- Flipping (Reflection): If you flip a shape over (like looking in a mirror), its size and shape remain the same. The new shape is congruent.
You can combine these moves too! If you slide a shape, then turn it, and then flip it, the final shape will still be congruent to the original one. These are the only ways to create a congruent shape from another.
Congruence is an Equivalence Relation
The idea of congruence has some cool properties:
- It's like looking in a mirror: Any shape is congruent to itself. It's the same shape, just in the same spot!
- It works both ways: If shape A is congruent to shape B, then shape B is also congruent to shape A.
- It's like a chain: If shape A is congruent to shape B, and shape B is congruent to shape C, then shape A is also congruent to shape C.
These three properties together mean that congruence is a type of "equivalence relation" in mathematics. It helps us group shapes that are exactly alike.
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