In Geometry, the word “congruent” means exactly the same, congruent can be taken to mean equal. When we look at circles, triangles, parallelograms or irregular shapes, we may need to determine if they are the same, or congruent.

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## Congruent Shapes

### What Is a Congruent Shape?

A congruent shape is a shape with all sides and all angles are equal. Congruent means exactly the same, so for shapes to be congruent, all sides and all angles must be exactly the same.

When considering two shapes, line them up so that the sides that look to be the same length are in the same position. They may be marked with a short straight line, the number of lines through a side should correspond with the same number of lines through a side on your second shape.

These sides are called corresponding sides. In Geometry, when looking for congruent shapes we want to find that all corresponding sides are congruent or equal.

A shape can only be proven to be congruent if you can prove that all sides and all angles are congruent.

Corresponding angles may also be marked with a curved line, or number of curved lines. These are the called corresponding angles and will be equal in a pair of congruent shapes.

Congruent shapes are often formed by a translation, or movement, of an original figure.

Three types of translations:

- Rotation
- Translation
- Reflection

All produce congruent figures, they are also known as rigid movements because they do not change the shape of the figure being moved.

Dilations change the shape of the figure and result in similar figures, which are not congruent.

A congruent shape will have all sides and angles exactly the same. Two congruent circles will have identical radii, and two congruent parallelograms will have 4 pairs of equal sides and 4 pairs of equal angles.

### List of Congruent Shapes

Pairs of any type of shape or figure can be proven congruent.

#### Triangles

Triangles are often used in problems involving congruency because of the angle sum theorem.

Triangles that are congruent will have three pairs of equal sides and three pairs of equal angles.

Triangles can be proven with three well-known theorems:

- Angle-Side-Angle
- Side-Angle-Side
- Side-Side-Side

The triangle angle sum theorem says that all triangles interior angles have a sum of 180 degrees, so proving all angles congruent does not prove that a triangle is congruent, it only proves that they are similar and their sides will have a proportional relationship.

#### Quadrilaterals

All 4 sided shapes, known as quadrilaterals, can be proven congruent.

- Rectangle
- Square
- Rhombus
- Parallelogram
- Kite
- Trapezoid (trapezium)

It is important to know the properties of each of these figures in order to prove them congruent.

For example, parallelograms have one or two sets of parallel sides, knowing the sets of congruent angles contained in parallel lines will make proving these figures congruent very simple.

#### Irregular Shapes

- When considering irregular shapes, it is very helpful to put both figures in the same orientation and to mark corresponding sides. These shapes can have any number of sides and angles.
- They may be proven congruent by proving that all sides and all angles are congruent. Marking sides and angles will help you ensure that you have not missed one!

#### Circles

- Circles can be congruent too! You can prove them congruent by proving that they have congruent radii or circumference.
- Proving circles congruent is often just one step in a larger congruency proof.