In the field of mathematics, specifically in geometry, congruence and similarity are considered as two related terms. Two shapes are said to be congruent if they are of the same shape and size, Whereas similar shapes have the same shape, but they are not necessarily of the same size.

## **Congruent vs Similarity**

The difference between congruent and similarity is that congruent figures can overlap each other, but similar figures are smaller or larger than one another. They differ in shape and size. Congruent figures are superimposing while similar figures resemble each other.

The congruent objects have a different orientation from the physical coordinates in a three-dimensional space. For instance, the two equilateral triangles have the same length, and their sides are also the same.

The similarity is about two objects having the same shape but not necessarily the same size. Similarities can be found in many cases where the two objects could be very closely compared and are also precise. For example, two circles are always similar because they have the same shape. If the circles have radii of different lengths, however, they are not said to not be congruent.

## **Comparison Table Between Congruent and Similarity**

Parameters of Comparison | Congruent | Similarity |

Definition | All sides and angles are of equal measurement | The shape is similar, but the size is not the same |

Orientation | It can be different, but figures will superimpose | Shapes would match but will not superimpose |

Principle | Congruent figures follow the mathematical principle of the S.S.S theorem, where the measurements of all sides and angles are the same. | No such rule is followed by a similar or identical figure. The shape, sides, and angles of the two figures can be different. |

Precision | Congruent figures are geometrically very precise and superimposing figures. | Similar is the term to define the identical figures that bear a resemblance to each other in shape largely. |

Meaning in broader terms | Congruent can also be used as the adjective to describe the objects or experiences showing accordant or agreeing. | It can be used as an adjective to describe objects and things that are similar. |

**What is Congruent?**

The word congruent derives from the Latin word ‘congruo’, which means ‘I agree’. Two congruent objects can match or map exactly to each other. They are of the same size and have the same shape. They follow the theorem of side/side/side where all sides are the same, and all the angles are the same. The figures can be superimposed on one another and can be shown in a different way on a plane or 3D space.

In a 3D space, figures may show multiple special coordinates and maybe orientate differently around XYZ axes. However, they can still superimpose themselves by rotating their figure. The two congruent shapes can move at different angles and are also easily mapped by translation, rotation, and reflection of the shape.

Congruent objects are meticulous in measurement and shape, and size. At first glimpse to the uninformed, the two compared shapes may look to be different because of the way of placing them. However, when they are mapped or rotated, they are exact replicas of each other, or we can say a mirror image of each other and hence will be congruent.

**What is Similarity?**

The word similarity derives from the Latin word ‘similis’, which means like, resembling, or similar. The similarity is about two objects having the same shape but not necessarily the same size.

Two different right-angle triangles, for instance, are similar, but their size can be different. They can be called similar shapes but not mapped to one another. Similarly, two similar objects will have the same shape, but the one may be a scaled-up or a scaled-down form of the other. The orientation of the shape could be different, but they will remain similar to each other.

Similarities can be found in many cases where the two objects could be very closely compared and are also precise. Similarities can be related to nature and have a natural connection to the surroundings. Leaves on the same tree would be similar but could be different in size and color in autumn. Objects that are similar to one another are similar in quantity and character.

**Main Differences Between Congruent and Similarity**

- Congruence is more likely a mathematical term in geometry where two shapes are of the exact size, shape, and form with all sides and angles are of the same measurement, whereas Similarity is the term that refers to the figures that look alike, but their dimensions are different.
- Congruent figures can superimpose each other, but their orientation can vary, so the shape is set in, rotated, and fitted into each other, whereas similar figures remain the same, need not be rotated.
- Congruence can be an adjective used to define objects or ideas that can be superimposed or coincidental. On the other hand, the similarity is a term used in daily life which encounters similar objects and activities.
- Congruent objects or experiences are not easily applied in everyday human resources, whereas similarities are part of life and are a comparable term when we look around us both mathematically and in art and literature.
- Groups of objects or classes of animals can be similar though they don’t need to overlap each other. For example, in the case of cats, their breed, color, and habitat would be different, but the species will be the same, never congruent.

**Conclusion**

Congruent and similar are the terms of the mathematical and the geometric field. Congruent figures are equal in dimension and can superimpose each other, whereas similar figures look identical but do not superimpose each other.

Incongruent figures of the two shapes can be congruent if at different locations on the same plane, with different orientations, if the distance between any two points is the same as the distance between the corresponding two points. For similarity to the dimensions need to be the same, the objects may be scaled up or scaled-down. The orientation and location of two objects are often different, but it has no impact on similarity.

**References**

- https://journals.aom.org/doi/abs/10.5465/255434
- https://conbio.onlinelibrary.wiley.com/doi/abs/10.1111/j.1523-1739.2004.00337.x

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