What is the difference between congruence and similarity? Compare these two examples as part of your answer. (Use the measurements shown in the diagrams to write a well-developed answer; I'm looking for a paragraph of 3-5 sentences.)

What is the difference between congruent and similarities

Congruent shapes have identical measurements and coincide with each other when superimposed. Two congruent objects are the same size and shape, but their orientation or placing in a space can be different.  This does not change the fact that they are the same because they possess the same physical properties, the same angles as well as the same measurements.

Similarity means closely resembling each other but not quite the same.  Mathematically a shape can be similar in its basic shape, a circle for example, but different in size.  The difference in size aspect means that a similar shape can never be congruent.

What is Congruent?

The word congruent comes from the Latin word ‘congruo’ which means ‘I agree’.  When two objects are congruent they can be matched or mapped exactly to each other.  They are the same size and have the same shape.  They fit into the ‘s.s.s theorem’ of side/side/side/ all three sides are the same and all three of the angles are the same. They can be superimposed on one another but can be set differently in terms of their orientation on a plane or in a 3D space.

In a 3D space they may have different special co-ordinates and be orientated differently round their XYZ axes.  However, they are still congruent because all their sides are equal.  All their angles are equal, and their shape is the same.   The art of mapping two congruent shapes is based on the translation, rotation and reflection of the shape and the shape should be able to move through different angles or turn over to be mapped exactly.

Congruent objects are exact in measurement and shape and size.  At first glance to the uninformed the two compared shapes may appear to be different because of the way they have been placed.  However, when they are mapped or rotated they are exact replicas of each other and therefore will be congruent.

What is Similarity?

The word similarity comes from the Latin ‘similis’ which means like, resembling or similar.  Similarity in the mathematical world requires two objects to have the same shape but not necessarily the same size.

Two different circles for instance are both circles and therefore similar but their size makes them different.  They can be compared as similar shapes, but not mapped to one another. Two objects that are similar will have the same shape but the one could be a scaled up or a scaled down version of the other.  The orientation of the shape could be different, but they will remain similar.  Mathematically objects are similar if they have the same shape but not necessarily the same size.

Similarities are found in instances where comparing two objects could be very closely compared, for instance Siamese twins would be very similar and definitely appear to be identical.  Similarities will be corresponding in their meaning like synonyms as they have a similar aspects and purpose. Synonyms are useful words that contribute to the diversity of our language and descriptions of people places and things.  Similarities can relate to nature and have a natural connection in their surroundings. Leaves on the same tree for example would be similar but could be different colors in autumn.  Objects that are akin to one another are similar in quantity and character. Groups of objects or classes of animals can be similar.  Cats for example are all cats, but their breed and color and habitat would make them similar in different ways, but not the same and never congruent.

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