Congruence is a relationship between two figures that preserves both size and angle, while similarity is a relationship between two figures that only preserves size. Congruence, on the other hand, is when two figures have the same size and shape. So, two triangles are congruent if they have the same side lengths and angle measures.

## What is congruence?

In mathematics, two figures are congruent if they have the same size and shape. That means that all the sides and angles of one figure match up with the corresponding sides and angles of the other figure.

## What is similarity?

Similarity is when two figures have the same shape but may be different sizes. For example, two triangles are similar if they have the same angle measures, even if one is twice as big as the other.

## How can you determine congruence and similarity?

In order to determine congruence, you need to be able to identify the corresponding angles and sides of two figures. If the corresponding angles and sides are equal, then the figures are Congruent. As long as the figures maintain the same size and shape, they will be considered congruent.

Similarity can be determined by looking at the ratio of the corresponding sides of two figures. If the ratios are equal, then the figures are similar. The key difference between similarity and congruence is that similar figures can be different sizes, while congruent figures must be equal in size.

## How to use congruence and similarity in mathematical proofs

Congruence refers to the equality of size and shape of two figures. In other words, if two figures can be superimposed on each other, they are said to be congruent. On the other hand, similarity refers to the similarity of size and shape of two figures. In other words, if two figures have the same shape but different sizes, they are said to be similar.

The main difference between congruence and similarity is that congruence is a stronger condition than similarity. That is, all similar figures are congruent but not all congruent figures are necessarily similar.

When using either concept in mathematical proofs, it is important to be clear about which one is being used. For example, in a proof by contradiction, if we assume that two similar figures are not congruent, then we can reach a contradiction (e.g., one of the sides must be longer in one figure than the other). However, this would not work if we assumed that two congruent figures were not similar – we would simply have two different-sized copies of the same figure, which is not necessarily a contradiction.

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