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The (congruent) symbol is used in geometry to state that two shapes are identical to eachother in shape and size. For example, the congruent symbol is typically used in an expression like this:
In plain language, this means the triangle defined by the points , and is congruent to the triangle defined by the points , and .
The equals symbol or equal sign is used in mathematics to assert that two expressions have the same value. It is also used in boolean logic as an operator, evaluating to true or false based on the two input expressions.