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(C) An equilateral triangle has $3$ sides of equal length and $3$ angles of $60^{\circ}$ each.Let the vertices of the equilateral triangle be $A, B,$

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Six

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(C) An equilateral triangle has $3$ sides of equal length and $3$ angles of $60^{\circ}$ each.Let the vertices of the equilateral triangle be $A, B,$ and $C$.$A$ correspondence of a triangle with itself is a permutation of its vertices.The total number of permutations of $3$ vertices is given by $3! = 3 	imes 2 	imes 1 = 6$.These $6$ correspondences are:$1. (A 	o A, B 	o B, C 	o C)$$2. (A 	o A, B 	o C, C 	o B)$$3. (A 	o B, B 	o A, C 	o C)$$4. (A 	o B, B 	o C, C 	o A)$$5. (A 	o C, B 	o A, C 	o B)$$6. (A 	o C, B 	o B, C 	o A)$Since all sides are equal and all angles are $60^{\circ}$,any permutation of the vertices results in a correspondence that preserves the ratios of corresponding sides (all ratios are $1$) and the equality of corresponding angles. Thus,all $6$ correspondences are similarities.

$...$ correspondences of an equilateral triangle with itself are similarity.

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