With the help of the definition of the similarity of triangles,prove that all equilateral triangles are similar.

(N/A) Two triangles are said to be similar if:
$1$. Their corresponding angles are equal.
$2$. Their corresponding sides are in the same ratio (or proportion).
Let there be two equilateral triangles,$\triangle ABC$ and $\triangle PQR$.
In an equilateral triangle,all angles are equal to $60^{\circ}$ and all sides are equal.
Therefore,for $\triangle ABC$: $\angle A = \angle B = \angle C = 60^{\circ}$ and $AB = BC = CA = a$.
For $\triangle PQR$: $\angle P = \angle Q = \angle R = 60^{\circ}$ and $PQ = QR = RP = b$.
Step $1$: Comparing angles:
$\angle A = \angle P = 60^{\circ}$,$\angle B = \angle Q = 60^{\circ}$,and $\angle C = \angle R = 60^{\circ}$.
Thus,the corresponding angles are equal.
Step $2$: Comparing sides:
$\frac{AB}{PQ} = \frac{a}{b}$,$\frac{BC}{QR} = \frac{a}{b}$,and $\frac{CA}{RP} = \frac{a}{b}$.
Thus,$\frac{AB}{PQ} = \frac{BC}{QR} = \frac{CA}{RP} = \frac{a}{b}$.
Since the corresponding angles are equal and the corresponding sides are in the same ratio,by the definition of similarity,$\triangle ABC \sim \triangle PQR$.
Hence,all equilateral triangles are similar.