[ Rectangles and squares. Properties and characteristics ]  [ Equilateral (equiangular) triangle ]

On the side $A B$ of the square $A B C D$, an equilateral triangle $A B M$ is constructed. Find the angle $D M C$.

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Find the angles of the isosceles triangle $D A M$.

## Solution

Let point $M$ be located outside the square $A B C D$. Then the angle at vertex $A$ of the isosceles triangle $DAM$ is $90^{\circ}+60^{\circ}=150^{\circ}$, so $\angle A M D=\left(180^{\circ}-150^{\circ}\right): 2=15^{\circ}$.

Similarly, $\angle B M C=15^{\circ}$. Therefore, $\angle D M C=60^{\circ}-15^{\circ}-15^{\circ}=30^{\circ}$.

If point $M$ is located inside the given square, then similarly we find that $\angle D M C=150^{\circ}$.

## Answer

$30^{\circ}$ or $150^{\circ}$.