The $N$-tiling of the triangle $ABC$ with the triangle $T$ is a process of cutting the triangle $ABC$ into $N$ congruent smaller triangles. The smaller triangle $T$ is called the tile. So far, little is known about the possible values of the number $N$, which is the main subject of the master's degree. When the tile $T$ is similar to the triangle $ABC$, we can prove that three forms of the number $N$ are possible. When $N$ is a perfect square, any triangle can be $N$-tiled. However, the tile $T$ is a right triangle if $N \in \{e^2+f^2, 3n^2; n, e, f \in {\mathbb N}\}$. The tile $T$ has commensurable angles if each one of them is a rational multiple of number $\pi$. Furthermore, let a triangle $ABC$ be $N$-tiled with the tile $T$, which has commensurable angles and is not similar to the triangle $ABC$. If the triangle $ABC$ is equilateral, it has $T$ angles $({\pi \over 6}, {\pi \over 3}, {\pi \over 2})︁$ or $({\pi \over 12}, {\pi \over 3}, {7\pi \over 12})︁$ and $N = 6n^2$ or it has $T$ angles $({\pi \over 6}, {\pi \over 6}, {2\pi \over 3})︁$ and $N = 3m^2$. However, if $ABC$ is an isosceles triangle with base angle $\alpha$ and tiled with the tile $T$, which is similar to one half of the triangle $ABC$, then $N$ is an even number. Moreover, the possible values of $N$ are analyzed, if not all angles of the tile $T$ are commensurable. We can prove that $N \ge 8$, when the triangle $ABC$ is $N$-tiled with the tile that is not similar to the triangle and has angles that are not all commensurable. Finally, we prove, based on above examples, that the 7-tiling of the triangle with the congruent tiles does not exist.
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