5. In the figure on the side, the measures of the angles at vertices $A, B, D, F, G$ are indicated. The angles at vertices $C$ and $E$ have the same measure, denoted by $x$. What is the value of this measure $x$?
(A) $41^{\circ}$
(B) $39^{\circ}$
(C) $40^{\circ}$
(D) $37^{\circ}$
(E) $38^{\circ}$

![](https://cdn.mathpix.com/cropped/2024_04_17_94b06074249d24c20b4bg-1.jpg?height=392&width=575&top_left_y=356&top_left_x=2256)
(5) The correct answer is $(A)$.

Let $H, K, L$ and $M$ be the vertices of the intersection of segment $A G$ with segments $B C$, $C D$, $D E$ and $E F$, respectively. Since the sum of the interior angles of a triangle is $180^{\circ}$, it follows that the angles at $H$ (which are opposite and equal) are $73^{\circ}$, and those at $M$ are $75^{\circ}$. The angles at $K$ then equal $180^{\circ}-73^{\circ}-x^{\circ}=107^{\circ}-x^{\circ}$, and those at $L$ equal $180^{\circ}-75^{\circ}-x^{\circ}=105^{\circ}-x^{\circ}$. Consequently, by summing the interior angles of triangle $K D L$, we get $50+107^{\circ}+105^{\circ}-2 x=180^{\circ}$, which simplifies to $x=41^{\circ}$.

Question proposed by Paolo Francini.