Consider the following figure with $D B$ and $D C$ being the tangents to $\Gamma$ at $B$ and $C$ respectively. Additionally, we have $\widehat{A B C}=62^{\circ}$ and $\widehat{A C B}=43^{\circ}$. Find the measure of the angle $B D C$.

![](https://cdn.mathpix.com/cropped/2024_05_10_bca2b404556ff27bef16g-42.jpg?height=502&width=1031&top_left_y=403&top_left_x=455)
As the sum of the angles in a triangle is $180, \widehat{B A C}=180^{\circ}-$ $62^{\circ}-43^{\circ}=75^{\circ}$. According to the tangent theorem, $\widehat{C B D}=\widehat{B C D}=75^{\circ}$. Therefore, $\widehat{B D C}=$ $180^{\circ}-75^{\circ}-75^{\circ}=30^{\circ}$.