B2. The sizes of the external angles at the hypotenuse of a right-angled triangle are in the ratio 13 : 17. What are the internal angles of this triangle?
B2. I. method

We write the ratio of the exterior angles of the triangle $\alpha^{\prime}: \beta^{\prime}=13: 17$. We consider the relationship between the interior and exterior angles of the triangle $\alpha+\alpha^{\prime}=180^{\circ}$ and the relationship between the exterior angle and the corresponding interior angles $\beta^{\prime}=90^{\circ}+\alpha$. We substitute into the ratio and get the equation $\frac{180^{\circ}-\alpha}{90^{\circ}+\alpha}=\frac{13}{17}$. We eliminate the fraction and get $17\left(180^{\circ}-\alpha\right)=13\left(90^{\circ}-\alpha\right)$. We simplify and calculate $\alpha=63^{\circ}$ and $\beta=27^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_ec35f4a613e63419eaedg-14.jpg?height=69&width=1687&top_left_y=565&top_left_x=230)

![](https://cdn.mathpix.com/cropped/2024_06_07_ec35f4a613e63419eaedg-14.jpg?height=57&width=1687&top_left_y=620&top_left_x=230)

![](https://cdn.mathpix.com/cropped/2024_06_07_ec35f4a613e63419eaedg-14.jpg?height=57&width=1687&top_left_y=668&top_left_x=230)

![](https://cdn.mathpix.com/cropped/2024_06_07_ec35f4a613e63419eaedg-14.jpg?height=72&width=1688&top_left_y=712&top_left_x=227)

![](https://cdn.mathpix.com/cropped/2024_06_07_ec35f4a613e63419eaedg-14.jpg?height=46&width=1685&top_left_y=774&top_left_x=228)

![](https://cdn.mathpix.com/cropped/2024_06_07_ec35f4a613e63419eaedg-14.jpg?height=51&width=1685&top_left_y=820&top_left_x=228)

II: method

We use the relationship for the sum of the exterior angles $\alpha^{\prime}+\beta^{\prime}+\gamma^{\prime}=360^{\circ}$ and consider the equality $\gamma=\gamma^{\prime}=90^{\circ}$. We set up the relationship $13 k+17 k+90^{\circ}=360^{\circ}$. We simplify $30 k=270^{\circ}$ and calculate $k=9^{\circ}$. We calculate $\alpha^{\prime}=13 \cdot 9^{\circ}=117^{\circ}$ and $\beta^{\prime}=17 \cdot 9^{\circ}=153^{\circ}$. Then we calculate $\alpha=180^{\circ}-\alpha^{\prime}=63^{\circ}$ and $\beta=180^{\circ}-\beta^{\prime}=27^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_ec35f4a613e63419eaedg-14.jpg?height=66&width=1687&top_left_y=1206&top_left_x=230)

![](https://cdn.mathpix.com/cropped/2024_06_07_ec35f4a613e63419eaedg-14.jpg?height=57&width=1687&top_left_y=1259&top_left_x=230)

![](https://cdn.mathpix.com/cropped/2024_06_07_ec35f4a613e63419eaedg-14.jpg?height=54&width=1685&top_left_y=1309&top_left_x=228)

![](https://cdn.mathpix.com/cropped/2024_06_07_ec35f4a613e63419eaedg-14.jpg?height=60&width=1687&top_left_y=1352&top_left_x=230)

![](https://cdn.mathpix.com/cropped/2024_06_07_ec35f4a613e63419eaedg-14.jpg?height=48&width=1687&top_left_y=1415&top_left_x=230)

![](https://cdn.mathpix.com/cropped/2024_06_07_ec35f4a613e63419eaedg-14.jpg?height=58&width=1682&top_left_y=1456&top_left_x=233)