Problem 2. When lines $a$ and $b$ intersect, one of the angles formed is seven times larger than the other. At the intersection point, a normal is drawn to line $a$. Calculate the angle between lines $b$ and $c$.
Solution. Let's denote them as $\alpha$ and $\beta$, the acute and obtuse angles respectively, that are formed by the intersection of lines $a$ and $b$ (see diagram). From the condition, we have $\beta=7 \alpha$, and since $\alpha+\beta=180^{\circ}$, it follows that $8 \alpha=180^{\circ}$, hence

![](https://cdn.mathpix.com/cropped/2024_06_05_dc4d34c0a9dd2a3ba572g-03.jpg?height=223&width=341&top_left_y=1571&top_left_x=1144)

$$
\alpha=180^{\circ}: 8=22.5^{\circ}=22^{\circ} 30^{\prime}
$$

Finally, the angle between lines $b$ and $c$ is

$$
90^{\circ}-\alpha=90^{\circ}-22^{\circ} 30^{\prime}=67^{\circ} 30^{\prime} \text {, or } 90^{\circ}+\alpha=90^{\circ}+22^{\circ} 30^{\prime}=112^{\circ} 30^{\prime}
$$