1. **State the problem:** We are given points A(2, 3), B(1, -2), and point C unknown. The angle between lines AB and BC is 45° (acute). We need to find the slope of AB and then use it to find the slope of BC. 2. **Find the slope of AB:** The slope formula is $$m=\frac{y_2 - y_1}{x_2 - x_1}$$ where $(x_1,y_1)$ and $(x_2,y_2)$ are coordinates of two points. For AB, $A(2,3)$ and $B(1,-2)$: $$m_{AB} = \frac{-2 - 3}{1 - 2} = \frac{-5}{-1} = 5$$ 3. **Use the angle between two lines formula:** If $m_1$ and $m_2$ are slopes of two lines, the angle $\theta$ between them satisfies: $$\tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$$ Given $\theta = 45^\circ$, so $\tan 45^\circ = 1$. 4. **Set up the equation for $m_{BC}$:** Let $m_{BC} = m$. $$1 = \left| \frac{5 - m}{1 + 5m} \right|$$ This gives two cases: Case 1: $$\frac{5 - m}{1 + 5m} = 1$$ Case 2: $$\frac{5 - m}{1 + 5m} = -1$$ 5. **Solve Case 1:** $$5 - m = 1 + 5m$$ $$5 - 1 = 5m + m$$ $$4 = 6m$$ $$m = \frac{4}{6} = \frac{2}{3}$$ 6. **Solve Case 2:** $$5 - m = -1 - 5m$$ $$5 + 1 = -5m + m$$ $$6 = -4m$$ $$m = \frac{6}{-4} = -\frac{3}{2}$$ 7. **Determine which slope corresponds to the acute angle:** Since the angle is acute, the slopes must form an angle less than 90°. The slope of AB is 5 (positive and steep). The slope $m = -\frac{3}{2}$ is negative, which would form an obtuse angle with 5. The slope $m = \frac{2}{3}$ is positive and closer to 5, so it corresponds to the acute angle. **Final answers:** - Slope of AB: $5$ - Slope of BC: $\frac{2}{3}$