Let A(1, 3) and B(5, 1) be two points. If a line with slope m intersects AB at an angle of $$45^\circ$$, then the possible values of m are

Slope of line AB = $$m_{AB}=\dfrac{1-3}{5-1}=-\dfrac{2}{4}=-\dfrac{1}{2}$$. And, the slope of the other line is m. The angle between the two lines is 45 degrees, and we know the formula -

$$\tan\theta=\left|\dfrac{m_2-m_1}{1+m_1m_2}\right|$$

$$\tan\left(45\right)=\left|\dfrac{m-\left(-\frac{1}{2}\right)}{1+m\left(-\frac{1}{2}\right)}\right|$$

$$1=\left|\dfrac{2m+1}{2-m}\right|$$

$$\left|2m+1\right|=\left|m-2\right|$$

There are two possible cases for this.

Case-1: $$2m+1=m-2$$ => $$m=-3$$

Case-2: $$2m+1=-m+2$$ => $$3m=1$$ => $$m=\dfrac{1}{3}$$

Thus, the two possible values for m = -3 and m = 1/3.