To check whether the lines are perpendicular or parallel, we will use the following rules:
1) For parallel lines, the slopes are equal.
2) For perpendicular lines, the product of the slopes is equal to -1.
Slope of line AB:
Using
[tex]\begin{gathered} (x_1,y_1)=(2,3) \\ (x_2,y_2)=(-1,4) \end{gathered}[/tex]The slope is given as
[tex]\begin{gathered} m_A=\frac{y_2-y_1}{x_2-x_1} \\ m_A=\frac{4-3}{-1-2} \\ m_A=-\frac{1}{3} \end{gathered}[/tex]Slope of line CD:
Using
[tex]\begin{gathered} (x_1,y_1)=(-5,3) \\ (x_2,y_2)=(-4,6) \end{gathered}[/tex]The slope is given as
[tex]\begin{gathered} m_B=\frac{y_2-y_1}{x_2-x_1} \\ m_B=\frac{6-3}{-4-(-5)} \\ m_B=\frac{3}{-4+5} \\ m_B=3 \end{gathered}[/tex]Comparing both slopes, we can observe that
[tex]\begin{gathered} m_A\times m_B=-1 \\ \text{Given that} \\ -\frac{1}{3}\times3=-1 \end{gathered}[/tex]Therefore, both lines are PERPENDICULAR.