In the given figure, how many pairs of lines are perpendicular to each other?

1. Identify the vectors for each line segment of the polygon

Let the vertices be $A(1, 2)$, $B(2, 4)$, $C(2.5, 4.5)$, $D(2, 5)$, $E(5, 5.5)$, $F(6, 4)$, $G(5, 3)$, $H(6, 2)$, $I(6, 1)$, $J(4, 1)$.
The vectors for the segments are:
$AB = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$
$BC = \begin{pmatrix} 0.5 \\ 0.5 \end{pmatrix}$
$CD = \begin{pmatrix} -0.5 \\ 0.5 \end{pmatrix}$
$DE = \begin{pmatrix} 3 \\ 0.5 \end{pmatrix}$
$EF = \begin{pmatrix} 1 \\ -1.5 \end{pmatrix}$
$FG = \begin{pmatrix} -1 \\ -1 \end{pmatrix}$
$GH = \begin{pmatrix} 1 \\ -1 \end{pmatrix}$
$HI = \begin{pmatrix} 0 \\ -1 \end{pmatrix}$
$IJ = \begin{pmatrix} -2 \\ 0 \end{pmatrix}$
$JA = \begin{pmatrix} -3 \\ 1 \end{pmatrix}$

2. Check adjacent segments for perpendicularity using dot product or slope

Two segments are perpendicular if their dot product is zero (or slopes are negative reciprocals).

1. $BC \cdot CD = (0.5)(-0.5) + (0.5)(0.5) = -0.25 + 0.25 = 0$. So $BC \perp CD$.
2. $FG \cdot GH = (-1)(1) + (-1)(-1) = -1 + 1 = 0$. So $FG \perp GH$.
3. $HI \cdot IJ = (0)(-2) + (-1)(0) = 0$. So $HI \perp IJ$.
   Checking other adjacent pairs:
   $AB \cdot BC = 1(0.5) + 2(0.5) = 1.5 \neq 0$
   $CD \cdot DE = -1.5 + 0.25 \neq 0$
   $DE \cdot EF = 3 - 0.75 \neq 0$
   $EF \cdot FG = -1 + 1.5 \neq 0$
   $GH \cdot HI = 0 + 1 \neq 0$
   $IJ \cdot JA = 6 + 0 \neq 0$
   $JA \cdot AB = -3 + 2 \neq 0$
   There are 3 pairs of perpendicular adjacent lines.