How to explain the perpendicularity of two lines to a High School student? Today I was teaching my friend from High School about linear functions. One of the exercises we had to do was finding equations of perpendicular and parallel lines. Explaining parallel equations was quite easy, if we have the equation $y = ax + b$ it's not hard to show with a couple of examples that changing the parameter $b$ only "moves" the line up or down but doesn't change the angle, thus lines $k$ and $\ell$ are parallel iff $a_k = a_{\ell}$. 
However, I couldn't find a clear way to explain why those lines are perpendicular iff $a_k \times a_{\ell}= -1$. Of course, it's obvious if we use the fact that $a = \tan (\alpha)$ with $\alpha$ being the angle at which line intersects the X axis and that $\tan (\alpha) = - \cot (\frac{\pi}{2} + \alpha)$. But this forces us to introduce trigonometry and rises oh so many questions about the origin of the equation above. Does anyone know a good, simple explanation that's easy to remember?

Give a good answer to this question may be also a way to give a rigorous definition of "perpendicular"  and of what is the "measure" (in radians) of an angle (a not simple task at elementary level) without using angles measured in degrees.
I sketch the reasoning in  steps:
1) Let's $O$ the intersection point of two straight lines $r$ and $s$ and take an orthogonal coordinate system with center $O$ such that the equations of the lines are $ r  \rightarrow y=ax $;   $s \rightarrow y=bx$.
2)Take two points AB in r such that P is the middle point of AB and two points CD in the same way. Using the midpoint formula students can find that:
$$
A\equiv (x_A,ax_A) \qquad B\equiv (-x_A,-ax_A) 
$$
$$
C\equiv (x_C,bx_C) \qquad D\equiv (-x_C,-bx_C) 
$$
3) Define: the stright lines $r$ and $s$ are perpendicular iff $\overline {AC}=\overline{BC}$
4) Proof: $r$ is perpendicular to $s$ iff $ab=-1$.  We have:
$$
\overline {AC}=\overline{BC} \Rightarrow \overline {AC}^2=\overline{BC}^2 \iff 
$$
$$
\iff (x_A-x_C)^2+(ax_A-bx_C)^2=(-x_A-x_C)^2+(-ax_A-bx_C)^2 \iff
$$
and, with simple calculations ,we have
$$
 x_Ax_C(1+ab)=0
$$
so, if $x_A,x_C \ne 0$, we have $ab=-1$ 
and we have obtained the result without the use of "angles" and giving a rigorous definition of perpendicularity.
5) At last, using a circle with center $O$ and radius $ r=\overline {AB}$, and noting that equal chords subtend equal arcs, we can define the "measure" of the angle $\angle AOC$ to be (the arc) $\wideparen{AB}/r=\pi/2$. 
