I am trying to solve a homework problem and as part of this problem, I need to show that a certain situation is impossible. I have the following situation. Given some line, I have $2$ points $H$ and $I$ on a perpendicular to said line. The rays traced from $H$ and $I$ both intersect the line at the same points $K$ and $L$ as follows:
My hypothesis is that $\angle KHL = \angle KIL$. Intuitively, I know that this forces the $2$ triangles $\triangle KHL$ and $\triangle KIL$ to be equal, but I can't seem to be able to show it. I tried using the law of sines since I know the opposite angle to the side I know has equal length for both triangles, but since I don't know any other side lengths I couldn't seem to continue. I also tried using Pythagoras on triangles $\triangle KJI$ and $\triangle KJH$, but I ran into a similar problem of not knowing how to relate the same angle hypothesis into this.
Is there a simple argument as to why these two triangles have to be equal that I'm missing? Or could anyone point me in the right direction? Thank you!
Hint: $\angle KIJ = \angle HKI + \angle KHJ$.
Conclude that $ \angle HKI + \angle HLI = 0 $, but these angles have the same sign, so must both be 0.