A pyramid in a prism - How to prove that this angle is right?

Question

I have been struggling to prove that angle CBE in this solid is a right angle. First of all, I thought about the possibiliy that angles AED and BEC are congruent, because they "share" the same vertex and AD = BC, this does not work however, because triangles BCE and ADE have different areas. Now, I am stuck. Could you shed some light on this problem for me?

Answer 1

The vector defined by $BC$ is $\{0,1,0 \}$ and the vector defined by $BE$ is $\{ -1,0, 1 \}$. The "dot product" of two vectors is the sum of the products of their paired components, here:

$0 \cdot (-1) + 1 \cdot 0 + 0 \cdot 1 = 0$.

This means the vectors are perpendicular.

Answer 2

$CB$, by construction, is orthogonal to plane $ABFE$, thus it is orthogonal to any line in this plane, in particular to line $BE$.