geometry
What we normally do when we find the angle between a slanting edge and a pyramid is find angle PQE, or PAE.
But why is angle BAE $\neq$ PQE or PAE?
Line AB is also a side of a square base ABCD and AE is part of a slanting edge.
It seems odd to me. The angle must be the same?
Many thanks in advance
The angle between two planes $\alpha$ and $\beta$, having a line of intersection $r$, is defined as the angle formed by two lines $a\subset \alpha$ and $b\subset\beta$, both perpendicular to $r$ at the same point $M$.
In your case, to find the angle between $OAB$ and $OBC$ you can for instance draw the altitudes $AM$ and $CM$ of those two faces: then the angle is $\angle AMC$. To compute it you can find the lengths of $AM$, $CM$ and $AC$ (by some standard trigonometry) and then use the cosine law to get the angle.
The angle between $OAB$ and $ABCD$ is even easier: if $OK$ is another altitude of face $OAB$, then the angle is just $\angle OKH$, where $H$ is the center of square base.
Your best bet is to setup an xyz-coordinate system as shown.
After determining the co-ordinates of each point in your figure (eg B = (0, 1, 0)), write down the equation of EM and BP.
The angle between them can then be found.
http://www.nabla.hr/PC-LinePlaneIn3DSp2.htm is a website that provides examples.
Best Answer
Imagine that the height of the pyramid is very small. Then $\angle PQE$ and $\angle PAE$ each go toward 0 (though not at the same rate), whereas $\angle BAE$ goes towards $45^\circ$.