Question

If the angles of a right right triangle at A form a Geometric progression, what would be the ratio?

Answer 1

I will assume you meant a right-angled triangle.

Since it is a right-angled triangle, the largest angle must be 90°
So let the angles be
a , ar, and ar^2
a + ar + ar^2 = 180
and ar^2 = 90
a = 90/r^2

sub into the first equation
a(1 + r + r^2) = 180
90/r^2 ( 1 + r + r^2) = 180
1/r^2 + 1/r + 1 = 2
times r^2
1 + r + r^2 = 2r^2
r^2 - r - 1 = 0
r = (1 ± √5)/2 , but since the angles are positive we want
r = (1 + √5)/2

which just happens to be one of my favourite numbers, namely the Golden Ratio

btw, the 3 angles would be
34.377° , 55.623° and 90°