Hello spectralseaSuppose that is the foot of the perpendicular from to (or produced, if necessary). (The attached diagram shows the case where meets between and .)

Let .

Then we have:

is a rectangle (all angles right-angles)Now suppose that is measured positive in the same sense as .

(opposite sides of rectangle)

(angle sum of quadrilateral )

and are supplementary angles

if and only if is acute

But

if and only if is acuteGrandad

if and only if