slopes

Dave Hewitt (now at Loughborough University) introduced me to this problem and his impressive work in seeking a generality

the problem appears in several places on the net








draw a line of three touching squares

draw the diagonal that is (1,1) (i.e. PQ)
then the diagonal (2,1) (i.e. PR)
then the diagonal (3,1)

show that the sum of the two smaller (angle of elevation) angles (a + b) is the third angle of elevation (of PQ, i.e 45 degrees).

you aren't allowed to use trigonometry (tangent rules or sine rules)
just the basics (angles on parallel lines, angles in triangles etc)

having done that, attempt to generalise the problem:

e.g.
given an (n,1) rectangle and an (m, 1) rectangle [with m greater than n]
can you find a (k,1) rectangle so that the two smaller angles of elevation sum to the third?