Constructing perpendiculars and triangles

Let segments of lengths a, b, c be given, with c the largest. To construct the triangle: lay off AB = c on a line; draw a circle of radius b centred at A and a circle of radius a centred at B; join their intersection C to A and B. Analysis: there is a solution only if the circles intersect, which needs a + b > c (the triangle inequality). If a side c and the adjacent angles α, β are given: take AB = c, lay off α at A and β at B into the same half-plane; the point where the other sides meet is C. The justification comes from the SSS or ASA criteria.