Tilings Of Right Trapezoids With Similar Triangles

We say that a triangle Δ tiles the polygon P, if P can be decomposed into finitely many non-overlapping triangles similar to Δ. A tiling is called regular if there are two angles of the triangles, say a and β, such that at each vertex V of the tiling the number of triangles having V as a vertex and having angle a at V is the same as the number of triangles having angle β at V. Otherwise the tiling is called irregular.Denote by R(δ) the right trapezoid with acute angle δ. In this paper we discuss the tiling of right trapezoid with with similar triangles. First, We consider the regular tilings of convex polygons with similar triangles and prove the following theorem.Theorem2.4Right trapezoid R(8) has not any regular tiling with similar triangles.Second, we discuss the irregular tilings of right trapezoid R(δ) with similar triangles, where, we discuss the right triangle and non-right triangle respectively. We prove the following theorems.Theorem3.3If the R(δ) has an irregular tiling with similar right triangles of angles (α,β,π/2), then, with a suitable permutation of a and β, we get (a,β)=(δ,π/2-δ) or (α,β)=(δ/2,π/2-δ/2).Theorem3.4If a non-right triangles T’ tiles a right trapezoid, then the angles of T’ are given by one of the four triples (π/6, π/6,2π/3),(π/8,7r/4,5π/8),(π/4,π/3,5π/12), and (π/12,π/4,2π/3).