Dissections Of Right Trapezoids With Congruent Triangles

Suppose P is a plane conVex polygon,△is a triangle of angles decomposed into finitely many non-overlapping triangles similar to △(α,β,γ),we say that P has a triangular dissection similar to △,or △can tile P.When these triangles are all congruent,we say that this triangle can congruently tile P.Consider the dissection of the convex polygon P with similar triangles of angles(α,β,γ)and suppose η is an angle of P,if the number of α at η is a,the number of β at η is b,the number of γ at η isc then we say the composition of η is aα,bβ,cγ.Let R(δ) be a right trapezoid with an acute angle δ. In the dissection of R(δ),we mean the vertices on the hypotenuse are the interior vertices of hypotenuse. The main results of this paper are as follows:Firstly,we show that R(δ) has not any dissection with congruent non-right triangles and show that R(δ) has dissection with congruent right triangles.Secondly,if R(δ) has dissection with congruent right triangles,then the right trian-gles which congruenThirdly, in the dissection ofR(δ) with congruent△(δ,π/2-δ,π/2),if angle δ is con t△(δ/2,π/2-δ/2,π/2),if angle δ is not consist of δ/2 and angle π-δis not consist of(π/2-δ/2) then δ=π/3.Lastly,considering the dissecting configuration of R(3π/8)with triangle△(3π/8,π/8,π/2) when 3π/8 is consist of three π/8,5π/8 is consist of one π/8,one π/2 or 5π/8 is consist of two π/8,one 3π/8, in the dissection the number of vertices on the hypotenuse is at least 2.In the dissect ing configuration of R(2π/5) with triangle△(2π/5,π/10,π/2),when 2π/5 is consist of four π/10,3π/5 consist of one π/10,one π/2 or 3π/5 is consist of two π/10,one2π/5,in the dissection the num ber of vertices on the hypotenuse is at least 2.Considering the dissecting configura tion of R(π/8) with triangle△(π/8,3π/8,π/2),when 7π/8 is consist of seven π/8 or 7π/8 is consist four π/8,one 3π/8 or 7π/8 is consist of three π/8,one π/2,in the dissection the number of vertices on the hypotenuse is at least 2.Consider the dissecting configuration of R(π/5) with tri angle△(π/5,3π/10,π/2),when 4π/5 is consist of four π/5,in the dissection the number of vertices on the hypotenuse is at least 2.And considering the dissecting configuration of R(π/10) with triangle△(π/10,2π/5,π/2),when 9π/10 is consist of nine π/10 or 9π/10 is consist of five π/10,one 2π/5 or 9π/10 consist of four π/10 one π/10 in the dissection the number of vertices on the hypotenuse is east 2.