Task
In the picture given below, line segments AD and BC intersect at X.
Line segments AB and CD are drawn, forming two triangles AXB
and CXD.
D
B
In each part (a)-(d) below, some additional assumptions about the
picture are given. In each problem, determine whether the given
assumptions are enough to prove that the two triangles are similar;
and if so, what the correct correspondence of vertices is. If the two
triangles must be similar, prove this result by describing a
sequence of similarity transformations that maps one triangle to
the other. If not, explain why not.
a. The lengths AX and XD satisfy the equation 2AX = 3XD.
b. The lengths AX, BX, CX, and DX satisfy the equation AX = DX
c. Lines AB and CD are parallel.
d. Angle XAB is congruent to angle XCD.