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.
AX
b. The lengths AX, BX, CX, and DX satisfy the equation A
c. Lines AB and CD are parallel.
d. Angle XAB is congruent to angle XCD.
=
DX
CX

a. In the question, we are given 2AX = 3XD. An arbitrary triangle AXB can be drawn where one extends AX and choose a point D on the extended line in order for 2AX = 3XD. Also, extend BX and choose a point C on the line so that 2BX = XC. Even though they both satisfy the condition, they're not similar.

b. We are given that AX/BX = DX/CX. When this is rearranged, we'll have AX/DX = BX/CX. In this case, let AX/DX = k. DX and AX will align upon rotation of 180°. The triangle can then be dilated by a factor of k. This brings about the mapping of triangle DXC to AXB. Therefore, the original triangle DXC is similar to AXB.

c. Lines AB and CD are parallel. Here, the dilation moves line CD onto line AB. Here, since the dilation moves D to a point on ray XA, D must move to A. Therefore, the rotation of triangle DXC is similar to AXB. Therefore, DXC is similar to AXB.

d. Here, angle XAB is congruent to angle XCD. This shows similarity and in this case, XCD is similar to XAB.

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