Tyler was trying to prove that all rhombuses are similar. He thought, “If I draw 2 rhombuses, I can find dilations that will take one rhombus exactly onto the other.” Then he wrote a proof. All rhombuses are not similar. Which steps in Tyler’s proof have flaws? Explain what is wrong with each step that is wrong. Step 1: Let ABCD and WXYZ be rhombuses. By the definition of a rhombus, \large AB\cong BC\cong CD\cong DA and \large WX\cong XY\cong YZ\cong ZW.

Step 2: Dilate rhombus ABCD by the scale factor given by \large \frac{WX}{AB}. Side A'B' will be the same length as WX because of how I chose the scale factor. Because all sides of a rhombus are congruent, B'C' will be the same length as WX and therefore the same length as XY, C'D' will be the same length as YZ, and A'D' will be the same length as WZ. That means all the corresponding sides of A'B'C'D' and WXYZ will be the same length.

Step 3: If all the sides in 2 figures are proportional, then those 2 figures are similar, so there must be a sequence of transformations that take ABCD onto WXYZ using dilations and rigid motions.

Step 4: Translate A'B'C'D' by the directed line segment A'W so that A" and W coincide.

Step 5: Rotate A"B"C"D" by angle B"WX so that B'" and X coincide. Now segments A'"B'" and WX coincide.

Step 6: If needed, reflect A'"B'"C'"D'" over segment WX so that D'" and Z are on the same side of WX. Now segments A""D"" and WZ coincide.

Step 7: Once 3 vertices of the rhombus are lined up, the other vertex has to line up as well or else the shapes wouldn’t be rhombuses. So, we have shown we can use rigid motions and dilations to line up any 2 rhombuses, and therefore, all rhombuses are similar.