Question

1. tyler is trying to prove that all rhombuses are similar. he thinks, \if i draw 2 rhombuses, i can find dilations that will take one rhombus exactly onto the other.\ then he wrote a proof. step 1: let abcd and wxyz be rhombuses. by the definition of a rhombus, ab is congruent to bc, cd, and da, and wx is congruent to xy, yz, and zw. step 2: dilate rhombus abcd by the scale factor given by \\(\frac{wx}{ab}\\). side ab will be the same length as wx because of how i chose the scale factor. because all sides of a rhombus are congruent, bc will be the same length as wx and therefore the same length as xy, cd will be the same length as yz, and ad will be the same length as wz. that means all the corresponding sides of abcd 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 abcd by the directed line segment aw so that a and w coincide. step 5: rotate abcd by angle bwx so that b and x coincide. now segments ab and wx coincide. step 6: if needed, reflect abcd over segment wx so that d and z are on the same side of wx. now segments ad 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 wouldnt 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. all rhombuses are not similar. which steps in tylers proof have flaws? explain what is wrong with each step that is wrong.

Explanation:

1. Step 3: The claim that proportional sides guarantee similarity is only true for triangles. For quadrilaterals, corresponding angles must also be congruent for similarity.
2. Step 6: There is no guarantee that $A''D''$ will align with $WZ$ after reflection. The dilated rhombus has matching side lengths but may have different internal angles, so $D''$ will not coincide with $Z$.
3. Step 7: The assumption that the fourth vertex must align is false, as the two rhombuses can have different angle measures even with congruent side lengths after dilation.

Answer:

1. Step 3: Incorrect similarity condition for quadrilaterals.
2. Step 6: Cannot guarantee $A''D''$ aligns with $WZ$ due to mismatched angles.
3. Step 7: Fourth vertex will not align for rhombuses with different angles.