I have this statement:

Are two similar rhombuses, if they have the three corresponding pairs of angles equal?

My answer was yes. Following the fundamental theorem (AA), in case of the triangle only 2 equal angles are needed, therefore in a quadrilateral, 3 equal angles will be needed.

But my answer was wrong, and this was the message:

The condition given in A) does not allow to determine that the rhombus are similar because it is also necessary that their corresponding sides are proportional

But, then this contradicts the fundamental theorem, which says that only the angles are necessary, to determine the similarity.

  • $\begingroup$ In the title, you say "triangles". In the statement, you say "rhombuses". In the next paragraph, you say "quadrilaterals". Please tidy this up! $\endgroup$ – TonyK Aug 21 '18 at 17:03
  • $\begingroup$ I believe there is a diamond there as well. $\endgroup$ – Mohammad Zuhair Khan Aug 21 '18 at 17:05
  • $\begingroup$ Sorry! are rhombus $\endgroup$ – Eduardo S. Aug 21 '18 at 17:20
  • $\begingroup$ Why do you need three angles? A rhombus has two pairs of equal angles while the total is $2\pi$, so it seems one angle would be enough assuming each figure is a rhombus. $\endgroup$ – herb steinberg Aug 21 '18 at 18:24

in case of the triangle only 2 equal angles are needed,

True. Note that this is a theorem, not a definition of similarity.

therefore in a quadrilateral, 3 equal angles will be needed

Why would you say that? How do you use the triangle to justify this?

According to your logic, all rectangles are similar. But that is not what "similar" means.

Notice that when you are given the three angles of the quadrilateral, no two of them are the angles of a triangle composed of sides of the quadrilateral. Sure, you can draw a diagonal and have constructed a triangle from two sides of the quadrilateral, but two of the angles of that triangle will be smaller than the angles of the quadrilateral at those same vertices. Knowing the angles of the quadrilateral tells you very little about those two angles of the triangle.

  • $\begingroup$ However, diagonals of rhombus are bisectors so essentially three given angles of a rhombus define all angles of the corresponding triangles, no? $\endgroup$ – Vasya Aug 21 '18 at 17:06
  • $\begingroup$ But note that two rhombuses are similar if three of their angles are equal. In fact, two rhombuses are similar if one of their angles is equal. $\endgroup$ – TonyK Aug 21 '18 at 17:07
  • $\begingroup$ @TonyK Good point, I shifted unconsciously from quadrilateral to the more specific "rhombus." I think it's fixed now. $\endgroup$ – David K Aug 21 '18 at 17:08
  • 1
    $\begingroup$ @Vasya Right, I should have stuck with general quadrilaterals. I think your observation is probably the kind of answer that was desired in that exercise. $\endgroup$ – David K Aug 21 '18 at 17:09
  • $\begingroup$ @TonyK but you should know if the two rhombus are proportional $\endgroup$ – Eduardo S. Aug 21 '18 at 20:25

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