Figures are mathematically similar if they share a scale factor/ratio, and also they have to have congruent corresponding angles. This is true because if there is a triangle, and both have side lengths that are related by a scale factor but don’t have angles that are congruent, the figure is not similar. In order to acheive similarity, both of these reasons have to be true.

True or False…

Any two rectangles are similar.

This is false because both of these things aren’t true in this case. For example, if you had 2 rectangles, one with the dimensions 4x4x4x4 with 90° angles since it is a rectangle, and the second rectangle was 4x18293x4x18293, with 90° angles would not be related by a scale factor to the first rectangle.

Any two equilateral triangles are similar

This is true because all sides would be the same, as would the angles. If you had 10 equilateral triangles with different measurements, they would all be similar. 1x1x1, 1.5×1.5×1.5, 2x2x2, 2.5×2.5×2.5, 3x3x3, 3.5×3.5×3.5, 4x4x4, 4.5×4.5×4.5, 5x5x5, 5.5×5.5×5.5 are all related to each other. The ratios from the first one to the rest of them are 1:1.5, 1:2, 1:2.5, 1:3, 1:3.5, 1:4, 1:4.5, 1:5, 1:5.5, and they are therefore similar because all equilateral triangles have the same angles.