All triangles are not similar because similarity requires both corresponding angles and corresponding sides to be proportional. While all equilateral triangles are similar, not all triangles have the same angle measures or side lengths.
1. Angle Measures: Triangles can have different angle measures. The sum of the interior angles in a triangle is always 180 degrees. However, the individual angle measures can vary. For example, an equilateral triangle has three equal angles of 60 degrees each. But a scalene triangle has three different angle measures, with no equal angles. Therefore, equilateral triangles are similar to each other, but not to other types of triangles.
2. Side Lengths: Triangles can also have different side lengths. In order for triangles to be similar, the corresponding sides must be proportional. This means that the ratios of the lengths of corresponding sides should be the same. For example, if one triangle has side lengths of 2, 4, and 6 units, and another triangle has corresponding side lengths of 4, 8, and 12 units, then these two triangles are similar because the ratios of corresponding side lengths are all equal (2:4:6 = 4:8:12).
However, if the corresponding side lengths are not proportional, then the triangles are not similar. For instance, consider a right triangle with side lengths of 3, 4, and 5 units. If we compare it to an equilateral triangle with side lengths of 3, 3, and 3 units, we can see that the corresponding side lengths are not proportional. Therefore, these two triangles are not similar.
Not all triangles are similar because similarity requires both corresponding angles and corresponding sides to be proportional. Triangles can have different angle measures and side lengths, which means they may not meet the criteria for similarity. It is important to consider both angle measures and side lengths when determining the similarity of triangles.