Which of the following sets of shapes are NOT all similar to each other? (A) right triangles (B) spheres (C) \( 30-60-90 \) triangles (D) squares (E) cubes

To determine which set of shapes are **not** all similar to each other, let's analyze each option: 1. **Right Triangles (Option A):** - **Similarity Criteria:** Two triangles are similar if all their corresponding angles are equal. - **Analysis:** While all right triangles share a 90-degree angle, the other two angles can vary. For example, a right triangle with angles \(30^\circ-60^\circ-90^\circ\) is **similar** to another \(30^\circ-60^\circ-90^\circ\) triangle, but not to a right triangle with angles \(45^\circ-45^\circ-90^\circ\). Therefore, **not all right triangles are similar**. 2. **Spheres (Option B):** - **Similarity Criteria:** Spheres are similar if they share the same shape, differing only in size. - **Analysis:** All spheres are inherently similar because they are perfectly symmetrical and only differ by their radii. 3. **\(30-60-90\) Triangles (Option C):** - **Similarity Criteria:** Triangles are similar if their corresponding angles are equal. - **Analysis:** All \(30-60-90\) triangles have the same set of angles and are thus similar to each other, differing only in size. 4. **Squares (Option D):** - **Similarity Criteria:** Squares are similar if they have the same angles and proportional sides. - **Analysis:** All squares are similar since they have four equal angles and sides, differing only in size. 5. **Cubes (Option E):** - **Similarity Criteria:** Cubes are similar if they have the same shape, differing only in size. - **Analysis:** All cubes are inherently similar because they are perfectly symmetrical and only differ by their edge lengths. **Conclusion:** Only **Option A: right triangles** contain shapes that are not all similar to each other. **Answer:** Option A