To solve this question, we need to understand the concept of rotational symmetry. A regular octagon has rotational symmetry because it can be rotated around its center and still look the same. The number of degrees one must rotate a regular octagon to map it onto itself corresponds to the internal angle between two adjacent sides.
For a regular octagon with 8 sides, we can calculate the smallest rotation that will map it onto itself as \( \frac{360°}{8} = 45° \). This means that any multiple of 45 degrees would also map the octagon onto itself, because the octagon has 8 symmetrical positions as it is rotated full circle (360 degrees).
Now let's evaluate the given degrees of rotation:
- 20°: This is not a multiple of 45° (since \( 20 ÷ 45 \) does not give a whole number), hence, rotating the octagon 20 degrees will not carry it onto itself.
- 30°: Similar to 20°, 30° is not a multiple of 45°, so rotating the octagon 30 degrees will not carry it onto itself.
- 45°: Since this is exactly 45°, it is a multiple of 45° and will map the octagon onto itself (in fact, it's the smallest non-zero rotation that will do so).
- 705°: Although 705° is not immediately recognizable as a multiple of 45°, we can break it down. If we divide 705° by 45°, we get \( 705 ÷ 45 = 15.666... \) which is not a whole number. However, we know that 720° is a multiple of 45° since \( 720 ÷ 45 = 16 \). If we subtract 720° - 360°, we get 360°, which is a full rotation back to the starting position; therefore, 720° is equivalent to 0° for the purpose of rotational symmetry. So we need to see if 705° - 360° (a full rotation) gives us a multiple of 45.
\( 705° - 360° = 345° \), and if we divide 345° by 45°, we get \( 345 ÷ 45 = 7.666... \), which is also not a whole number. But if we continue to subtract full rotations (360°) until we get a number less than 360°, we'll find that \( 705° - 2 × 360° = 705° - 720° = -15° \). Since -15° is not a multiple of 45°, 705° also does not carry the octagon onto itself.
Thus, the only rotation from the list that will carry the octagon onto itself is 45°.
