a To find the measure of ∠ 1, we need to count how many times the kaleidoscope can be rotated and map onto itself before returning to its starting position. The ability of the object to rotate onto itself is called rotational symmetry.
From the diagram, we see that after 12 rotations, the kaleidoscope will return to its starting position. With this, we can use the formula.
12(m∠ 1)=180^(∘) ⇔ m∠ 1=15^(∘)
b To find the measure of ∠ 1, we need to count the number of lines of symmetry. Below we have drawn the lines of symmetry. Note that we have to avoid double-counting any of the lines.
From the diagram, we see that after 6 rotations, the kaleidoscope will return to its starting position. With this, we can use the formula.
6(m∠ 1)=180^(∘) ⇔ m∠ 1=30^(∘)
