To solve your questions regarding a regular pentagon:
Question 2
a. What are the degrees of the turns that will allow it to turn on itself?
A regular pentagon has 5 equal sides and 5 equal angles. The full rotation (360 degrees) can be divided evenly by the number of sides:
\[ \text{Degrees of each turn} = \frac{360^\circ}{5} = 72^\circ \]
Therefore, the turn angles that allow it to turn on itself are multiples of 72 degrees:
- 72 degrees (1 turn)
- 144 degrees (2 turns)
- 216 degrees (3 turns)
- 288 degrees (4 turns)
- 360 degrees (5 turns, which brings it back to the starting position)
In summary, the angles of turns that will allow the pentagon to rotate onto itself are 72 degrees, 144 degrees, 216 degrees, and 288 degrees, along with a full 360 degrees.
b. Since a regular pentagon can make turns less than 360 degrees onto itself, it has what kind of symmetry?
A regular pentagon has rotational symmetry because it can be rotated about its center by certain angles (multiples of 72 degrees) and still look the same.
In addition, it also has reflectional symmetry (also known as line symmetry), as you can draw lines through the vertices and midpoints of the sides, such that the halves on either side of the line are mirror images.
To summarize, a regular pentagon possesses:
- Rotational symmetry (specifically 5-fold symmetry)
- Reflectional symmetry (5 lines of symmetry)
Therefore, it has both rotational and reflectional symmetry.