Answer :
To determine which of the given angles is a possible angle of rotational symmetry for a regular polygon with 15 sides, we need to follow these steps:
1. Understanding Rotational Symmetry: A regular polygon has rotational symmetry if it can be rotated around its center by a certain angle and still look the same as it did before the rotation.
2. Calculating Basic Angle of Rotation: For a regular polygon with \( n \) sides, the angles at which it looks identical are multiples of \( \frac{360^\circ}{n} \). For a polygon with 15 sides, \( n = 15 \).
[tex]\[ \text{Angle Step} = \frac{360^\circ}{15} = 24^\circ \][/tex]
3. Finding All Possible Rotational Angles: The possible angles of rotational symmetry are the multiples of \( 24^\circ \). We list these angles from \( 24^\circ \) up to \( 336^\circ \):
[tex]\[ 24^\circ, 48^\circ, 72^\circ, 96^\circ, 120^\circ, 144^\circ, 168^\circ, 192^\circ, 216^\circ, 240^\circ, 264^\circ, 288^\circ, 312^\circ, 336^\circ \][/tex]
4. Checking Given Options Against Possible Angles: We compare the given angles \( 12^\circ, 45^\circ, 72^\circ, 90^\circ \) with the calculated possible angles:
- \( 12^\circ \) is not in the list.
- \( 45^\circ \) is not in the list.
- \( 72^\circ \) is in the list.
- \( 90^\circ \) is not in the list.
Therefore, the only possible angle of rotational symmetry among the given options for a regular polygon with 15 sides is:
[tex]\[ 72^\circ \][/tex]
1. Understanding Rotational Symmetry: A regular polygon has rotational symmetry if it can be rotated around its center by a certain angle and still look the same as it did before the rotation.
2. Calculating Basic Angle of Rotation: For a regular polygon with \( n \) sides, the angles at which it looks identical are multiples of \( \frac{360^\circ}{n} \). For a polygon with 15 sides, \( n = 15 \).
[tex]\[ \text{Angle Step} = \frac{360^\circ}{15} = 24^\circ \][/tex]
3. Finding All Possible Rotational Angles: The possible angles of rotational symmetry are the multiples of \( 24^\circ \). We list these angles from \( 24^\circ \) up to \( 336^\circ \):
[tex]\[ 24^\circ, 48^\circ, 72^\circ, 96^\circ, 120^\circ, 144^\circ, 168^\circ, 192^\circ, 216^\circ, 240^\circ, 264^\circ, 288^\circ, 312^\circ, 336^\circ \][/tex]
4. Checking Given Options Against Possible Angles: We compare the given angles \( 12^\circ, 45^\circ, 72^\circ, 90^\circ \) with the calculated possible angles:
- \( 12^\circ \) is not in the list.
- \( 45^\circ \) is not in the list.
- \( 72^\circ \) is in the list.
- \( 90^\circ \) is not in the list.
Therefore, the only possible angle of rotational symmetry among the given options for a regular polygon with 15 sides is:
[tex]\[ 72^\circ \][/tex]
