A line of symmetry divides a figure into mirror halves. Count the mirror-fold lines in each figure: flower $6$, rangoli $4$, butterfly $1$, and none for the pinwheel or cloud.
Yes. The flower has $6$ lines of symmetry, the rangoli has $4$, and the butterfly has $1$. The pinwheel and the cloud have no line of symmetry.
The $4$ lines are the two diagonals and the two lines through the midpoints of opposite sides.
A square has $4$ lines of symmetry.
For a non-square rectangle, folding along a diagonal does not make the two triangular parts overlap exactly as mirror halves.
No.
Points on the mirror line remain fixed. Points not on the mirror line move to their mirror-image positions across that line.
Along diagonal $AC$, points $A$ and $C$ stay fixed while $B$ and $D$ exchange positions. Along the horizontal line of symmetry, the top and bottom corresponding points exchange positions.
An isosceles triangle has one symmetry line, an equilateral triangle has three, and a scalene triangle has none. If a triangle had two symmetry lines, it would force all three sides equal, giving three symmetry lines.
a. An isosceles triangle; b. an equilateral triangle; c. a scalene triangle. No, a triangle with exactly two lines of symmetry is not possible.
For $5$ equal radial arms, the smallest angle is $360^\circ\div5=72^\circ$. For $6$ equal radial arms, the smallest angle is $360^\circ\div6=60^\circ$. The remaining angles are multiples of the smallest angle up to $360^\circ$.
a. $72^\circ,144^\circ,216^\circ,288^\circ,360^\circ$; b. $60^\circ,120^\circ,180^\circ,240^\circ,300^\circ,360^\circ$.
The smallest angle is $360^\circ\div7=51\frac{3}{7}^\circ$.
$51\frac{3}{7}^\circ$; no, it is not a whole number.
Repeated rotations by the smallest angle produce the next matching positions, so all angles of symmetry are multiples of the smallest one.
Yes.
A full turn of $360^\circ$ always brings any figure back to its original position.
True.
Repeated turns by the smallest angle must exactly complete $360^\circ$, so the smallest angle must divide $360$.
True.
A regular hexagon has reflection symmetry and rotational symmetry. An equilateral triangle also has $3$ lines of symmetry and rotational symmetry of order $3$.
Examples: a regular hexagon and an equilateral triangle.
The equilateral triangle has $3$ lines and $3$ rotational angles. A non-equilateral isosceles triangle has one line of symmetry and no non-trivial rotational symmetry. A slanted parallelogram has $180^\circ$ rotational symmetry but no reflection symmetry. A kite has reflection symmetry but no rotational symmetry.
a. Equilateral triangle; b. isosceles triangle that is not equilateral; c. a non-rectangular parallelogram; d. a kite.
Angles of symmetry are multiples of the smallest angle: $2\times60^\circ,3\times60^\circ,\ldots,6\times60^\circ$.
$120^\circ,180^\circ,240^\circ,300^\circ,360^\circ$.
If there are two symmetry angles less than $60^\circ$ and $60^\circ$ is also an angle, then $60^\circ$ is the third multiple of the smallest angle. Thus the smallest angle is $60^\circ\div3=20^\circ$.
$20^\circ$.
$45^\circ$ works because $360\div45=8$. $17^\circ$ does not work because $360$ is not divisible by $17$.
a. Yes; b. No.
The outer boundary has three identical sectors arranged around the centre, giving three reflection axes and rotational symmetry every $120^\circ$.
a. Yes, $3$ lines of symmetry. b. Yes, with angles $120^\circ,240^\circ,360^\circ$.
A regular polygon with $n$ sides has $n$ lines of symmetry. For triangle through decagon, this gives $3$ through $10$.
$3,4,5,6,7,8,9,10$.
A regular $n$-gon has rotational symmetry of order $n$, so the number of angles of symmetry is $n$.
$3,4,5,6,7,8,9,10$.
The sequence begins with $3$ symmetries and then has $6$ lines and $6$ rotational angles in the later Koch snowflake stages.
Lines of symmetry: $3,6,6,6,6$. Angles of symmetry: $3,6,6,6,6$.
The Ashoka Chakra has $24$ equally spaced spokes, giving symmetry every $360^\circ\div24=15^\circ$ and $24$ corresponding reflection axes.
$24$ lines of symmetry and $24$ angles of symmetry.