SymmetryClass 6 Mathematics NCERT Solutions

Final Answer: Yes, lines of symmetry are present in some figures. The flower has 6 lines of symmetry, the rangoli has 4, and the butterfly has 1. The pinwheel and the cloud have no lines of symmetry.

A figure has more than one angle of symmetry if it has rotational symmetry (i.e., an angle of symmetry between $0^\circ$ and $360^\circ$).

Final Answer: The equilateral triangle, square, rectangle, regular pentagon, and regular hexagon have more than one angle of symmetry.

The order of rotational symmetry is the number of times a figure maps onto itself during a full $360^\circ$ rotation.

Given: A circle divided into 12 equal sectors.

Given: The smallest angle of symmetry is $60^\circ$.

Solution: All other angles of symmetry will be multiples of the smallest angle, up to $360^\circ$. The angles are: $2 \times 60^\circ = 120^\circ$ $3 \times 60^\circ = 180^\circ$ $4 \times 60^\circ = 240^\circ$ $5 \times 60^\circ = 300^\circ$ $6 \times 60^\circ = 360^\circ$

Final Answer: The other angles of symmetry are $120^\circ, 180^\circ, 240^\circ, 300^\circ, \text{ and } 360^\circ$.

Given: $60^\circ$ is an angle of symmetry. There are two angles of symmetry smaller than $60^\circ$.

Solution: Let the smallest angle of symmetry be $\alpha$. Then all other angles of symmetry must be multiples of $\alpha$. Since $60^\circ$ is an angle of symmetry, $\alpha$ must be a factor of 60. The angles of symmetry are $\alpha, 2\alpha, 3\alpha, \dots$. We are given that there are two angles less than $60^\circ$. These must be $\alpha$ and $2\alpha$. The next angle, $3\alpha$, must be greater than or equal to $60^\circ$. So, we have $2\alpha < 60^\circ$ and $3\alpha \geq 60^\circ$. From $3\alpha \geq 60^\circ$, we get $\alpha \geq 20^\circ$. From $2\alpha < 60^\circ$, we get $\alpha < 30^\circ$. So, $20^\circ \leq \alpha < 30^\circ$. Since $\alpha$ must be a factor of 60, the only factor of 60 in this range is 20.

Final Answer: The smallest angle of symmetry is $20^\circ$.

Condition: For a figure to have rotational symmetry with the smallest angle $\alpha$, $\alpha$ must be a factor of $360^\circ$.

Analysis: The outer boundary of the new Parliament Building is shaped like an equilateral triangle.

Rule: A regular polygon with $n$ sides has $n$ lines of symmetry.

Final Answer: The number of lines of symmetry for a regular n-gon is $n$. The number sequence obtained is 3, 4, 5, 6, 7, ..., which is the sequence of natural numbers starting from 3.

Rule: A regular polygon with $n$ sides has rotational symmetry of order $n$, which means it has $n$ angles of symmetry.

Final Answer: The number of angles of symmetry for a regular n-gon is $n$. The number sequence obtained is 3, 4, 5, 6, 7, ..., which is the sequence of natural numbers starting from 3.

Final Answer: The sequence for the number of lines of symmetry is 3, 6, 6, 6, .... The sequence for the number of angles of symmetry is 3, 6, 6, 6, ....

Analysis of Ashoka Chakra: The Ashoka Chakra is a wheel with 24 spokes, all equally spaced.

Final Answer: The Ashoka Chakra has 24 lines of symmetry and 24 angles of symmetry.

The kolam design is symmetrical. It has:

Therefore, the kolam has 4 lines of symmetry.

Possibility of a Triangle with Two Lines of Symmetry: No, it is not possible to draw a triangle with exactly two lines of symmetry. If a triangle has two lines of symmetry, it implies that two pairs of sides are equal and two pairs of angles are equal. This forces the third pair of sides and angles to also be equal, making the triangle equilateral, which has three lines of symmetry.

For each figure, the given drawing is in one of the four quadrants created by the vertical and horizontal lines of symmetry. To complete the figure, the drawing must be reflected into the other three quadrants.

This process creates a larger figure that is symmetrical with respect to both the vertical and horizontal blue lines.

Winning Strategy: The winning strategy is for the second player (Player 2). The strategy is based on symmetry.

To Find: The smallest angle of symmetry for a figure with 7 angles of symmetry.

Solution: If a figure has 7 angles of symmetry, it means its order of rotational symmetry is 7. The smallest angle of rotation that maps the figure onto itself is found by dividing a full circle ($360^\circ$) by the order of symmetry.

Smallest angle = $\frac{360^\circ}{7}$

To convert this to a mixed fraction, we perform the division: $360 \div 7 = 51$ with a remainder of $3$.

So, the fraction is $51\frac{3}{7}^\circ$.

Final Answer: The smallest angle of symmetry is $51\frac{3}{7}^\circ$. No, the number of degrees is not a whole number.

Analysis: A square can be folded in the following ways to make the two halves overlap perfectly:

There are no other lines along which a square can be folded to have overlapping halves.

Final Answer: No, there is no other way to fold the square. The square shape has 4 lines of symmetry.

Analysis: Consider a rectangle that is not a square. If you fold it along one of its diagonals, the two triangular halves formed will not overlap completely. The vertices of one triangle will not fall on the corresponding vertices of the other triangle.

Final Answer: No, a rectangle's diagonal is not a line of symmetry (unless the rectangle is also a square).

Given: A square with vertices labeled A (top-left), B (top-right), C (bottom-right), and D (bottom-left).

Answer: True

Reason: A rotation of $360^\circ$ is a full turn, which always brings any figure back to its original position. Therefore, $360^\circ$ is an angle of symmetry for every figure.

Answer: True

Reason: If the smallest angle of symmetry is $\alpha$, then rotating the figure repeatedly by $\alpha$ must eventually bring it back to the starting position after a full $360^\circ$ turn. This means that for some integer $n$, $n \times \alpha = 360^\circ$. This is the definition of $\alpha$ being a factor of 360.