What is the smallest angle of rotational symmetry for a square 45º 90º 180º 360º?

A quadrilateral is a four-sided polygon.

All quadrilaterals are coplanar and have two diagonals.

Quadrilaterals may be drawn in a "convex" form or a "concave" form. The quadrilaterals we will be using will be convex, where the diagonals lie within the figure. A concave quadrilateral contains an angle greater than 180º, and at least one of the diagonals lies partially, or entirely, outside of the figure (as shown below). The concave figure "caves in" upon itself.

The sum of the measures of the four angles of any quadrilateral equals 360º.

In the diagram at the right, m∠A + m∠B + m∠C + m∠D = 360º. Every quadrilateral can be decomposed into two triangles, each of whose angles' measures sum to 180º. m∠A + m∠1 + m∠ 3 = 180º m∠C + m∠2 + m∠4 = 180º m∠A+m∠1+m∠3+m∠C+m∠2+m∠4 = 360º

There are several "types" of quadrilaterals determined by their characteristics, or properties.
We will be examining three types of quadrilaterals: parallelograms, trapezoids and kites.

A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

	Definition: A parallelogram has two pairs of parallel sides. • The opposite sides of a parallelogram are congruent. • A parallelogram has no reflectional symmetry. • A parallelogram has rotational symmetry of 180º (Order 2).
	Definition: A rectangle is a parallelogram with 4 right angles. • A rectangle has two pairs of parallel sides. • A rectangle has opposite sides congruent. • A rectangle also has four right angles. • A rectangle has two axes of reflectional symmetry. • A rectangle has rotational symmetry of 180º (Order 2).
	Definition: A rhombus is a parallelogram with 4 sides. • A rhombus has two pairs of parallel sides. • A rhombus has all four sides congruent. • A rhombus has 2 (diagonals) axes of reflectional symmetry. • A rhombus has rotational symmetry of 180º (Order 2).
	Definition: A square is a parallelogram with 4 sides and 4 right angles. • A square has two pairs of parallel sides. • A square has all four sides congruent. • A square has four right angles. • A square has four axes of reflectional symmetry. • A square has rotational symmetry of 90º (Order 4).

Read more about symmetries at Symmetry in Geometry.

A trapezoid is a quadrilateral with at least one pair of parallel sides.

	Definition: A trapezoid has at least 1 pair of parallel sides. • A trapezoid has no axes of reflectional symmetry. • A trapezoid has no rotational symmetry (Order 1). Note: Having no rotational symmetry means that the figure must be rotated a full 360º to again appear in it original position. This may be referred to as the identity symmetry.
	Definition: An isosceles trapezoid is a trapezoid with congruent base angles. • An isosceles trapezoid has congruent legs. • An isosceles trapezoid has at least 1 axis of reflectional symmetry. (If it has 2, it is a rectangle.) • An isosceles trapezoid (with only 1 pair of parallel sides) has no rotational symmetry.

Note: The definition of an isosceles triangle states that the triangle has two congruent "sides". But the definition of isosceles trapezoid stated above, mentions congruent base "angles", not sides (or legs). Why? If an isosceles trapezoid is defined to be "a trapezoid with congruent legs", a parallelogram will be an isosceles trapezoid. If this occurs, the other properties that an isosceles trapezoid can possess can no longer hold, since they will not be true for a parallelogram. If we define an isosceles trapezoid to be a trapezoid with congruent base angles, we can easily prove the sides (legs) to also be congruent, a parallelogram will NOT be an isosceles trapezoid, and all of the commonly known properties of an isosceles trapezoid will still be true.

A kite is a quadrilateral whose four sides are drawn such that there are two distinct sets of adjacent, congruent sides.

• A quadrilateral kite resembles a flying kite. • A kite has two pairs of adjacent sides of equal measure. Notice that these sides are not "opposite" sides. • The kite at the right has one vertical axis of reflectional symmetry, . • The kite has no rotational symmetry.

A kite also resembles a triangle reflected over one of its sides, forming a quadrilateral.
Notice ΔABD being reflected over its side from B to D, forming ΔCBD.

Copy the figures with punched holes and find the axes of symmetry for the following:

The axes of symmetry in the given figures are as follows.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Page No 269:

Given the line(s) of symmetry, find the other hole(s):

(a)

(b)

(c)

(d)

(e)

Page No 269:

In the following figures, the mirror line (i.e., the line of symmetry) is given as a dotted line. Complete each figure performing reflection in the dotted (mirror) line. (You might perhaps place a mirror along the dotted line and look into the mirror for the image). Are you able to recall the name of the figure you complete?

The given figures can be completed as follows.

(a) It will be a square.

(b) It will be a triangle.

(c) It will be a rhombus.

(d) It will be a circle.

(e) It will be a pentagon.

(f) It will be an octagon.

Page No 269:

The following figures have more than one line of symmetry. Such figures are said to have multiple lines of symmetry.

Identify multiple lines of symmetry, if any, in each of the following figures:

(a) The given figure has 3 lines of symmetry. Hence, it has multiple lines

of symmetry.

(b) The given figure has 2 lines of symmetry. Hence, it has multiple lines

of symmetry.

(c) The given figure has 3 lines of symmetry. Hence, it has multiple lines

of symmetry.

(d)The given figure has 2 lines of symmetry. Hence, it has multiple lines

of symmetry.

(e) The given figure has 4 lines of symmetry. Hence, it has multiple lines

of symmetry.

(f) The given figure has only 1 line of symmetry.

(g) The given figure has 4 lines of symmetry. Hence, it has multiple lines

of symmetry.

(h) The given figure has 6 lines of symmetry. Hence, it has multiple lines

of symmetry.

Page No 270:

Copy the figure given here.

Take any one diagonal as a line of symmetry and shade a few more squares to make the figure symmetric about a diagonal. Is there more than one way to do that? Will the figure be symmetric about both the diagonals?

We can shade a few more squares so as to make the given figure symmetric about any of its diagonals.

Yes, the figure is symmetric about both the diagonals. There is more than one way so as to make the figure symmetric about a diagonal as we can choose any of its 2 diagonals.

Page No 270:

Copy the diagram and complete each shape to be symmetric about the mirror line (s):

The given figures can be completed about the given mirror lines as follows.

(a)	(b)	(c)	(d)

Page No 270:

State the number of lines of symmetry for the following figures:

(a) An equilateral triangle

(b) An isosceles triangle

(c) A scalene triangle

(d) A square

(e) A rectangle

(f) A rhombus

(g) A parallelogram

(h) A quadrilateral

(i) A regular hexagon

(j) A circle

(a) There are 3 lines of symmetry in an equilateral triangle.

(b)There is only 1 line of symmetry in an isosceles triangle.

(c) There is no line of symmetry in a scalene triangle.

(d)There are 4 lines of symmetry in a square.

(e) There are 2 lines of symmetry in a rectangle.

(f)There are 2 lines of symmetry in a rhombus.

(g) There is no line of symmetry in a parallelogram.

(h) There is no line of symmetry in a quadrilateral.

(i) There are 6 lines of symmetry in a regular hexagon.

(j)There are infinite lines of symmetry in a circle. Some of these are represented as follows.

Page No 270:

What letters of the English alphabet have reflectional symmetry (i.e., symmetry related to mirror reflection) about.

(a) a vertical mirror

(b) a horizontal mirror

(c) both horizontal and vertical mirrors

(a) A, H, I, M, O, T, U, V, W, X, Y are the letters having a reflectional

symmetry about a vertical mirror.

(b) B, C, D, E, H, I, K, O, X are the letters having a reflectional symmetry about a horizontal mirror.

(c) H, I, O, X are the letters having a reflectional symmetry about both the vertical mirror and the horizontal mirror.

Page No 270:

Give three examples of shapes with no line of symmetry.

A scalene triangle, a parallelogram, and a trapezium do not have any line of symmetry.

Page No 270:

What other name can you give to the line of symmetry of

(a) an isosceles triangle?

(b)a circle?

(a) An isosceles triangle has only 1 line of symmetry.

Therefore, this line of symmetry is the median and also the altitude of this isosceles triangle.

(b) There are infinite lines of symmetry in a circle. Some of these are represented as follows.

It can be concluded that each line of symmetry is the diameter for this circle.

Page No 274:

Which of the following figures have rotational symmetry of order more than 1:

(a) The given figure has its rotational symmetry as 4.

(b) The given figure has its rotational symmetry as 3.

(c) The given figure has its rotational symmetry as 1.

(d) The given figure has its rotational symmetry as 2.

(e) The given figure has its rotational symmetry as 3.

(f) The given figure has its rotational symmetry as 4.

Hence, figures (a), (b), (d), (e), and (f) have rotational symmetry of order more than 1.

Page No 274:

Give the order of rotational symmetry for each figure:

(a) The given figure has its rotational symmetry as 2.

(b) The given figure has its rotational symmetry as 2.

(c) The given figure has its rotational symmetry as 3.

(d) The given figure has its rotational symmetry as 4.

(e) The given figure has its rotational symmetry as 4.

(f) The given figure has its rotational symmetry as 5.

(g) The given figure has its rotational symmetry as 6.

(h) The given figure has its rotational symmetry as 3.

Page No 275:

Name any two figures that have both line symmetry and rotational symmetry.

Equilateral triangle and regular hexagon have both line of symmetry and rotational symmetry.

Page No 275:

Draw, wherever possible, a rough sketch of

(i) a triangle with both line and rotational symmetries of order more than 1.

(ii) a triangle with only line symmetry and no rotational symmetry of order more than 1.

(iii) a quadrilateral with a rotational symmetry of order more than 1 but not a line symmetry.

(iv) a quadrilateral with line symmetry but not a rotational symmetry of order more than 1.

(i) Equilateral triangle has 3 lines of symmetry and rotational symmetry of

order 3.

(ii) Isosceles triangle has only 1 line of symmetry and no rotational symmetry of order more than 1.

(iii) A parallelogram is a quadrilateral which has no line of symmetry but a rotational symmetry of order 2.

(iv)A kite is a quadrilateral which has only 1 line of symmetry and no rotational symmetry of order more than 1.

Page No 275:

If a figure has two or more lines of symmetry, should it have rotational symmetry of order more than 1?

Yes. If a figure has two or more lines of symmetry, then it will definitely have its rotational symmetry of order more than 1.

Page No 275:

Fill in the blanks:

The given table can be completed as follows.

Page No 276:

Name the quadrilaterals which have both line and rotational symmetry of order more than 1.

Square, rectangle, and rhombus are the quadrilaterals which have both line and rotational symmetry of order more than 1. A square has 4 lines of symmetry and rotational symmetry of order 4. A rectangle has 2 lines of symmetry and rotational symmetry of order 2. A rhombus has 2 lines of symmetry and rotational symmetry of order 2.

Page No 276:

After rotating by 60° about a centre, a figure looks exactly the same as its original position. At what other angles will this happen for the figure?

It can be observed that if a figure looks symmetrical on rotating by 60º, then it will also look symmetrical on rotating by 120º, 180º, 240º, 300º, and 360º i.e., further multiples of 60º.

Page No 276:

Can we have a rotational symmetry of order more than 1 whose angle of rotation is

(i) 45°?

(ii) 17°?

It can be observed that if the angle of rotation of a figure is a factor of 360º, then it will have a rotational symmetry of order more than 1.

It can be checked that 45º is a factor of 360º but 17º is not. Therefore, the figure having its angle of rotation as 45º will have its rotational symmetry of order more than 1. However, the figure having its angle of rotation as 17º will not be having its rotational symmetry of order more than 1.