Class 6 : Maths ( English ) – Lesson 9. Symmetry

EXPLANATION AND ANALYSIS

🌿 Explanation & Analysis

🔵 1. What is symmetry?
In daily life, we often notice objects that look the same on both sides—like a butterfly 🦋, a leaf 🍃, or the human face 🙂. This balanced appearance is called symmetry.
In mathematics, symmetry means that a figure can be divided into two identical parts such that one part is the mirror image of the other.

🧠 If a figure looks exactly the same after being folded along a line, that line shows symmetry.

✏️ Note: Symmetry is about balance and matching, not about size or colour.

💡 Concept: A figure is symmetrical if it can be divided into two identical mirror-image halves.

🔵 2. Line of symmetry
A line of symmetry is a line that divides a figure into two equal and identical parts. When the figure is folded along this line, both halves coincide exactly.

Examples in idea:
🔵 A square has several lines of symmetry
🔵 A rectangle has fewer lines of symmetry
🔵 Some shapes have only one line of symmetry

🧠 The line of symmetry acts like a mirror.

✏️ Note: A line of symmetry can be vertical, horizontal, or slanting.

🔵 3. Identifying symmetry by folding
One simple way to check symmetry is paper folding.
If a shape is folded and both halves match perfectly, the fold line is a line of symmetry.

🟢 This method helps students understand symmetry visually.
🟡 It avoids guesswork and builds clarity.

✔️ Folding is a practical way to test symmetry.

🔵 4. Shapes with one line of symmetry
Some figures have only one line of symmetry.

Examples:
🔵 An isosceles triangle
🔵 A kite
🔵 Certain letters like A and T

🧠 These figures are balanced only in one direction.

💡 Concept: A figure may have symmetry in one direction but not in others.

🔵 5. Shapes with more than one line of symmetry
Some shapes are highly symmetrical and have more than one line of symmetry.

Examples:
🔵 Square
🔵 Rectangle
🔵 Circle

🧠 A square has several lines of symmetry because all its sides and angles are equal.

✏️ Note: The more regular a shape is, the more lines of symmetry it usually has.

🔵 6. Shapes with no line of symmetry
Not all shapes are symmetrical.

Examples:
🔴 A scalene triangle
🔴 An irregular shape
🔴 Certain letters like F, Z

🧠 These shapes cannot be divided into two identical mirror-image halves.

✔️ Absence of symmetry is also an important idea.

🔵 7. Symmetry in letters of the English alphabet
English letters show different types of symmetry.

🟢 Letters with vertical symmetry: A, M, T, U
🟡 Letters with horizontal symmetry: B, C, D
🔴 Letters with no symmetry: F, G, J

🧠 Studying letters makes symmetry fun and familiar.

✏️ Note: Capital letters are usually used when checking symmetry.

🔵 8. Symmetry in numbers
Some numbers also show symmetry.

Examples:
🔵 0 and 8 show symmetry
🔵 3 and 7 do not show symmetry

🧠 This shows that symmetry appears even in symbols and digits.

🔵 9. Reflection symmetry
Symmetry is also called reflection symmetry because one half of a figure reflects the other, just like in a mirror 🪞.

🟢 The mirror line is the line of symmetry.
🟡 Each point on one side has an equal-distance matching point on the other side.

💡 Concept: Reflection symmetry means both sides are mirror images.

🔵 10. Drawing the mirror image
To draw the mirror image of a figure:
🔵 Draw a vertical or horizontal line (mirror line)
🔵 Measure the distance of points from the line
🔵 Mark the same distance on the opposite side

🧠 This helps in understanding reflection clearly.

✏️ Note: Every point and its image are equally distant from the mirror line.

🔵 11. Symmetry in nature
Nature is full of symmetry 🌍.

Examples:
🦋 Butterfly wings
🌼 Flowers
🍁 Leaves

🧠 Symmetry in nature shows balance and beauty.

✔️ Mathematics helps us understand these natural patterns.

🔵 12. Symmetry in everyday objects
Many man-made objects show symmetry.

Examples:
🏠 Buildings
🪟 Windows
🚗 Vehicles
⚖️ Logos and designs

🧠 Designers use symmetry to make objects look attractive and balanced.

🔵 13. Importance of symmetry
Symmetry is important because it:
🟢 Makes shapes easy to recognise
🟢 Helps in design and construction
🟢 Improves visual balance
🟢 Develops spatial understanding

✔️ Symmetry connects mathematics with art and design.

🔵 14. Learning symmetry through activities
Symmetry is best learned by doing.

Activities include:
🔵 Paper folding
🔵 Mirror placement
🔵 Drawing half figures and completing them

🧠 These activities make learning active and enjoyable.

💡 Concept: Doing activities strengthens understanding more than memorising.

🔵 15. Symmetry as a foundation concept
Symmetry prepares students for future topics such as:
🔵 Geometry
🔵 Patterns
🔵 Tessellations
🔵 Coordinate geometry

✔️ A strong understanding of symmetry helps in higher mathematics.

🧠 Symmetry trains the eye to notice balance and structure.

Summary

Symmetry means balance and sameness on both sides of a figure. A figure is symmetrical if it can be divided into two identical mirror-image halves by a line called the line of symmetry. This line can be vertical, horizontal, or slanting. Folding is a simple method to identify symmetry.

Different shapes have different numbers of lines of symmetry. Some shapes have one line of symmetry, some have many, and some have none. Letters and numbers also show symmetry, making the concept easy to understand and relate to daily life. Symmetry is also known as reflection symmetry because each half reflects the other.

Symmetry is seen widely in nature and everyday objects, showing beauty and balance. It plays an important role in design, construction, and art. Learning symmetry through activities helps students develop strong visual and logical thinking skills. It forms a foundation for many advanced mathematical ideas.

📝 Quick Recap

🔵 Symmetry means identical mirror-image halves
🟢 A line of symmetry divides a figure equally
🟡 Some shapes have many lines, some one, some none
🔴 Letters, numbers, nature, and objects show symmetry
✔️ Symmetry builds visual balance and mathematical thinking

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TEXTBOOK QUESTIONS

🌿 LINE OF SYMMETRY

🔒 ❓ Q1. Figure it Out
Do you see any line of symmetry in the figures at the start of the chapter?
What about in the picture of the cloud?

📌 ✅ Answer:
🔹 A line of symmetry is a line that divides a figure into two identical halves.
🔹 In the figures shown at the start of the chapter, some figures are drawn symmetrically.
🔹 These figures can be folded along a certain line so that both halves overlap exactly.
🔹 Such figures have at least one line of symmetry.

🔹 In the picture of the cloud:
🔹 The cloud does not have a perfect regular shape.
🔹 Its left and right sides are not identical.
🔹 So, the cloud picture does not have a clear line of symmetry.

🔒 ❓ Q2. Figure it Out
For each of the following figures, identify the line(s) of symmetry if it exists.

📌 ✅ Answer:

🔹 Figure A (first irregular shape):
🔹 The shape is uneven.
🔹 No line can divide it into two identical halves.
🔹 So, it has no line of symmetry.

🔹 Figure B (pentagon-like shape):
🔹 The left and right sides are mirror images of each other.
🔹 A vertical line passing through the top vertex and the midpoint of the bottom side divides it equally.
🔹 So, it has one line of symmetry (vertical).

🔹 Figure C (trapezium-like shape):
🔹 The left and right sides are not equal or mirror images.
🔹 No vertical, horizontal, or slanted line can divide it into two identical halves.
🔹 So, it has no line of symmetry.

🔹 Figure D (L-shaped figure):
🔹 The arms of the L are of unequal lengths.
🔹 No line divides it into two identical mirror halves.
🔹 So, it has no line of symmetry.

🔹 Figure E (triangular shape):
🔹 The sides and angles are all unequal.
🔹 It is a scalene triangle.
🔹 A scalene triangle has no line of symmetry.

🔒 ❓ Figure it Out

🔒 ❓ Q1. Figure it Out — Punching Game
In each of the following figures, a hole was punched in a folded square sheet of paper and then the paper was unfolded. Identify the line along which the paper was folded.
Figure (d) was created by punching a single hole. How was the paper folded?

📌 ✅ Answer:
🔹 When a paper is folded, the fold line becomes a line of symmetry after unfolding.
🔹 The holes appear in mirror-image positions on either side of the fold line.

🔹 Figure (a):
🔸 The two holes are at the same height and are mirror images left–right.
🔸 So, the paper was folded along a vertical line through the middle of the square.

🔹 Figure (b):
🔸 The two holes are close together near the top-right corner and symmetric about a slant.
🔸 So, the paper was folded along a diagonal line.

🔹 Figure (c):
🔸 The holes are one above and one below, placed symmetrically.
🔸 So, the paper was folded along a horizontal line through the middle.

🔹 Figure (d):
🔸 Four holes appear, one near each corner, all symmetric.
🔸 Only one hole was punched, so the paper must have been folded twice.
🔸 First fold: vertically through the middle.
🔸 Second fold: horizontally through the middle.
🔹 After punching one hole and unfolding, four symmetric holes are formed.

🔒 ❓ Q2.
Given the line(s) of symmetry, find the other hole(s).

📌 ✅ Answer:

🔹 Figure (a):
🔸 The dashed diagonal line is the line of symmetry.
🔸 The other hole will be at the mirror position across the diagonal, at the same distance on the opposite side.

🔹 Figure (b):
🔸 The dashed horizontal line is the line of symmetry.
🔸 The other hole will appear directly above the given hole, at the same vertical distance.

🔹 Figure (c):
🔸 The dashed vertical line inside the triangle is the line of symmetry.
🔸 The other hole will appear at the same height on the opposite side of the line.

🔹 Figure (d):
🔸 The dashed slanted line is the line of symmetry of the circle.
🔸 The other hole will be at the mirror point across the slanted line, at equal distance from the centre.

🔹 Figure (e):
🔸 The dashed diagonal line is the line of symmetry.
🔸 The missing hole will be placed at the symmetric position across the diagonal, maintaining equal distance.

🔹 In all cases:
🔸 The new hole must be placed so that both holes are equal distances from the line of symmetry.

🔒 ❓ Q3. Paper Cutting (Understanding Folds)
Consider a vertical fold and a horizontal fold as shown. Use these ideas to answer the following.

📌 ✅ Answer:
🔹 A vertical fold creates mirror images left–right about the fold line.
🔹 A horizontal fold creates mirror images top–bottom about the fold line.
🔹 After opening the paper, every cut appears as a reflected copy across the fold line(s).

🔒 ❓ Q4. Predict the shape of the hole when the paper is opened. Then verify.

📌 ✅ Answer:

🔹 (a)
🔹 The paper is folded and an irregular curved cut is made on one side of the fold.
🔹 After opening, the hole will be a symmetric shape formed by the curve and its mirror image across the fold line.
🔹 So the final hole looks like two matching curved edges joined symmetrically.

🔹 (b)
🔹 The paper is folded and a V-shaped / zig-zag cut is made.
🔹 On opening, the cut appears on both sides of the fold.
🔹 The final hole is a symmetric zig-zag pattern with left–right mirror symmetry.

🔹 (c)
🔹 The paper is folded vertically, then horizontally, and rectangular cuts are made.
🔹 Since there are two folds, each cut appears four times after opening.
🔹 The final hole pattern is a four-fold symmetric rectangular design, matching the shown shape.

🔹 (d)
🔹 The paper is folded vertically and step-like rectangular cuts are made near the fold.
🔹 On opening, each cut reflects to the other side.
🔹 The final hole becomes an I-shaped symmetric figure, with left–right symmetry.

🔹 Key idea for all parts:
🔸 Each fold creates a line of symmetry.
🔸 The number of times a cut appears after opening depends on the number of folds.
🔸 The final shape is obtained by reflecting the cut across every fold line.

🔒 ❓ Q5.
Suppose you have to get each of these shapes with some folds and a single straight cut. How will you do it?

🔒 ❓ (a) The hole in the centre is a square.

📌 ✅ Answer:
🔹 Goal: To obtain a square hole at the centre using folds + one straight cut.

🔹 Step 1: Take a square sheet of paper.
🔹 Step 2: Fold the paper vertically through the centre so that left and right halves coincide.
🔹 Step 3: Fold the paper horizontally through the centre so that top and bottom halves coincide.
🔹 Now the paper is folded into four equal layers, with the centre of the sheet at one corner of the folded paper.

🔹 Step 4: On the folded paper, make one straight cut parallel to the edges, cutting a small square shape at the folded corner.
🔹 Step 5: Open the paper fully.

🔹 Observation:
🔹 The cut gets reflected across both fold lines.
🔹 Four identical cuts join together to form a square hole at the centre.

🔹 Verification (square properties):
🔸 All four sides of the hole are equal.
🔸 All four angles are 90°.
🔹 So, the hole in the centre is a square.

🔒 ❓ (b) The hole in the centre is a square (tilted / diamond-shaped).

📌 ✅ Answer:
🔹 Goal: To obtain a tilted square (diamond-shaped) hole using folds + one straight cut.

🔹 Step 1: Take a square sheet of paper.
🔹 Step 2: Fold the paper along one diagonal.
🔹 Step 3: Fold it again along the other diagonal, bringing all four corners together.
🔹 Now the paper is folded into four equal layers, and the centre of the original sheet lies at the folded tip.

🔹 Step 4: Make one straight cut across the folded tip.
🔹 Step 5: Open the paper completely.

🔹 Observation:
🔹 The straight cut is reflected across both diagonal fold lines.
🔹 The resulting hole has four equal sides but is rotated with respect to the paper edges.

🔹 Verification (square properties):
🔸 All sides of the hole are equal.
🔸 All interior angles are right angles.
🔹 Hence, the central figure is also a square, though it appears tilted.

🔒 ❓ Q6.
How many lines of symmetry do these shapes have?

🔒 ❓ (a) Given shapes

📌 ✅ Answer:
🔹 First shape (tilted square / diamond):
🔹 It is a square, only rotated.
🔹 A square has
🔸 2 diagonals as lines of symmetry
🔸 2 lines joining midpoints of opposite sides
🔹 Total number of lines of symmetry = 4

🔹 Second shape (star-like symmetric shape):
🔹 All arms are identical and evenly spaced.
🔹 Each arm can be folded onto the opposite arm.
🔹 There is one line of symmetry through each pair of opposite arms.
🔹 Total number of lines of symmetry = 8

🔒 ❓ (b) A triangle with equal sides and equal angles.

📌 ✅ Answer:
🔹 This is an equilateral triangle.
🔹 Each line joining a vertex to the midpoint of the opposite side is a line of symmetry.
🔹 There are three such lines.
🔹 Number of lines of symmetry = 3

🔒 ❓ (c) A hexagon with equal sides and equal angles.

📌 ✅ Answer:
🔹 This is a regular hexagon.
🔹 Lines of symmetry pass through
🔸 opposite vertices (3 lines)
🔸 midpoints of opposite sides (3 lines)
🔹 Total number of lines of symmetry = 6

🔒 ❓ Q7.
Trace each figure and draw the line(s) of symmetry, if any.

📌 ✅ Answer:

🔹 First figure (three diamond shapes forming a peak):
🔹 The left and right halves are mirror images.
🔹 One vertical line of symmetry passes through the centre.

🔹 Second figure (row of diamonds):
🔹 The figure is symmetric about a vertical line through the middle.
🔹 One vertical line of symmetry exists.

🔹 Third figure (stacked diamonds):
🔹 The figure is symmetric left–right.
🔹 One vertical line of symmetry exists.

🔹 Fourth figure (criss-cross diamond pattern):
🔹 The figure is symmetric about
🔸 a vertical line
🔸 a horizontal line
🔹 So, it has 2 lines of symmetry.

🔹 Square spiral figure:
🔹 The spiral keeps changing direction.
🔹 No folding line can divide it into identical halves.
🔹 So, it has no line of symmetry.

🔹 Octagon-like grid figure:
🔹 The figure is symmetric about
🔸 a vertical line
🔸 a horizontal line
🔹 So, it has 2 lines of symmetry.

🔹 Irregular pentagon on grid:
🔹 Sides and angles are unequal.
🔹 No mirror line exists.
🔹 So, it has no line of symmetry.

🔹 Star-like figure on grid:
🔹 All arms are equal and evenly placed.
🔹 The figure can be folded along
🔸 vertical
🔸 horizontal
🔸 two diagonal lines
🔹 Total number of lines of symmetry = 4

🔒 ❓ Q8.
Find the lines of symmetry for the kolam shown.

📌 ✅ Answer:
🔹 The kolam is arranged in a regular and balanced pattern.
🔹 It is symmetric about
🔸 one vertical line through the centre
🔸 one horizontal line through the centre
🔸 two diagonal lines through the centre
🔹 Therefore, the kolam has 4 lines of symmetry.

🔒 ❓ Q9. Draw the following.

🔒 ❓ (a) A triangle with exactly one line of symmetry.

📌 ✅ Answer:
🔹 An isosceles triangle.
🔹 It has exactly one line of symmetry passing through the vertex between the equal sides and the midpoint of the base.

🔒 ❓ (b) A triangle with exactly three lines of symmetry.

📌 ✅ Answer:
🔹 An equilateral triangle.
🔹 All sides and all angles are equal.
🔹 It has 3 lines of symmetry, one through each vertex and the midpoint of the opposite side.

🔒 ❓ (c) A triangle with no line of symmetry.

📌 ✅ Answer:
🔹 A scalene triangle.
🔹 All sides and angles are unequal.
🔹 It has no line of symmetry.

🔒 ❓ Is it possible to draw a triangle with exactly two lines of symmetry?

📌 ✅ Answer:
🔹 No, it is not possible.
🔹 A triangle can have only
🔸 3 lines of symmetry (equilateral triangle)
🔸 1 line of symmetry (isosceles triangle)
🔸 0 lines of symmetry (scalene triangle)
🔹 No triangle has exactly two lines of symmetry.

🔒 ❓ Q10. Draw the following. In each case, the figure should contain at least one curved boundary.

🔒 ❓ (a) A figure with exactly one line of symmetry.

📌 ✅ Answer:
🔹 A semicircle.
🔹 The diameter is the only line of symmetry.
🔹 So, it has exactly one line of symmetry.

🔒 ❓ (b) A figure with exactly two lines of symmetry.

📌 ✅ Answer:
🔹 An ellipse (oval).
🔹 It has
🔸 one vertical line of symmetry
🔸 one horizontal line of symmetry
🔹 Hence, it has exactly two lines of symmetry.

🔒 ❓ (c) A figure with exactly four lines of symmetry.

📌 ✅ Answer:
🔹 A circle.
🔹 Vertical, horizontal, and two diagonal diameters act as symmetry lines.
🔹 Hence, it has four or more lines of symmetry.

🔒 ❓ Q11.
Copy the following on squared paper. Complete them so that the blue line is a line of symmetry.

📌 ✅ Answer:
🔹 Figure (a):
🔹 Already completed correctly. The red figure is a mirror image across the blue vertical line.

🔹 Figure (b):
🔹 Reflect each point and line segment of the red figure across the horizontal blue line.
🔹 Corresponding points must be at equal distances from the line.

🔹 Figure (c):
🔹 Reflect the red figure across the slanting blue line.
🔹 Rotating the book helps in visualising the reflection.

🔹 Figure (d):
🔹 Draw the mirror image of the red figure on the other side of the vertical blue line.

🔹 Figure (e):
🔹 Reflect the red polygon across the horizontal blue line so that the two halves overlap exactly.

🔹 Figure (f):
🔹 Reflect the red figure across the slanting blue line.
🔹 Rotating the book helps to complete it accurately.

🔒 ❓ Q12.
Copy the following drawing on squared paper. Complete each one so that the resulting figure has the two blue lines as lines of symmetry.

📌 ✅ Answer:

🔹 Figure (a):
🔹 The two blue lines are diagonals intersecting at the centre.
🔹 Complete the red segment by drawing its mirror image across both diagonals.
🔹 After completion, folding along either diagonal will make both halves overlap exactly.

🔹 Figure (b):
🔹 The blue lines are diagonals crossing at the centre.
🔹 Reflect the given red zig-zag first across one diagonal, then across the other.
🔹 The final figure must look identical in all four diagonal regions.

🔹 Figure (c):
🔹 One blue line is vertical and the other is horizontal.
🔹 Reflect the red stepped shape across the vertical line, then across the horizontal line.
🔹 All four quadrants must show identical shapes.

🔹 Figure (d):
🔹 The blue lines meet at right angles at the centre.
🔹 Complete the figure by reflecting the red shape across both lines of symmetry.
🔹 The completed shape will match on left–right and top–bottom folds.

🔹 Figure (e):
🔹 The blue lines are vertical and horizontal.
🔹 Reflect the red shape across the vertical line and then across the horizontal line.
🔹 Check by folding along each blue line.

🔹 Figure (f):
🔹 One blue line is vertical and the other is horizontal.
🔹 Reflect the red polygon across both blue lines.
🔹 The completed figure will be symmetric in all four parts.

🔒 ❓ Q13.
Copy the following on a dot grid. For each figure draw two more lines to make a shape that has a line of symmetry.

📌 ✅ Answer:

🔹 Figure (a):
🔹 Add two lines so that the left side becomes a mirror image of the right side.
🔹 A vertical line through the centre will be the line of symmetry.

🔹 Figure (b):
🔹 Extend two line segments so that the top and bottom parts match.
🔹 The horizontal middle line becomes the line of symmetry.

🔹 Figure (c):
🔹 Add two lines to balance the slanting arms equally.
🔹 A vertical line through the centre acts as the line of symmetry.

🔹 Figure (d):
🔹 Complete the shape so that both sides of a vertical line look identical.
🔹 The vertical centre line is the line of symmetry.

🔹 Figure (e):
🔹 Add two lines to balance the shape on either side of a slanted line.
🔹 That slanted line becomes the line of symmetry.

🔹 Figure (f):
🔹 Add two matching line segments so that the shape mirrors across a vertical line.
🔹 The vertical line through the centre is the line of symmetry.

🌿 ROTATIONAL SYMMETRY

🔒 ❓ Figure it Out – Q1
Find the angles of symmetry for the given figures about the point marked ●.

📌 ✅ Answer:

🔹 Figure (a):
🔹 The figure coincides with itself after rotations of
🔸 90°
🔸 180°
🔸 270°
🔹 Hence, the angles of symmetry are 90°, 180°, 270°.

🔹 Figure (b):
🔹 The figure matches itself only after a half turn.
🔹 Angle of symmetry = 180°.

🔹 Figure (c):
🔹 The figure coincides with itself only after a half turn.
🔹 Angle of symmetry = 180°.

🔒 ❓ Q2
Which of the following figures have more than one angle of symmetry?

📌 ✅ Answer:

🔹 Circle with a cross:
🔹 Matches itself after
🔸 90°
🔸 180°
🔸 270°
🔹 So, it has more than one angle of symmetry.

🔹 Triangle shown:
🔹 It matches itself only after a full turn.
🔹 So, it has only one angle of symmetry (360°).

🔹 Circle divided into three equal sectors:
🔹 Matches itself after
🔸 120°
🔸 240°
🔹 So, it has more than one angle of symmetry.

🔹 Four-petal curved figure:
🔹 Matches itself after
🔸 90°
🔸 180°
🔸 270°
🔹 So, it has more than one angle of symmetry.

🔹 Crossed straight lines:
🔹 Matches itself after
🔸 90°
🔸 180°
🔸 270°
🔹 So, it has more than one angle of symmetry.

🔹 Five-point star:
🔹 Matches itself after
🔸 72°
🔸 144°
🔸 216°
🔸 288°
🔹 So, it has more than one angle of symmetry.

🔹 Double semicircle shape:
🔹 Matches itself only after 180°.
🔹 So, it has only one angle of symmetry.

🔒 ❓ Q3
Give the order of rotational symmetry for each figure.

📌 ✅ Answer:

🔹 First figure (tilted line with arrow ends):
🔹 It matches itself only once in a full rotation.
🔹 Order of rotational symmetry = 1.

🔹 Second figure (crossed lines):
🔹 Matches itself after
🔸 90°
🔸 180°
🔸 270°
🔹 Order of rotational symmetry = 4.

🔹 Third figure (six-point star):
🔹 Matches itself every 60°.
🔹 Order of rotational symmetry = 6.

🔹 Fourth figure (three-armed figure):
🔹 Matches itself after
🔸 120°
🔸 240°
🔹 Order of rotational symmetry = 3.

🔹 Fifth figure (plus shape):
🔹 Matches itself after
🔸 90°
🔸 180°
🔸 270°
🔹 Order of rotational symmetry = 4.

🔹 Sixth figure (regular pentagon):
🔹 Matches itself every 72°.
🔹 Order of rotational symmetry = 5.

🌿 SYMMETRIESOF A CIRCLE

🔒 ❓ Figure it Out

🔒 ❓ 1. Colour the sectors of the circle below so that the figure has:
i) 3 angles of symmetry
ii) 4 angles of symmetry
iii) What are the possible numbers of angles of symmetry you can obtain by colouring the sectors in different ways?

📌 ✅ Answer:

🔹 The circle is divided into 12 equal sectors.

🔸 (i) 3 angles of symmetry
➡️ Colour the sectors in a repeating pattern of every 4th sector using the same colour.
➡️ The figure will match itself after rotation of 120°.
➡️ Hence, there are 3 angles of symmetry.

🔸 (ii) 4 angles of symmetry
➡️ Colour the sectors in a repeating pattern of every 3rd sector.
➡️ The figure matches after rotation of 90°.
➡️ Hence, there are 4 angles of symmetry.

🔸 (iii) Possible numbers of angles of symmetry
➡️ Since the circle has 12 equal sectors, symmetry is possible for any number that divides 12.
➡️ Possible numbers are:
🔹 1, 2, 3, 4, 6, 12

🔒 ❓ 2. Draw two figures other than a circle and a square that have both reflection symmetry and rotational symmetry.

📌 ✅ Answer:

🔹 Equilateral triangle
➡️ It has 3 lines of reflection symmetry.
➡️ It has 3 angles of rotational symmetry.

🔹 Regular hexagon
➡️ It has 6 lines of reflection symmetry.
➡️ It has 6 angles of rotational symmetry.

🔒 ❓ 3. Draw, wherever possible, a rough sketch of:

🔒 ❓ (a) A triangle with at least two lines of symmetry and at least two angles of symmetry.

📌 ✅ Answer:

🔹 This is not possible.
➡️ A triangle can have either 1 line of symmetry (isosceles) or 3 lines of symmetry (equilateral).
➡️ No triangle has exactly two lines of symmetry.

🔒 ❓ (b) A triangle with only one line of symmetry but not having rotational symmetry.

📌 ✅ Answer:

🔹 Isosceles triangle (not equilateral)
➡️ It has one line of reflection symmetry.
➡️ It does not match itself under rotation (except 360°).

🔒 ❓ (c) A quadrilateral with rotational symmetry but no reflection symmetry.

📌 ✅ Answer:

🔹 Parallelogram (not rectangle or square)
➡️ It has rotational symmetry of order 2 (180°).
➡️ It has no line of reflection symmetry.

🔒 ❓ (d) A quadrilateral with reflection symmetry but not having rotational symmetry.

📌 ✅ Answer:

🔹 Kite (not a rhombus)
➡️ It has one line of reflection symmetry.
➡️ It does not have rotational symmetry.

🔒 ❓ 4. In a figure, 60° is the smallest angle of symmetry. What are the other angles of symmetry of this figure?

📌 ✅ Answer:

🔹 Smallest angle of symmetry = 60°
➡️ Total angle in one rotation = 360°

🔸 Other angles of symmetry are multiples of 60°:
🔹 120°
🔹 180°
🔹 240°
🔹 300°
🔹 360°

🔒 ❓ 5. In a figure, 60° is an angle of symmetry. The figure has two angles of symmetry less than 60°. What is its smallest angle of symmetry?

📌 ✅ Answer:

🔹 Let the smallest angle be x.
➡️ Angles less than 60° are: x and 2x

🔸 Given that 60° is also an angle of symmetry, so:
➡️ 3x = 60°

🔹 Solving:
➡️ x = 20°

🔹 Smallest angle of symmetry = 20°

🔒 ❓ 6. Can we have a figure with rotational symmetry whose smallest angle of symmetry is:

🔒 ❓ (a) 45°?

📌 ✅ Answer:

🔹 Yes
➡️ Because 360 ÷ 45 = 8
➡️ A figure with 8-fold rotational symmetry is possible.

🔒 ❓ (b) 17°?

📌 ✅ Answer:

🔹 No
➡️ 360 ÷ 17 is not a whole number.
➡️ Hence, a figure cannot have 17° as its smallest angle of symmetry.

🔒 ❓ Question 7.
This is a picture of the new Parliament Building in Delhi.

a. Does the outer boundary of the picture have reflection symmetry? If so, draw the lines of symmetries. How many are they?

📌 ✅ Answer:
🔹 The outer boundary shown is a regular polygon–like shape (top view).
🔹 Yes, it has reflection symmetry.
🔹 Each line of symmetry passes through the centre and divides the shape into two equal mirror halves.
🔹 Since the outer boundary has six equal sides, it has 6 lines of symmetry.

🔸 Therefore, the outer boundary has 6 lines of reflection symmetry.

🔒 ❓ Question 7(b).
Does it have rotational symmetry around its centre? If so, find the angles of rotational symmetry.

📌 ✅ Answer:
🔹 Yes, the outer boundary has rotational symmetry about its centre.
🔹 It matches itself after rotation through equal angles.
🔹 Since the shape has 6 identical sides, the smallest angle of rotation is:

🔸 360° ÷ 6 = 60°

🔹 Hence, the angles of rotational symmetry are:
🔸 60°, 120°, 180°, 240°, 300°, 360°

🔹 So, it has rotational symmetry of order 6.

🔒 ❓ Question 8.
How many lines of symmetry do the shapes in the first shape sequence in Chapter 1, Table 3, the Regular Polygons, have? What number sequence do you get?

📌 ✅ Answer:
🔹 In regular polygons, the number of sides = number of lines of symmetry.

🔹 Examples:
🔸 Equilateral triangle → 3 lines of symmetry
🔸 Square → 4 lines of symmetry
🔸 Pentagon → 5 lines of symmetry
🔸 Hexagon → 6 lines of symmetry

🔹 The number sequence obtained is:
🔸 3, 4, 5, 6, 7, …

🔹 This is a natural number sequence starting from 3.

🔒 ❓ Question 9.
How many angles of symmetry do the shapes in the first shape sequence in Chapter 1, Table 3, the Regular Polygons, have? What number sequence do you get?

📌 ✅ Answer:
🔹 For a regular polygon, the number of angles of rotational symmetry = number of sides.

🔹 Examples:
🔸 Triangle → 3 angles of symmetry
🔸 Square → 4 angles of symmetry
🔸 Pentagon → 5 angles of symmetry

🔹 The number sequence obtained is:
🔸 3, 4, 5, 6, 7, …

🔹 This sequence is the same as the number of sides of regular polygons.

🔒 ❓ Question 10.
How many lines of symmetry do the shapes in the last shape sequence in Chapter 1, Table 3, the Koch Snowflake sequence, have? How many angles of symmetry?

📌 ✅ Answer:
🔹 The Koch snowflake is based on an equilateral triangle.
🔹 No matter how many steps are added, its overall shape remains symmetric.

🔹 Lines of symmetry:
🔸 The Koch snowflake has 3 lines of reflection symmetry.

🔹 Angles of symmetry:
🔸 It matches itself after rotations of 120°.
🔸 So, it has 3 angles of rotational symmetry (120°, 240°, 360°).

🔒 ❓ Question 11.
How many lines of symmetry and angles of symmetry does the Ashoka Chakra have?

📌 ✅ Answer:
🔹 The Ashoka Chakra has 24 equally spaced spokes.

🔹 Lines of symmetry:
🔸 Each spoke gives one line of symmetry.
🔸 Total lines of symmetry = 24

🔹 Angles of rotational symmetry:
🔸 Smallest angle of rotation = 360° ÷ 24 = 15°
🔸 Hence, it has 24 angles of symmetry.

🔹 Therefore:
🔸 Lines of symmetry = 24
🔸 Angles of symmetry = 24

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OTHER IMPORTANT QUESTIONS

(CBSE MODEL QUESTION PAPER)

ESPECIALLY MADE FROM THIS CHAPTER ONLY

🔵 Section A — Very Short Answer

(Q1–Q6 | 1 × 6 = 6 marks)

🔵 Question
Q1. What is symmetry?

🟢 Answer
✔️ Symmetry means that a figure can be divided into two identical mirror-image halves.

🔵 Question
Q2. What is a line of symmetry?

🟢 Answer
✔️ A line of symmetry is a line that divides a figure into two equal and identical parts.

🔵 Question
Q3. Name one object from daily life that shows symmetry.

🟢 Answer
✔️ A butterfly shows symmetry.

🔵 Question
Q4. Does a scalene triangle have a line of symmetry? (Yes/No)

🟢 Answer
✔️ No, a scalene triangle has no line of symmetry.

🔵 Question
Q5. Which English letter has vertical symmetry: A or F?

🟢 Answer
✔️ The letter A has vertical symmetry.

🔵 Question
Q6. What type of symmetry is also called mirror symmetry?

🟢 Answer
✔️ Reflection symmetry is also called mirror symmetry.

🟢 Section B — Short Answer–I

(Q7–Q12 | 2 × 6 = 12 marks)

🟢 Question
Q7. Explain the meaning of symmetry using an example.

🟢 Answer
🔵 Symmetry means balance on both sides of a figure.
🔵 For example, a butterfly has two wings that look exactly the same on both sides.

🟢 Question
Q8. What is meant by folding test for symmetry?

🟢 Answer
🔵 In the folding test, a figure is folded along a line.
🔵 If both parts match exactly, the figure is symmetrical along that line.

🟢 Question
Q9. Write two shapes that have more than one line of symmetry.

🟢 Answer
🔵 A square has more than one line of symmetry.
🔵 A circle has many lines of symmetry.

🟢 Question
Q10. Name two shapes that have no line of symmetry.

🟢 Answer
🔵 A scalene triangle has no line of symmetry.
🔵 An irregular shape has no line of symmetry.

🟢 Question
Q11. Write two capital letters of the English alphabet that have symmetry.

🟢 Answer
🔵 The letters A and M have symmetry.

🟢 Question
Q12. Why is symmetry important in designing objects?

🟢 Answer
🔵 Symmetry makes objects look balanced and attractive.
🔵 It also helps in proper design and stability.

🟡 Section C — Short Answer–II

(Q13–Q22 | 3 × 10 = 30 marks)

🟡 Question
Q13. What is a line of symmetry? Explain with one example.

🟢 Answer
🔵 A line of symmetry is a line that divides a figure into two identical mirror-image parts.
🔵 When the figure is folded along this line, both halves match exactly.
✔️ Example: A rectangle has a vertical line of symmetry.

🟡 Question
Q14. How can you check whether a figure is symmetrical or not?

🟢 Answer
🔵 Fold the figure along a line.
🔵 If both parts coincide exactly, the figure is symmetrical.
🔵 If they do not match, the figure is not symmetrical.

🟡 Question
Q15. Write three shapes that have only one line of symmetry.

🟢 Answer
🔵 An isosceles triangle
🔵 A kite
🔵 A semicircle

🟡 Question
Q16. Write the number of lines of symmetry in the following shapes: square and rectangle.

🟢 Answer
🔵 A square has 4 lines of symmetry.
🔵 A rectangle has 2 lines of symmetry.

🟡 Question
Q17. Why does a circle have many lines of symmetry?

🟢 Answer
🔵 All points on a circle are equally distant from its centre.
🔵 Any diameter divides the circle into two equal halves.
✔️ Therefore, a circle has many lines of symmetry.

🟡 Question
Q18. Write three capital letters of the English alphabet that have no line of symmetry.

🟢 Answer
🔵 F
🔵 G
🔵 J

🟡 Question
Q19. What is reflection symmetry?

🟢 Answer
🔵 Reflection symmetry means one half of a figure is the mirror image of the other half.
🔵 The mirror line is called the line of symmetry.

🟡 Question
Q20. Draw any shape with two lines of symmetry and name it.

🟢 Answer
🔵 A rectangle has two lines of symmetry.
🔵 One vertical and one horizontal line divide it into equal halves.

🟡 Question
Q21. Explain symmetry seen in nature with one example.

🟢 Answer
🔵 Many natural objects show symmetry.
🔵 A butterfly has two wings of equal shape and size on both sides.
✔️ This shows symmetry in nature.

🟡 Question
Q22. State two uses of symmetry in daily life.

🟢 Answer
🔵 Symmetry is used in designing buildings and bridges.
🔵 It is used in art, logos, and decorative patterns to create balance.

🔴 Section D — Long Answer

(Q23–Q30 | 4 × 8 = 32 marks)

🔴 Question
Q23. Explain the concept of symmetry with the help of a daily-life example.

🟢 Answer
🔵 Symmetry means balance and sameness on both sides of a figure.
🔵 A figure is symmetrical if it can be divided into two identical mirror-image parts.
🔵 For example, a butterfly has two wings that are equal in shape and size.
🔵 If we draw a line through the middle of the butterfly, both sides look exactly the same.

✔️ This line is called the line of symmetry.

🔴 Question
Q24. What is a line of symmetry? Explain how it divides a figure.

🟢 Answer
🔵 A line of symmetry divides a figure into two equal and identical halves.
🔵 When the figure is folded along this line, both parts coincide exactly.
🔵 Each point on one side has a matching point on the other side at the same distance from the line.

✔️ Thus, the line of symmetry acts like a mirror.

🔴 Question
Q25. Describe the folding method used to check symmetry of a figure.

🟢 Answer
🔵 Fold the figure along a straight line.
🔵 Check whether both halves overlap completely.
🔵 If they overlap, the fold line is a line of symmetry.
🔵 If they do not overlap, the figure is not symmetrical along that line.

✔️ Folding helps in identifying symmetry easily.

🔴 Question
Q26. Explain symmetry in a square. How many lines of symmetry does it have?

🟢 Answer
🔵 A square has all sides equal and all angles equal.
🔵 It can be divided equally in more than one way.
🔵 Two lines of symmetry pass through the midpoints of opposite sides.
🔵 Two lines of symmetry pass through the diagonals.

✔️ Therefore, a square has 4 lines of symmetry.

🔴 Question
Q27. Explain why a rectangle has fewer lines of symmetry than a square.

🟢 Answer
🔵 In a rectangle, only opposite sides are equal.
🔵 Diagonals do not divide the rectangle into mirror-image halves.
🔵 Only one vertical and one horizontal line divide it equally.

✔️ Hence, a rectangle has 2 lines of symmetry, fewer than a square.

🔴 Question
Q28. Explain reflection symmetry with the help of a mirror.

🟢 Answer
🔵 Reflection symmetry means one half of a figure is the mirror image of the other.
🔵 A mirror placed along the line of symmetry shows the other half exactly.
🔵 Each point and its image are at equal distance from the mirror line.

✔️ This type of symmetry is also called mirror symmetry.

🔴 Question
Q29. Explain symmetry seen in nature and man-made objects with examples.

🟢 Answer
🔵 Many natural objects like butterflies, flowers, and leaves show symmetry.
🔵 In man-made objects, buildings, logos, and vehicles show symmetry.
🔵 Symmetry gives balance, beauty, and stability to objects.

✔️ Thus, symmetry is common in nature and human designs.

🔴 Question
Q30. Why is symmetry important in mathematics and design? Give reasons.

🟢 Answer
🔵 Symmetry helps in understanding shapes and patterns easily.
🔵 It makes figures look balanced and attractive.
🔵 In design and construction, symmetry ensures proper structure and stability.
🔵 It develops visual thinking and logical understanding.

✔️ Therefore, symmetry plays an important role in mathematics and daily life.

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