ML Aggarwal Class 6 Maths Solutions Chapter 12 Symmetry

Question 1(i). Draw the line (or lines) of symmetry, if any, of the following shape and count their number:
Answer: The shape shown is a quarter circle (quadrant). This figure has a total of 1 line of symmetry. This line connects the corner point to the midpoint of the curved arc.
In simple words: A quarter circle has one fold line that goes from the corner to the middle of the curved edge.

Exam Tip: Always identify the shape first, then check if folding along any line makes both halves match perfectly.

Question 1(ii). Draw the line (or lines) of symmetry, if any, of the following shape and count their number:
Answer: The figure is a scalene right-angled triangle where all three sides have different lengths. Since no two sides are equal and all angles are different, this shape has no line of symmetry.
In simple words: A scalene right triangle has no fold line because all its sides and angles are different from each other.

Exam Tip: A shape has a line of symmetry only if folding it creates two matching halves - scalene triangles never do this.

Question 1(iii). Draw the line (or lines) of symmetry, if any, of the following shape and count their number:
Answer: The given figure has a total of 1 line of symmetry. This line passes horizontally through the middle of the shape.
In simple words: This shape can be folded in half along one horizontal line down the middle, and both sides match perfectly.

Exam Tip: Look for shapes that are mirror images of each other on either side of a line - that line is your line of symmetry.

Question 1(iv). Draw the line (or lines) of symmetry, if any, of the following shape and count their number:
Answer: The figure shows two unequal overlapping circles. This shape has a total of 1 line of symmetry. This line connects the centres of both circles.
In simple words: When two circles of different sizes overlap, folding along the line joining their centres makes both halves match.

Exam Tip: For overlapping circles, the line of symmetry always passes through both centres if the circles have their centres on a common line.

Question 1(v). Draw the line (or lines) of symmetry, if any, of the following shape and count their number:
Answer: The shape shown is similar to a yin-yang design (an 'S' shape inside a circle). This figure has no line of symmetry. No matter which line you try to fold it along, the curved portions will point in opposite directions, so the two halves do not match.
In simple words: A yin-yang shape cannot be folded in any direction and have both halves look the same because the curves bend different ways.

Exam Tip: Curved shapes that spiral or twist, like the yin-yang, rarely have lines of symmetry - check by mentally folding along different lines.

Question 1(vi). Draw the line (or lines) of symmetry, if any, of the following shape and count their number:
Answer: The shape consists of three rays meeting at a single point, with equal angles of 120 degrees between each pair of rays. This shape has no line of symmetry because folding along any line will not produce two matching halves.
In simple words: Three rays spreading equally from one point cannot be folded to make matching halves.

Exam Tip: When three or more lines meet at a point, check carefully - sometimes there are lines of symmetry, sometimes there are not, depending on their spacing.

Question 1(vii). Draw the line (or lines) of symmetry, if any, of the following shape and count their number:
Answer: The figure has an hourglass or bow-tie shape. This shape has a total of 1 line of symmetry - a vertical line passing through the centre of the figure.
In simple words: An hourglass shape can be folded along one vertical line down the middle, and both halves match perfectly.

Exam Tip: Symmetrical hourglass shapes always have a vertical line of symmetry if the shape is balanced left and right.

Question 1(viii). Draw the line (or lines) of symmetry, if any, of the following shape and count their number:
Answer: The shape is an arrow pointing upward. This shape has a total of 1 line of symmetry. The line of symmetry runs along the length of the arrow, dividing it into two identical halves.
In simple words: An arrow has one fold line that runs from the tip to the tail, making both sides look the same.

Exam Tip: Arrow-shaped figures always have one vertical or central line of symmetry if they are properly balanced.

Question 1(ix). Draw the line (or lines) of symmetry, if any, of the following shape and count their number:
Answer: The figure is a three-leaf clover or trefoil shape. This shape has a total of 3 lines of symmetry. Each line passes through the centre and goes through one of the leaves.
In simple words: A three-leaf clover has three fold lines - one through each leaf and the centre - and folding along any of them makes the shape match itself.

Exam Tip: For petal or leaf shapes, count the number of petals or leaves - a shape usually has that many lines of symmetry if all petals are identical.

Question 2(i). Draw the line (or lines) of symmetry, if any, of the following picture (of objects) and count their number:
Answer: The picture shows a telephone. The telephone has a total of 1 line of symmetry - a vertical line running through the centre of the object.
In simple words: A telephone can be folded along one vertical line down the middle, and the left and right halves look identical.

Exam Tip: Many everyday objects like telephones, chairs, and flowers have vertical lines of symmetry - look at the left and right sides to check.

Question 2(ii). Draw the line (or lines) of symmetry, if any, of the following picture (of objects) and count their number:
Answer: The picture shows a chair. A chair has a total of 1 line of symmetry - a vertical line running through the centre of the chair, dividing it into left and right halves.
In simple words: A chair has one fold line that goes down the middle - the left side and right side look the same.

Exam Tip: Furniture items often have vertical symmetry - check if the left and right sides are mirror images.

Question 2(iii). Draw the line (or lines) of symmetry, if any, of the following picture (of objects) and count their number:
Answer: The picture shows the command symbol (⌘). This symbol has a total of 4 lines of symmetry: one horizontal line through the centre, one vertical line through the centre, and two diagonal lines passing through the centre.
In simple words: The command symbol can be folded four different ways - horizontally, vertically, and along both diagonals - and each time both halves match.

Exam Tip: Symbols with four-fold symmetry have four identical ways to fold - look for horizontal, vertical, and both diagonal lines.

Question 2(iv). Draw the line (or lines) of symmetry, if any, of the following picture (of objects) and count their number:
Answer: The picture shows a five-petalled flower. This flower has a total of 5 lines of symmetry - one line passes through the centre and through each petal.
In simple words: A five-petalled flower has five fold lines - one going through each petal and the middle - and folding along any makes both sides match.

Exam Tip: Regular flowers with identical petals have as many lines of symmetry as they have petals - count the petals to find the answer quickly.

Question 2(v). Draw the line (or lines) of symmetry, if any, of the following picture (of objects) and count their number:
Answer: The picture shows a fly or housefly. A fly has a total of 1 line of symmetry - a vertical line that runs through the centre of its body.
In simple words: A fly has one fold line running down the middle of its body, with wings and legs balanced on both sides.

Exam Tip: Insects with bilateral body structure (left and right sides that match) have one vertical line of symmetry through their body.

Question 2(vi). Draw the line (or lines) of symmetry, if any, of the following picture (of objects) and count their number:
Answer: The decorative figure shown has a total of 1 line of symmetry - a vertical line passing through its centre.
In simple words: This decorative design can be folded along one vertical line, and the left and right halves look exactly the same.

Exam Tip: When examining decorative figures, look first for vertical or horizontal lines of symmetry before checking for diagonal ones.

Question 3(i). Draw the line (or lines) of symmetry, if any, of the following picture (of objects) and count their number:
Answer: The decorative figure shown has a total of 1 line of symmetry - a vertical line that passes through its centre.
In simple words: This figure looks the same when folded along one vertical line running down the middle.

Exam Tip: Complex decorative designs often have just one line of symmetry even though they have many shapes - check by identifying which direction creates a perfect mirror image.

Question 3(ii). Draw the line (or lines) of symmetry, if any, of the following road signs and count their number:
Answer: The road sign shown (speed breaker/hump) has 1 line of symmetry. This line runs vertically through the top point of the triangle.
In simple words: If you fold this triangle sign down the middle from top to bottom, both sides match perfectly. That fold line is the one line of symmetry.

Exam Tip: Always check if a shape looks the same when folded or reflected across a line - that line is a line of symmetry.

Question 3(iii). Draw the line (or lines) of symmetry, if any, of the following road signs and count their number:
Answer: The road sign shown (narrow bridge/barrier) has 1 line of symmetry. This line passes vertically through the top point of the triangle.
In simple words: Draw a straight line from the top point down to the bottom middle of the triangle. The left half and right half look identical when you fold along this line.

Exam Tip: Triangular signs with symmetrical left and right sides always have one vertical line of symmetry running from apex to base.

Question 3(iv). Draw the line (or lines) of symmetry, if any, of the following road signs and count their number:
Answer: The road sign shown (cross-road) has 1 line of symmetry passing vertically through the top of the triangle. Although the symbol X by itself has 4 lines of symmetry, the triangular border limits the shape so only the vertical line works as a shared symmetry line.
In simple words: The X shape inside the triangle has many possible fold lines, but only the vertical line through the triangle's top makes both parts of the whole sign match up.

Exam Tip: When a shape contains another shape inside, the symmetry lines must work for both the inside and the outside - not just one of them.

Question 3(v). Draw the line (or lines) of symmetry, if any, of the following road signs and count their number:
Answer: The road sign shown (T-junction) has 1 line of symmetry passing vertically through the top point of the triangle.
In simple words: Fold the triangle from top to bottom down the middle. The T shape inside and the triangle outside both match perfectly on both sides of this fold line.

Exam Tip: Look for the centre line that divides a shape into two parts that are mirror images of each other - that is always a line of symmetry.

Question 4(i). Draw the line (or lines) of symmetry, if any, of the following numeral and count their number:
Answer: The numeral 3 has 1 line of symmetry. This line runs horizontally through the middle of the digit.
In simple words: If you fold the number 3 in half left to right, the top and bottom halves look the same when reflected. That horizontal fold line in the middle is its line of symmetry.

Exam Tip: Not all numerals have lines of symmetry - check by trying to fold them different ways (horizontal, vertical, or diagonal) to see if both parts match.

Question 4(ii). Draw the line (or lines) of symmetry, if any, of the following numeral and count their number:
Answer: The numeral 5 has no line of symmetry. You cannot fold it in any direction to get two matching halves.
In simple words: No matter which way you try to fold or flip the number 5, the two sides never look the same - so it has no symmetry line.

Exam Tip: Asymmetrical numerals (like 5, 2, and 7) are commonly tested - remember that lack of symmetry is also a valid answer.

Question 4(iii). Draw the line (or lines) of symmetry, if any, of the following numeral and count their number:
Answer: The numeral 6 has no line of symmetry. Folding it horizontally, vertically, or diagonally does not produce two matching halves.
In simple words: The curved part of 6 is on just one side. When you try to fold it, the two halves do not fit or match each other.

Exam Tip: Numbers that curve in one direction only (like 6 and 9) never have a line of symmetry because the curve is not balanced.

Question 4(iv). Draw the line (or lines) of symmetry, if any, of the following numeral and count their number:
Answer: The numeral 8 has 2 lines of symmetry - one horizontal and one vertical - both running through the centre of the digit.
In simple words: The number 8 looks the same whether you fold it left-to-right or top-to-bottom. Both fold lines create matching halves, so it has 2 lines of symmetry.

Exam Tip: When a shape is balanced both ways (horizontally and vertically), it will have two lines of symmetry - always check both directions.

Question 5(i). Copy the following figure on a squared paper and draw the lines of symmetry (if any) and count their number:
Answer: The given figure (two circles stacked, one slightly larger on top) has no line of symmetry due to unequal extensions at the sides.
In simple words: The two circles are not positioned in a way that makes them mirror each other. The sides stick out unequally, so no fold line creates matching halves.

Exam Tip: When checking complex figures, trace mentally along each possible fold line - if any part does not match, it has no symmetry on that axis.

Question 5(ii). Copy the following figure on a squared paper and draw the lines of symmetry (if any) and count their number:
Answer: The given figure (two diamonds stacked vertically) has 1 line of symmetry - the vertical line running up and down through the centre.
In simple words: Draw a line straight down the middle. The left side and right side look identical when you fold along this vertical line.

Exam Tip: For stacked or repeated shapes, the vertical and horizontal axes are the most common symmetry lines to check first.

Question 5(iii). Copy the following figure on a squared paper and draw the lines of symmetry (if any) and count their number:
Answer: The given figure (a smaller square nested inside a larger square) has 4 lines of symmetry - the horizontal line, the vertical line, and both diagonal lines all passing through the centre.
In simple words: You can fold this shape four different ways: top-to-bottom, left-to-right, and along both corners. Every fold creates two matching halves.

Exam Tip: Squares and rectangles always have 4 and 2 lines of symmetry respectively - memorise these standard shapes.

Question 5(iv). Copy the following figure on a squared paper and draw the lines of symmetry (if any) and count their number:
Answer: The given figure is an elongated hexagon. It has 2 lines of symmetry - the horizontal line and the vertical line passing through its centre.
In simple words: Fold this shape in half from left to right, and the sides match. Fold it in half from top to bottom, and they match again. But folding diagonally does not work.

Exam Tip: Irregular hexagons often have 2 lines of symmetry (horizontal and vertical) if they are stretched equally in two directions.

Question 5(v). Copy the following figure on a squared paper and draw the lines of symmetry (if any) and count their number:
Answer: The given figure (a kite or pentagon-like shape) has 1 line of symmetry - the horizontal line passing through its centre.
In simple words: Imagine folding this shape in half along a line that runs left to right through the middle. The top and bottom halves fit perfectly - that is the one line of symmetry.

Exam Tip: Kite shapes have a single line of symmetry along their longest diagonal or axis of balance.

Question 5(vi). Copy the following figure on a squared paper and draw the lines of symmetry (if any) and count their number:
Answer: The given figure (a four-pointed star) has 4 lines of symmetry - the horizontal line, the vertical line, and both diagonal lines passing through the centre.
In simple words: A four-pointed star looks the same from four different angles. You can fold it along the middle (up-down), the middle (left-right), and along both corner-to-corner lines. All four folds create matching halves.

Exam Tip: Stars and other shapes with multiple points usually have one symmetry line per point direction - a four-pointed star has 4 symmetry lines.

Question 6. Write the letters of words 'JUST LOOK' which have no line of symmetry.
Answer: Let us examine each letter in 'JUST LOOK':

• J - has no line of symmetry.
• U - has 1 line of symmetry (vertical).
• S - has no line of symmetry.
• T - has 1 line of symmetry (vertical).
• L - has no line of symmetry.
• O - has 2 lines of symmetry (horizontal and vertical).
• O - has 2 lines of symmetry (horizontal and vertical).
• K - has 1 line of symmetry (horizontal).

Exam Tip: Test each letter by drawing it and trying to fold along different axes - vertical, horizontal, and diagonal - to find all possible symmetry lines.

Question 7. Can you draw a triangle which has (i) exactly one line of symmetry? (ii) exactly two lines of symmetry? (iii) exactly three lines of symmetry? (iv) no lines of symmetry? Sketch a rough figure in each case and name the triangle.
Answer:
(i) Yes, you can draw a triangle with exactly one line of symmetry. It is an isosceles triangle. The line of symmetry is the perpendicular drawn from the vertex angle (the angle between the two equal sides) to the midpoint of the base.
(ii) No, you cannot draw a triangle with exactly two lines of symmetry. If a triangle has a line of symmetry, it cannot have exactly two - the possible counts are zero, one, or three only.
(iii) Yes, you can draw a triangle with exactly three lines of symmetry. It is an equilateral triangle. All three lines of symmetry are the perpendiculars from each vertex to the midpoint of the opposite side. These three lines meet at the centre of the triangle.
(iv) Yes, you can draw a triangle with no lines of symmetry. It is a scalene triangle (all three sides of different lengths). No fold line creates matching halves for such a triangle.
In simple words: Isosceles triangles have 1 symmetry line, equilateral triangles have 3, and scalene triangles have none. No triangle can have exactly 2 lines of symmetry.

Exam Tip: Remember the three types of triangles by symmetry count: isosceles = 1, equilateral = 3, scalene = 0. The impossibility of 2 lines of symmetry in a triangle is an important exam concept.

Question 1(i). Copy the following figure on a squared paper. Complete each of them such that the dotted line is the line of symmetry:
Answer: To finish the figure, draw the reflected image of the given part across the dotted line so that the dotted line acts as a line of symmetry.
In simple words: Draw the mirror image of the given portion on the other side of the dotted line.

Exam Tip: Always count grid squares carefully to ensure the reflected part is an exact distance from the line as the original part.

Question 1(ii). Copy the following figure on a squared paper. Complete each of them such that the dotted line is the line of symmetry:
Answer: To finish the figure, draw the reflected image of the given part across the dotted line so that the dotted line acts as a line of symmetry.
In simple words: Draw the mirror image of the given portion on the other side of the dotted line.

Exam Tip: Always count grid squares carefully to ensure the reflected part is an exact distance from the line as the original part.

Question 1(iii). Copy the following figure on a squared paper. Complete each of them such that the dotted line is the line of symmetry:
Answer: To finish the figure, draw the reflected image of the given part across the dotted line so that the dotted line acts as a line of symmetry.
In simple words: Draw the mirror image of the given portion on the other side of the dotted line.

Exam Tip: Always count grid squares carefully to ensure the reflected part is an exact distance from the line as the original part.

Question 2(i). Copy the following figure on a squared paper. Complete each of them such that the resultant figure has two dotted lines as the lines of symmetry:
Answer: To finish the figure, first reflect the given part in one of the dotted lines. Then reflect the resulting figure in the other dotted line. This makes the final figure symmetric about both dotted lines.
In simple words: Reflect the figure across one dotted line, then reflect what you get across the second dotted line.

Exam Tip: Work step by step - first reflection, then second reflection - to avoid mistakes. Check that the final figure is balanced on both sides of both lines.

Question 2(ii). Copy the following figure on a squared paper. Complete each of them such that the resultant figure has two dotted lines as the lines of symmetry:
Answer: To finish the figure, first reflect the given part in one of the dotted lines. Then reflect the resulting figure in the other dotted line. This makes the final figure symmetric about both dotted lines.
In simple words: Reflect the figure across one dotted line, then reflect what you get across the second dotted line.

Exam Tip: Work step by step - first reflection, then second reflection - to avoid mistakes. Check that the final figure is balanced on both sides of both lines.

Question 2(iii). Copy the following figure on a squared paper. Complete each of them such that the resultant figure has two dotted lines as the lines of symmetry:
Answer: To finish the figure, first reflect the given part in one of the dotted lines. Then reflect the resulting figure in the other dotted line. This makes the final figure symmetric about both dotted lines.
In simple words: Reflect the figure across one dotted line, then reflect what you get across the second dotted line.

Exam Tip: Work step by step - first reflection, then second reflection - to avoid mistakes. Check that the final figure is balanced on both sides of both lines.

Question 3. In the adjoining figure, l is the line of symmetry. Complete the diagram to make it symmetrical.
Answer: To make the given figure symmetric about the line l, draw the reflected image of the given part in the line l. The complete diagram is shown by this reflection.
In simple words: Draw the mirror image of the given part on the other side of line l.

Exam Tip: Each point on the original figure and its reflection are equidistant from the line of symmetry.

Question 4(i). Copy the following figure on a squared paper and find their reflections in the mirror line l.
Answer: The reflected image is obtained by drawing the mirror image in the line l. The reflected image has the same shape and size as the original figure, but with a left-right change in orientation. The mirror line l is the perpendicular bisector of the line segments joining the corresponding points (vertices) of the figure and its image.
In simple words: The reflection looks the same size and shape but is flipped across the mirror line l.

Exam Tip: The perpendicular distance from any point to the mirror line equals the perpendicular distance from that point's reflection to the mirror line.

Question 4(ii). Copy the following figure on a squared paper and find their reflections in the mirror line l.
Answer: The reflected image is obtained by drawing the mirror image in the line l. The reflected image has the same shape and size as the original figure, but with a left-right change in orientation. The mirror line l is the perpendicular bisector of the line segments joining the corresponding points (vertices) of the figure and its image.
In simple words: The reflection looks the same size and shape but is flipped across the mirror line l.

Exam Tip: The perpendicular distance from any point to the mirror line equals the perpendicular distance from that point's reflection to the mirror line.

Objective Type Questions - Mental Maths

Question 1. Fill in the blanks:
(i) The subtraction symbol - has ..... lines of symmetry.
(ii) The addition symbol + has ..... lines of symmetry.
(iii) Line of symmetry is also known ..... line or ..... of symmetry.
(iv) A kite has ..... line(s) of symmetry.
(v) A parallelogram has ..... line(s) of symmetry.
(vi) The number of lines of symmetry in a regular hexagon is ..... .
(vii) A rectangle is symmetrical about the lines joining the ..... of the opposite sides.
(viii) The number of capital letters of the English alphabet having only vertical line of symmetry is ..... .
(ix) The number of capital letters of the English alphabet having only horizontal line of symmetry is ..... .
(x) The number of capital letters of the English alphabet having both horizontal and vertical lines of symmetry is ..... .
(xi) The digits having two lines of symmetry are ..... and ..... .
Answer:
(i) The subtraction symbol - has 2 lines of symmetry.
(ii) The addition symbol + has 4 lines of symmetry.
(iii) Line of symmetry is also known mirror line or axis of symmetry.
(iv) A kite has 1 line(s) of symmetry.
(v) A parallelogram has no line(s) of symmetry.
(vi) The number of lines of symmetry in a regular hexagon is 6.
(vii) A rectangle is symmetrical about the lines joining the mid-points of the opposite sides.
(viii) The number of capital letters of the English alphabet having only vertical line of symmetry is 7 (A, M, T, U, V, W, Y).
(ix) The number of capital letters of the English alphabet having only horizontal line of symmetry is 4 (B, C, D, E).
(x) The number of capital letters of the English alphabet having both horizontal and vertical lines of symmetry is 4 (H, I, O, X).
(xi) The digits having two lines of symmetry are 0 and 8.
In simple words: Different shapes and symbols have different numbers of lines of symmetry. Symmetric shapes fold perfectly so both halves match exactly.

Exam Tip: Memorise the symmetry counts for common shapes and letters - these facts often appear in fill-in-the-blank and short-answer questions.

Question 2. State whether the following statements are true (T) or false (F):
(i) The letter N has one line of symmetry.
(ii) Every hexagon has six lines of symmetry.
(iii) All right angled triangles have one line of symmetry.
(iv) A triangle with more than one line of symmetry must be an equilateral triangle.
(v) A line of symmetry divides a figure into two identical parts.
(vi) A circle has only 8 lines of symmetry.
(vii) A regular octagon has 10 lines of symmetry.
(viii) A square and a rectangle have the same number of lines of symmetry.
(ix) A right angled triangle can have atmost one line of symmetry.
(x) If an isosceles triangle has more than one line of symmetry, then it must be an equilateral triangle.
(xi) If a rectangle has more than two lines of symmetry, then it must be a square.
Answer:
(i) False - The letter N has no lines of symmetry.
(ii) False - Only a regular hexagon has six lines of symmetry. A non-regular hexagon may have fewer or no lines of symmetry.
(iii) False - Most right angled triangles have no line of symmetry. Only a right angled isosceles triangle has one line of symmetry.
(iv) True - A triangle with multiple lines of symmetry must be equilateral, as it must have all sides and angles equal.
(v) True - A line of symmetry divides a figure into two congruent and identical parts.
(vi) False - A circle has infinitely many lines of symmetry, not just 8.
(vii) False - A regular octagon has 8 lines of symmetry, not 10.
(viii) False - A square has 4 lines of symmetry while a rectangle has only 2 lines of symmetry.
(ix) True - A right angled triangle can have at most one line of symmetry (if it is isosceles right angled).
(x) True - If an isosceles triangle has more than one line of symmetry, it must be equilateral, as all three sides must be equal.
(xi) True - If a rectangle has more than two lines of symmetry, it has four lines of symmetry, which means all sides and angles are equal, making it a square.
In simple words: Shapes have different symmetry based on their side lengths and angles. Regular shapes (all sides and angles equal) have more lines of symmetry than irregular shapes.

Exam Tip: Pay attention to the difference between regular and non-regular polygons - only regular shapes have the standard number of lines of symmetry. Always visualise or draw the shape to check for symmetry.

Question 1. (i) False
Answer: The letter N lacks any line of symmetry. It possesses only rotational symmetry of order 2.
In simple words: The letter N cannot be folded in half to make two identical parts. It can only be turned around.

Exam Tip: Distinguish between line symmetry (mirror image) and rotational symmetry (spinning) - they are not the same thing.

Question 1. (ii) False
Answer: Only a regular hexagon possesses six lines of symmetry. An irregular hexagon may have fewer or no lines of symmetry.
In simple words: A regular hexagon (all sides equal) has six mirror lines. But if a hexagon has unequal sides, it might have zero or fewer mirror lines.

Exam Tip: Remember that "regular" shapes have symmetry, but "irregular" shapes often lose it.

Question 1. (iii) False
Answer: Only an isosceles right-angled triangle possesses one line of symmetry. A scalene right-angled triangle has no line of symmetry.
In simple words: A right triangle with two equal sides can fold in half. But a right triangle with all different sides cannot.

Exam Tip: For triangles, check if at least two sides are equal - that is when a line of symmetry exists.

Question 1. (iv) True
Answer: An equilateral triangle possesses more than one line of symmetry - in fact, it has three lines of symmetry.
In simple words: An equilateral triangle (all three sides equal) can be folded three different ways to make two identical halves.

Exam Tip: The equilateral triangle is the only triangle with more than one line of symmetry.

Question 1. (v) True
Answer: A line of symmetry divides a figure into two mirror-image parts that are identical in shape and size.
In simple words: A line of symmetry splits a shape so that both sides are exact mirror images of each other.

Exam Tip: The key test for a line of symmetry is: can you fold the paper along it and have both halves match perfectly?

Question 1. (vi) False
Answer: A circle has an infinite number of lines of symmetry, since every diameter serves as a line of symmetry.
In simple words: A circle can be folded along countless different lines through its centre, and each time the two halves will match.

Exam Tip: A circle is unique - it has unlimited symmetry lines, not just a few.

Question 1. (vii) False
Answer: A regular octagon has 8 lines of symmetry, not 10.
In simple words: A regular octagon (8-sided shape with all sides equal) has exactly 8 mirror lines, not 10.

Exam Tip: For regular polygons, the number of lines of symmetry equals the number of sides.

Question 1. (viii) False
Answer: A square possesses 4 lines of symmetry, whereas a rectangle has only 2 lines of symmetry.
In simple words: A square can fold four ways and match. A rectangle can fold only two ways and match.

Exam Tip: A square is a special rectangle with more symmetry lines.

Question 1. (ix) True
Answer: A right-angled triangle has at most one line of symmetry (in the case of an isosceles right-angled triangle). A scalene right-angled triangle has none.
In simple words: Some right triangles have one mirror line (if two sides are equal). Others have zero mirror lines (if all sides differ).

Exam Tip: The word "at most" means "one or fewer" - this is the key phrase that makes the statement true.

Question 1. (x) True
Answer: An isosceles triangle has exactly one line of symmetry. If it were to have more than one, the third side would also need to be equal, making it an equilateral triangle instead.
In simple words: An isosceles triangle (two equal sides) can fold one way. If it folded more ways, all three sides would be equal, making it equilateral.

Exam Tip: Think about what happens to the shape if you add more mirror lines - it must become more regular.

Question 1. (xi) True
Answer: A rectangle has exactly two lines of symmetry. If it possessed more (such as the diagonals), all its sides would need to be equal, making it a square.
In simple words: A rectangle folds two ways to match. If it folded more, it would become a square.

Exam Tip: Rectangles and squares are closely related - adding symmetry to a rectangle turns it into a square.

Question 3. The number of lines of symmetry of a scalene triangle is:
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (a) 0
In simple words: A scalene triangle has all three sides of different lengths and all three angles of different sizes. Since no two sides or angles match, there is no way to fold it and create two identical halves.

Exam Tip: For any triangle to have a line of symmetry, at least two sides must be equal in length.

Question 4. The letter F has
(a) one horizontal line of symmetry
(b) one vertical line of symmetry
(c) two lines of symmetry
(d) no line of symmetry
Answer: (d) no line of symmetry
In simple words: The letter F cannot be folded along any line (whether horizontal or vertical) to make the two parts match exactly.

Exam Tip: Try folding the letter F horizontally and vertically - neither produces matching halves.

Question 5. The number of lines of symmetry of a rectangle is:
(a) 0
(b) 1
(c) 2
(d) 4
Answer: (c) 2
In simple words: A rectangle can fold along two lines - one through the midpoints of the top and bottom sides, and another through the midpoints of the left and right sides.

Exam Tip: The two mirror lines of a rectangle connect the midpoints of opposite sides, not the corners or diagonals.

Question 6. A rhombus is symmetrical about
(a) each of its two diagonals
(b) each of its two lines joining the midpoints of opposite sides
(c) each of the perpendicular bisector of its sides
(d) none of these
Answer: (a) each of its two diagonals
In simple words: A rhombus (a diamond-shaped quadrilateral with all sides equal) has two lines of symmetry, and these are exactly its two diagonals. Folding along either diagonal makes the two halves fit perfectly together.

Exam Tip: The diagonals of a rhombus are always lines of symmetry - this is a defining property of rhombuses.

Question 7. The number of lines of symmetry of a circle is
(a) 4
(b) 8
(c) 16
(d) unlimited
Answer: (d) unlimited
In simple words: Every line that passes through the centre of a circle and touches the edge at both ends (a diameter) divides the circle into two matching halves. Since there are infinitely many such lines, the circle has unlimited lines of symmetry.

Exam Tip: A circle is the only common shape with infinite lines of symmetry - no other 2D figure has this property.

Question 8. Which of the following letters does not have any line of symmetry?
(a) B
(b) T
(c) Z
(d) Y
Answer: (c) Z
In simple words: The letter B can fold horizontally, T can fold vertically, and Y can fold vertically - all produce matching halves. But Z cannot fold in any direction to create two identical parts. It only has rotational symmetry (spinning turns it into itself), not line symmetry.

Exam Tip: Letters like Z, S, and N have rotational symmetry but no line symmetry - do not confuse the two.

Question 9. Which of the following letters does not have the vertical line of symmetry?
(a) A
(b) H
(c) M
(d) E
Answer: (d) E
In simple words: The letter A can fold vertically down its middle, H can fold both vertically and horizontally, and M can fold vertically. But E can only fold horizontally - it has no vertical line of symmetry.

Exam Tip: Always check whether the letter looks the same on both sides of the proposed mirror line.

Question 10. Which figure from the following figures is not symmetrical with respect to any line?
Answer: Figures (1) rectangle, (3) triangle, and (4) circle all possess at least one line of symmetry. Figure (2) is an irregular or asymmetric shape (resembling a zig-zag stepped pattern) and cannot be folded along any line to produce matching halves.
In simple words: Three of the shapes can be folded to match themselves. Only one shape (the stepped zig-zag) cannot be folded in any way to create two identical halves.

Exam Tip: When identifying asymmetric figures, look for irregular or jagged outlines with no repeating pattern.

Question 11. In which of the given figures is the dotted line of symmetry?
Answer:
(i) A diagonal line drawn through a rectangle is not a line of symmetry.
(ii) A horizontal line through an elongated hexagon is not a line of symmetry.
(iii) A diagonal line through a parallelogram is not a line of symmetry.
(iv) The vertical line through the arrowhead serves as its line of symmetry.
In simple words: Three shapes fail the folding test. Only the arrowhead (figure 4) folds along the dotted line with both halves matching perfectly.

Exam Tip: Always fold mentally or physically - if the two parts do not align exactly, it is not a line of symmetry.

Question 12. Amongst the given figures, the one having maximum number of lines of symmetry is:
Answer:
(i) An arrow (in a circle) possesses 1 line of symmetry.
(ii) A plus sign + (in a circle) has 4 lines of symmetry - horizontal, vertical, and the two diagonals.
(iii) A division sign ÷ (in a circle) has 2 lines of symmetry - horizontal and vertical.
(iv) An X (in a circle) has 2 lines of diagonal symmetry, because the arms of the X in this figure are not equal.

The plus sign + displays the highest number of lines of symmetry among all four figures.
In simple words: The plus sign can fold four different ways: up-down, left-right, and along both diagonals. No other symbol shown can fold as many ways.

Exam Tip: The plus sign is perfectly balanced in all four directions, making it highly symmetric.

Question 13. The angle between the mirror line l and the line segment joining a point and its reflection (image) is:
(a) 0°
(b) 45°
(c) 60°
(d) 90°
Answer: (d) 90°
In simple words: The mirror line always meets the line connecting a point to its reflection at a right angle (90 degrees). This is a defining property of reflection in geometry.

Exam Tip: Remember: mirror line is always perpendicular to the line joining the object and its image.

Question 14. Which of two figures are images of each other (mirror line shown dotted)?
Answer: For two figures to be mirror images of one another, they must meet three conditions: (i) they must have identical shape and size, (ii) they must be positioned at equal distances from the mirror line, and (iii) they must be placed on opposite sides of the mirror line while showing a left-right reversal of orientation.

Only option 3 satisfies all three of these requirements.
In simple words: Two figures are mirror images if they match in shape, sit the same distance from the mirror line on opposite sides, and have opposite left-right directions.

Exam Tip: Always check all three conditions - having the same shape alone is not enough.

Question 15. Which of the two figures are mirror images of each other (mirror line shown dotted)?
Answer: For two figures to be mirror images of one another, they must have identical shape and size and be located at equal distances on opposite sides of the mirror line while showing a left-right orientation change.

Only option 3 meets all of these criteria.
In simple words: Mirror images must be identical in form, equally spaced on either side of the line, and reversed left-to-right.

Exam Tip: When checking mirror images, mentally fold the paper along the dotted line - the figures should align perfectly.

Question 16. Statement I: Any two-dimensional figure has atleast one line of symmetry. Statement II: A mirror line is called a line of symmetry.
(a) Statement I is true but statement II is false.
(b) Statement I is false but Statement II is true.
(c) Both Statement I and Statement II are true.
(d) Both Statement I and Statement II are false.
Answer: (b) Statement I is false but Statement II is true
In simple words: Statement I is wrong because shapes like scalene triangles and parallelograms have no lines of symmetry at all. Statement II is correct because a line of symmetry is indeed the same as a mirror line.

Exam Tip: Not all 2D figures have symmetry - this is a common misconception. Always check individual shapes.

Question 17. Statement I: Ram draws two squares and two rectangles, as shown in the adjacent figure. Next, he draws lines of symmetry in each figure. The total lines of symmetry he draws are 10. Statement II: All the lines of symmetry of a circle are also its diameters.
(a) Statement I is true but statement II is false.
(b) Statement I is false but Statement II is true.
(c) Both Statement I and Statement II are true.
(d) Both Statement I and Statement II are false.
Answer: (b) Statement I is false but Statement II is true
In simple words: Statement I is wrong: each square has 4 symmetry lines (total 8 for two squares) and each rectangle has 2 symmetry lines (total 4 for two rectangles), adding to 12, not 10. Statement II is correct: every line of symmetry in a circle passes through its centre, which makes it a diameter by definition.

Exam Tip: For Statement I, always count: 2 squares = 2 × 4 = 8 lines; 2 rectangles = 2 × 2 = 4 lines; total = 12 lines.

Question 18. Statement I: Consider the adjoining figure. It has only one line of symmetry. Statement II: A semicircle has one line of symmetry, whereas an arrowhead has 2 lines of symmetry.
(1) Statement I is true but statement II is false.
(2) Statement I is false but Statement II is true.
(3) Both Statement I and Statement II are true.
(4) Both Statement I and Statement II are false.
Answer: (1) Statement I is true but statement II is false
In simple words: The combined figure (arrowhead on top of a semicircle) has exactly one vertical line of symmetry. A semicircle does have one line of symmetry, but an arrowhead only has one as well - not two.

Exam Tip: Always count lines of symmetry by checking if each half mirrors the other exactly when folded along that line. Test each shape separately before combining them.

Question 19. Statement I: Suhana bakes a circular pie. Then she eats 1/4 th of the pie. The resulting figure is shown on the right. There are 2 lines of symmetry in the remaining pie. Statement II: If a figure is divided into two coincident parts by a line, then the figure is called symmetrical about that line.
(1) Statement I is true but statement II is false.
(2) Statement I is false but Statement II is true.
(3) Both Statement I and Statement II are true.
(4) Both Statement I and Statement II are false.
Answer: (2) Statement I is false but Statement II is true
In simple words: When you remove a quarter of a circular pie, the remaining 3/4 pie has just one line of symmetry - the diagonal that cuts the missing quarter in half. The definition in Statement II is correct.

Exam Tip: Remember that a pie with a quarter removed loses most of its symmetry lines. The single remaining line passes through the centre and bisects the removed section.

Question 20. Statement I: Two squares are placed edge to edge. There are 4 lines of symmetry in the resulting figure. Statement II: A smaller circle and a larger circle are drawn with the same centre. The resulting figure has 8 lines of symmetry.
(1) Statement I is true but statement II is false.
(2) Statement I is false but Statement II is true.
(3) Both Statement I and Statement II are true.
(4) Both Statement I and Statement II are false.
Answer: (4) Both Statement I and Statement II are false
In simple words: Placing two equal squares side by side creates a rectangle with only 2 lines of symmetry (not 4). Two circles sharing the same centre have countless lines of symmetry - not just 8.

Exam Tip: When combining shapes, count symmetry lines carefully. A 2-by-1 rectangle has fewer symmetry lines than a single square. Concentric circles have infinite symmetry.

Check Your Progress

Question 1(i). Draw the line (or lines) of symmetry, if any, of the following figure/picture and count their number:
Answer: The figure shown is an ellipse (oval shape). It contains 2 lines of symmetry - the major axis and the minor axis.
In simple words: An oval has two lines that split it into matching halves: one going the long way and one going the short way.

Exam Tip: Ovals always have exactly two perpendicular lines of symmetry at their longest and shortest points. Both pass through the centre.

Question 1(ii). Draw the line (or lines) of symmetry, if any, of the following figure/picture and count their number:
Answer: The figure is the division symbol (÷). It has 2 lines of symmetry - one running horizontally through the dash, and one running vertically through the centre.
In simple words: The division sign can be folded in two ways so both sides match exactly: once sideways and once up-and-down.

Exam Tip: Symbols with both a dot and a dash benefit from checking both horizontal and vertical folds. Both axes pass through the exact middle of the symbol.

Question 1(iii). Draw the line (or lines) of symmetry, if any, of the following figure/picture and count their number:
Answer: The given figure has no line of symmetry.
In simple words: This shape cannot be folded along any line so that both halves look the same.

Exam Tip: If a shape appears irregular or lopsided, test it by trying to fold it various ways - if no fold works, it has zero lines of symmetry.

Question 1(iv). Draw the line (or lines) of symmetry, if any, of the following figure/picture and count their number:
Answer: The given figure shows two equal overlapping circles. It contains 2 lines of symmetry - the line joining the centres of both circles and the perpendicular bisector of that line.
In simple words: Two identical circles overlapping have two mirror lines: one passing through both centres, and one at right angles to it.

Exam Tip: When two equal circles overlap, always identify the line connecting their centres first, then the perpendicular to that line - these are your two lines of symmetry.

Question 1(v). Draw the line (or lines) of symmetry, if any, of the following figure/picture and count their number:
Answer: The given figure is a yin-yang shape (an 'S' inside a circle). It has no line of symmetry.
In simple words: The yin-yang symbol curves in a way that prevents any fold from making both halves match perfectly.

Exam Tip: The yin-yang symbol looks balanced but has rotational symmetry, not line symmetry. No straight fold line will work for this shape.

Question 1(vi). Draw the line (or lines) of symmetry, if any, of the following figure/picture and count their number:
Answer: The given picture is a pair of scissors. It has 1 line of symmetry along the horizontal direction. When the scissors are folded along this line, both halves will match exactly.
In simple words: Scissors fold along a horizontal middle line, with the handles below and the blades above splitting into matching pieces.

Exam Tip: Scissors have a natural fold line running horizontally through the pivot point. This is the only line where both halves mirror each other perfectly.

Question 1(vii). Draw the line (or lines) of symmetry, if any, of the following figure/picture and count their number:
Answer: The given picture is a Christmas tree. It has 1 line of symmetry - the vertical line passing through its centre.
In simple words: A Christmas tree is balanced left and right, so one straight vertical line down its middle creates two identical halves.

Exam Tip: Christmas trees are symmetrical only along the vertical axis. The trunk and the triangular outline both contribute to this vertical symmetry.

Question 1(viii). Draw the line (or lines) of symmetry, if any, of the following figure/picture and count their number:
Answer: The given picture is a four-leaf clover. It has 4 lines of symmetry - the horizontal line, the vertical line, and the two diagonal lines all passing through the centre.
In simple words: A four-leaf clover looks the same four different ways: fold it left-right, top-bottom, or along either diagonal, and both sides match.

Exam Tip: Four-leaf clovers have maximum symmetry with four perpendicular lines. Always check horizontal, vertical, and both diagonals for four-petal shapes.

Question 1(ix). Draw the line (or lines) of symmetry, if any, of the following figure/picture and count their number:
Answer: The given picture is a flower with a small circle at its centre. It has 4 lines of symmetry - one passing through each pair of opposite petals, all running through the centre.
In simple words: A four-petaled flower splits into matching halves four different ways: once between each pair of opposite petals.

Exam Tip: For flowers with an even number of petals, count pairs of opposite petals. Each line of symmetry passes through the flower's centre and between one pair of opposite petals.

Question 2(i). Draw the line (or lines) of symmetry, if any, of the following picture of playing cards and count their number:
Answer: The given picture (ace of clubs) has 1 line of symmetry - the vertical line passing through the centre of the club symbol.
In simple words: A club symbol on a playing card is symmetric along one vertical line, so the left and right sides match when folded.

Exam Tip: Playing card symbols like clubs are designed with vertical symmetry only. The rectangular card border itself has more symmetry lines, but focus on the suit symbol.

Question 2(ii). Draw the line (or lines) of symmetry, if any, of the following picture of playing cards and count their number:
Answer: [Content to be completed based on the image that would appear here in the source PDF]
In simple words: [Placeholder - answer would depend on the specific card shown]

Exam Tip: Examine the suit symbol on the card carefully to identify all possible fold lines where left and right (or top and bottom) halves would match.

Question 2(iii). Draw the line (or lines) of symmetry, if any, of the following picture of playing cards and count their number:
Answer: The picture shown is the six of spades. This card has a single line of symmetry - the vertical line running through the middle of the card.
In simple words: If you fold the card along a vertical line through the centre, both sides match exactly.

Exam Tip: Always check both vertical and horizontal directions when looking for lines of symmetry in playing cards - count carefully and mark each line clearly on your sketch.

Question 2(iv). Draw the line (or lines) of symmetry, if any, of the following picture of playing cards and count their number:
Answer: The picture shown is the five of hearts. This card has a single line of symmetry - the vertical line passing through the middle heart.
In simple words: If you fold the card along a vertical line through the centre, the two sides form perfect mirror images.

Exam Tip: When counting lines of symmetry in playing cards, remember that a line must divide the shape so each half is a perfect mirror image of the other.

Question 2(v). Draw the line (or lines) of symmetry, if any, of the following picture of playing cards and count their number:
Answer: The picture shown is the four of diamonds. This card has two lines of symmetry - one vertical line and one horizontal line, both passing through the centre of the card.
In simple words: If you fold this card either up-and-down or left-and-right through the middle, both sides will match perfectly each time.

Exam Tip: Some shapes have multiple lines of symmetry - test each direction (vertical, horizontal, and diagonal) separately to make sure you find them all.

Question 3. Write the letters of the word 'ALGEBRA' which have no line of symmetry.
Answer: We check each letter in 'ALGEBRA':
A - has 1 line of symmetry (vertical).
L - has no line of symmetry.
G - has no line of symmetry.
E - has 1 line of symmetry (horizontal).
B - has 1 line of symmetry (horizontal).
R - has no line of symmetry.
A - has 1 line of symmetry (vertical).

Therefore, the letters that have no line of symmetry are: L, G, and R.
In simple words: Some letters look the same when you fold them, but L, G, and R never fold into matching halves.

Exam Tip: Test each letter by imagining folding it along both vertical and horizontal lines - if at least one fold creates a perfect match, the letter has a line of symmetry.

Question 4. On a squared paper, sketch the following:
(i) A triangle with a horizontal line of symmetry but no vertical line of symmetry.
(ii) A quadrilateral with both horizontal and vertical lines of symmetry.
(iii) A quadrilateral with a horizontal line of symmetry but no vertical line of symmetry.
(iv) A hexagon with exactly two lines of symmetry.
Answer:
(i) An isosceles triangle drawn sideways (with its line of symmetry going left to right) shows a horizontal line of symmetry but no vertical line of symmetry.

(ii) A rectangle is a quadrilateral that has both horizontal and vertical lines of symmetry.

(iii) A kite drawn sideways displays a horizontal line of symmetry but no vertical line of symmetry. When you fold along the horizontal line, both halves are perfect mirror images. However, folding along a vertical line does not create matching halves because the left and right sides are different sizes.

(iv) An elongated hexagon (not a regular hexagon) has exactly two lines of symmetry - one horizontal line and one vertical line running through its centre.
In simple words: Different shapes have different numbers of fold lines. A rectangle folds two ways, a kite folds only one way, and some hexagons fold two ways.

Exam Tip: When sketching on grid paper, always test your shape by imagining folding it - if the two halves match, that fold line is a line of symmetry.

Question 5(i). Copy the following figure on a squared paper and complete each figure such that the resultant figure is symmetrical about the dotted line (or lines):
Answer: To complete the figure, we need to draw the mirror image of the part already shown across the dotted line so that the whole shape becomes symmetrical about that line.
In simple words: Whatever you see on one side of the dotted line, draw it as a mirror image on the other side.

Exam Tip: Count grid squares carefully when reflecting a shape - each point on the original side must be the same distance from the dotted line as its mirror point on the other side.

Question 5(ii). Copy the following figure on a squared paper and complete each figure such that the resultant figure is symmetrical about the dotted line (or lines):
Answer: To complete the figure, we need to draw the mirror image of the part already shown across the dotted line so that the whole shape becomes symmetrical about that line.
In simple words: Draw the exact opposite of what you see on the other side of the dotted line to make it symmetrical.

Exam Tip: For diagonal lines of symmetry, remember that distances perpendicular to the line of symmetry must be equal on both sides.

Question 5(iii). Copy the following figure on a squared paper and complete each figure such that the resultant figure is symmetrical about the dotted line (or lines):
Answer: To complete the figure, we need to draw the mirror image of the part already shown across the dotted line so that the whole shape becomes symmetrical about that line.
In simple words: Reflect the given portion of the shape across both dotted lines to make it balanced and symmetrical.

Exam Tip: When multiple lines of symmetry are present, draw the reflection for each line separately and carefully - the final shape should look balanced from all directions.

Question 6. State whether the following statement is true or false. Justify your answer: "A straight line dividing a figure into two identical parts is necessarily a line of symmetry."
Answer: False.

A line of symmetry must divide a figure into two parts that are mirror images of each other. Two parts can be identical in shape and size without being mirror images. For instance, the diagonal of a parallelogram splits it into two triangles that are congruent (same shape and size), but when you fold the parallelogram along this diagonal, the two triangles do not overlap perfectly. This means the diagonal is not a line of symmetry, even though it divides the parallelogram into two identical parts.

Therefore, a straight line dividing a figure into two identical parts is not necessarily a line of symmetry.
In simple words: Just because a line splits a shape into two equal pieces does not mean it is a line of symmetry - the pieces must fold onto each other perfectly.

Exam Tip: Remember the difference between "congruent" (same shape and size) and "mirror images" (one is the reflection of the other) - all mirror images are congruent, but not all congruent shapes are mirror images.

Question 7. Kolam patterns often have beautiful bilateral symmetry (i.e., vertical or horizontal symmetric). How many lines of (reflection) symmetry can you draw for the adjoining Kolam design?
Answer: The Kolam design shown is symmetric with respect to both its vertical and horizontal axes. Therefore, the design has 2 lines of symmetry - one vertical and one horizontal, both passing through the centre.
In simple words: This Kolam pattern looks the same whether you fold it up-and-down or left-and-right.

Exam Tip: For artistic designs and patterns, always test both vertical and horizontal directions - and if available, diagonal directions too - to count all possible lines of symmetry.

Question 8. How many lines of symmetry does Ashoka Chakra have?
Answer: The Ashoka Chakra has 24 equally spaced spokes radiating from the centre. Each line that passes through the centre and connects any pair of opposite spokes serves as a line of symmetry. Therefore, the Ashoka Chakra has 24 lines of symmetry.
In simple words: The Ashoka Chakra's 24 spokes are evenly spread out, and any line through the middle connecting two opposite spokes folds the design perfectly in half.

Exam Tip: For circular designs with regularly spaced parts, count the number of equally spaced elements and that number tells you how many lines of symmetry pass through the centre.

Question 9. While old Parliament design was inspired by Chausath Yogini Temple, the new Parliament building has a beautiful triangular design. Does the outer boundary of the picture have reflection symmetry? If so, draw the lines of symmetries. How many such lines can be drawn?
Answer: Yes, the outer boundary of the picture has reflection symmetry. The boundary is based on an equilateral triangle shape. An equilateral triangle has 3 lines of symmetry - one from each vertex down to the midpoint of the opposite side. Therefore, the outer boundary has 3 lines of reflection symmetry.
In simple words: The triangular shape can be folded three different ways, each time creating perfect matching halves.

Exam Tip: For regular polygons, remember that an equilateral triangle has 3 lines of symmetry, a square has 4, a regular pentagon has 5, and so on - the number equals the number of sides.