CLASS 8 NCERT WORKSHEET CH-4 QUADRILATERALS WITH ANSWER KEY

 CLASS 8 NCERT WORKSHEET  CH-4 QUADRILATERALS WITH ANSWER KEY

Here is a worksheet based on the provided source PDF, "ch4 quadrilaterals class 8.pdf". This worksheet covers key definitions, properties, deductions, and problem-solving elements from the chapter.

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Quadrilaterals: Chapter 4 Worksheet

Instructions: Read the questions carefully and answer them based on the information provided in your textbook. Show your reasoning where required.

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Section 1: Introduction to Quadrilaterals

1. What is the derivation of the word 'quadrilateral'? Provide the meaning of its root words.
2. Observe Figures (i), (ii), and (iii) in the source. What makes these figures quadrilaterals, as opposed to Figures (iv) and (v)?

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Section 2: Rectangles and Squares

1. Define a rectangle based on its angles and opposite sides.
2. Describe two key properties of the diagonals of a rectangle, as derived through geometric deduction.
3. Explain how a carpenter can use the properties of diagonals to construct a rectangular frame, given one 8 cm long strip of wood.
   • What should be the length of the other strip?
   • Where should they both be joined?
4. A quadrilateral has diagonals that are equal in length and bisect each other.
   • If the angle between the diagonals is 60°, what type of quadrilateral is formed? Justify your answer by calculating all its interior angles.
   • Will the quadrilateral remain a rectangle if this angle is changed? Explain your reasoning using the general case with angle 'x' between the diagonals.
5. State the simplified definition of a rectangle based solely on its angles. Prove why a quadrilateral with all angles equal to 90° must have opposite sides of equal length.
6. List four properties of a rectangle.
7. Define a square.
8. Explain the relationship between squares and rectangles using a Venn diagram.
9. What additional condition, beyond equal and bisecting diagonals, is necessary for a quadrilateral to be a square (instead of just a rectangle)?
10. If the diagonals of a quadrilateral are equal in length, bisect each other, and intersect at right angles (90°), what type of quadrilateral is formed? Justify your answer.
11. List five properties of a square.
12. Problem Solving (from "Figure it Out"):
    • Draw a quadrilateral whose diagonals have equal lengths of 8 cm that bisect each other, and intersect at an angle of 90°. What shape do you get?
    • If a quadrilateral has four equal sides and one angle of 90°, will it be a square? Justify your answer using geometric reasoning.

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Section 3: Angles in a Quadrilateral

1. What is the sum of all interior angles in any quadrilateral? Explain how this can be deduced by dividing a quadrilateral into triangles.
2. Based on your answer to Question 1, explain why it is impossible for a quadrilateral to have three right angles with the fourth angle not being a right angle.

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Section 4: More Quadrilaterals with Parallel Opposite Sides (Parallelograms)

1. Define a parallelogram.
2. Is a rectangle a parallelogram? Justify your answer.
3. Illustrate the relationship between rectangles, squares, and parallelograms using a Venn diagram.
4. If one angle of a parallelogram is 30°, determine the measures of the remaining three angles. Explain the properties of adjacent and opposite angles in a parallelogram.
5. List four properties of a parallelogram.
6. Are the diagonals of a parallelogram always equal in length?
7. Prove that the diagonals of a parallelogram always bisect each other.
8. True/False: A quadrilateral whose diagonals bisect each other must be a parallelogram. Justify your answer.

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Section 5: Quadrilaterals with Equal Sidelengths (Rhombus)

1. Define a rhombus.
2. Explain why every rhombus is also a parallelogram.
3. List six properties of a rhombus.
4. Do the diagonals of a rhombus always intersect at a 90° angle? Prove your answer using congruence.
5. True/False: A quadrilateral whose diagonals are perpendicular to each other must be a rhombus. Justify your answer.
6. Problem Solving (from "Figure it Out"): Using diagonal properties, construct a rhombus whose diagonals are of lengths 4 cm and 5 cm.

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Section 6: Kite and Trapezium

1. Define a kite and list its key properties related to sides and diagonals.
2. Define a trapezium.
3. In a trapezium, if one pair of opposite sides are parallel, what can you say about the sum of the angles on the same side of a non-parallel transversal?
4. What is an isosceles trapezium? What special property do its base angles possess?
5. True/False: Isosceles trapeziums are parallelograms. Justify your answer.
6. Venn Diagram Challenge: Draw a Venn diagram showing the set of parallelograms, kites, rhombuses, rectangles, and squares, illustrating their relationships. Based on your diagram and source information, answer the following:
   • What is the quadrilateral that is both a kite and a parallelogram?
   • Can there be a quadrilateral that is both a kite and a rectangle?
   • Is every kite a rhombus? If not, what is the correct relationship between these two types of quadrilaterals?

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Section 7: General Properties and Review

1. Consider a quadrilateral where the opposite sides are equal. What type of quadrilateral is it? Justify your answer.
2. True/False Statements: For each statement, state whether it is True or False and provide a justification based on the source material:
   • A quadrilateral whose diagonals are equal and bisect each other must be a square.
   • A quadrilateral having three right angles must be a rectangle.
   • A quadrilateral in which the opposite angles are equal must be a parallelogram.

A quadrilateral in which all the angles are equal is a rectangle.

Quadrilaterals: Chapter 4 Worksheet - Answer Key

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Section 1: Introduction to Quadrilaterals

The word ‘quadrilateral’ is derived from Latin words: ‘quadri’ meaning four, and ‘latus’ referring to sides.
Figures (i), (ii), and (iii) are quadrilaterals because their angles are defined as the angles between their sides, as marked in the figures. Figures (iv) and (v) do not fit this description of a quadrilateral.

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• Section 2: Rectangles and Squares

A rectangle is defined as a quadrilateral in which:

The angles are all right angles (90°).
The opposite sides are of equal length.
Alternatively, a rectangle can be defined as a quadrilateral whose diagonals are equal and bisect each other.

Two key properties of the diagonals of a rectangle are:

They always have the same length.
They always intersect at their midpoints, meaning they bisect each other.

To construct a rectangular frame, the carpenter should:

Make the length of the other strip also 8 cm long, as the diagonals of a rectangle must have the same length.
They should both be joined at their midpoints, as the diagonals of a rectangle bisect each other.

A quadrilateral has diagonals that are equal in length and bisect each other:

If the angle between the diagonals is 60°:In ∆AOB, where OA=OB and the angle at O is 60°, the base angles 'a' are calculated as (180° - 60°)/2 = 60°.
Similarly, for ∆AOD, if the angle is 120° (linear pair to 60°), the base angles 'b' are (180° - 120°)/2 = 30°.
The interior angles of the quadrilateral are formed by a+b, which is 60° + 30° = 90°.
Since all angles are 90° and opposite sides are equal (due to congruent triangles formed by the diagonals), the quadrilateral formed is a rectangle.
Yes, the quadrilateral will remain a rectangle if this angle is changed. If the angle between the diagonals is 'x', the base angles of the isosceles triangles formed by the diagonals are a = 90° - x/2 and b = x/2. The sum of these angles, which forms each interior angle of the quadrilateral, is a + b = (90° - x/2) + x/2 = 90°. Therefore, no matter what the angles between the diagonals are, if the diagonals are equal and they bisect each other, the quadrilateral formed is a rectangle.

The simplified definition of a rectangle based solely on its angles is: A rectangle is a quadrilateral in which all the angles are 90°. To prove why a quadrilateral with all angles equal to 90° must have opposite sides of equal length:

Consider a quadrilateral ABCD with all angles measuring 90°.
Draw diagonal BD. In ∆BAD and ∆DCB, ∠BAD = ∠DCB = 90°.
By geometric reasoning (similar to Deduction 2), ∠1 (∠ADB) = ∠2 (∠CBD).
Therefore, ∆BAD ≅ ∆DCB by the AAS congruence condition.
As corresponding parts of congruent triangles, AD = CB and DC = BA, proving that the opposite sides have equal lengths.

Four properties of a rectangle are:

All the angles of a rectangle are 90°.
The opposite sides of a rectangle are equal.
The opposite sides of a rectangle are parallel to each other.
The diagonals of a rectangle are of equal length and they bisect each other.
A square is a quadrilateral in which all the angles are equal to 90°, and all the sides are of equal length.
Every square is also a rectangle, but every rectangle is not a square. This relationship can be represented by a Venn diagram where the set of squares is entirely contained within the set of rectangles.
Beyond equal and bisecting diagonals, the additional condition necessary for a quadrilateral to be a square is that the diagonals must intersect at right angles (90°).
If the diagonals of a quadrilateral are equal in length, bisect each other, and intersect at right angles (90°), the type of quadrilateral formed is a square. This is because equal and bisecting diagonals make it a rectangle, and the additional condition of perpendicular intersection ensures that all sides are equal, thus satisfying the definition of a square.

Five properties of a square are:

All the sides of a square are equal to each other.
The opposite sides of a square are parallel to each other.
The angles of a square are all 90°.
The diagonals of a square are of equal length and they bisect each other at 90°.
The diagonals of a square bisect the angles of the square (divide the angles into equal halves).

Problem Solving (from "Figure it Out"):

If a quadrilateral's diagonals have equal lengths of 8 cm, bisect each other, and intersect at an angle of 90°, the shape you get is a square.
Yes, if a quadrilateral has four equal sides and one angle of 90°, it will be a square.
Justification: A quadrilateral with four equal sides is a rhombus. In a rhombus, adjacent angles sum to 180° and opposite angles are equal. If one angle is 90°, its adjacent angles must also be 90° (180° - 90° = 90°), and its opposite angle must be 90°. Thus, all four angles are 90°. A quadrilateral with all sides equal and all angles 90° is a square.

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• Section 3: Angles in a Quadrilateral

The sum of all interior angles in any quadrilateral is 360°. This can be deduced by:

Drawing a diagonal (e.g., SM in quadrilateral SOME) which divides the quadrilateral into two triangles (∆SEM and ∆SOM).
The sum of angles in each triangle is 180°.
Adding the angles of both triangles: (∠1 + ∠2 + ∠3) + (∠4 + ∠5 + ∠6) = 180° + 180° = 360°.
Since these six angles combine to form the four angles of the quadrilateral, the total sum is 360°.
It is impossible for a quadrilateral to have three right angles with the fourth angle not being a right angle because the sum of all angles in any quadrilateral must be 360°. If three angles are 90° each, their sum is 270°. For the total sum to be 360°, the fourth angle must be 360° - 270° = 90°.

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• Section 4: More Quadrilaterals with Parallel Opposite Sides (Parallelograms)

A parallelogram is a quadrilateral in which opposite sides are parallel.
Yes, a rectangle is a parallelogram. A rectangle has opposite sides parallel, which satisfies the definition of a parallelogram. Specifically, a rectangle is a special type of parallelogram where all its angles are 90°.
The Venn diagram shows Rectangles and Squares both nested inside the larger set of Parallelograms. The Square set is also nested inside the Rectangle set.

If one angle of a parallelogram is 30° (e.g., ∠A = 30°):

Adjacent angles in a parallelogram add up to 180°. So, ∠D = 180° - 30° = 150° and ∠B = 180° - 30° = 150°.
Opposite angles in a parallelogram are equal. So, ∠C = ∠A = 30°.
The remaining three angles are 150°, 30°, and 150°.

Four properties of a parallelogram are:

The opposite sides of a parallelogram are equal.
The opposite sides of a parallelogram are parallel.
In a parallelogram, the adjacent angles add up to 180°, and the opposite angles are equal.
The diagonals of a parallelogram bisect each other.
No, the diagonals of a parallelogram need not be equal in length.

To prove that the diagonals of a parallelogram always bisect each other:

Consider a parallelogram EASY with diagonals AY and ES intersecting at O.
AE = YS because they are opposite sides of a parallelogram.
Angles ∠EAO and ∠YSO are equal as alternate angles (since AE||YS and AY is a transversal).
Angles ∠AEO and ∠SYO are equal as alternate angles (since AE||YS and ES is a transversal).
Therefore, ∆AOE ≅ ∆YOS by the ASA congruence condition.
As corresponding parts of congruent triangles, OA = OY and OE = OS. This means O is the midpoint of both diagonals, thus the diagonals bisect each other.
True. A quadrilateral whose diagonals bisect each other must be a parallelogram [58(iii)]. If the diagonals bisect each other, it can be proven through congruence (e.g., SAS for triangles formed by diagonals) that opposite sides are equal, which in turn implies they are parallel, making the quadrilateral a parallelogram.

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• Section 5: Quadrilaterals with Equal Sidelengths (Rhombus)

A rhombus is a quadrilateral in which all the sides have the same length.
Every rhombus is also a parallelogram because its opposite sides are parallel. This can be shown by drawing a diagonal (e.g., AE in rhombus GAME). Since the alternate interior angles formed by the diagonal and the sides are equal (due to all sides being equal), it implies that opposite sides are parallel (EM||GA and GE||AM).

Six properties of a rhombus are:

All the sides of a rhombus are equal to each other.
The opposite sides of a rhombus are parallel to each other.
In a rhombus, the adjacent angles add up to 180°, and the opposite angles are equal.
The diagonals of a rhombus bisect each other.
The diagonals of a rhombus bisect its angles.
The diagonals of a rhombus intersect each other at an angle of 90°.

Yes, the diagonals of a rhombus always intersect at a 90° angle. Proof using congruence:

Consider the rhombus GAME with diagonals intersecting at O.
In ∆GEO and ∆MEO:GE = ME (all sides of a rhombus are equal).
EO is common to both triangles.
GO = MO (diagonals of a rhombus bisect each other).
Therefore, ∆GEO ≅ ∆MEO by the SSS congruence condition.
Since they are congruent, their corresponding angles are equal: ∠GOE = ∠MOE.
These two angles form a linear pair, meaning ∠GOE + ∠MOE = 180°.
Thus, each angle must be 90°, so the diagonals intersect at right angles.
False. A quadrilateral whose diagonals are perpendicular to each other is not necessarily a rhombus [58(iv)]. A kite also has perpendicular diagonals, but a general kite does not have all four sides equal like a rhombus.
Problem Solving (from "Figure it Out"): To construct a rhombus whose diagonals are of lengths 4 cm and 5 cm, one would use the properties that diagonals bisect each other and intersect at 90°. Draw a 5 cm line segment and mark its midpoint. Then, draw a 4 cm line segment perpendicular to the first one at its midpoint, ensuring it is bisected (2 cm on each side of the intersection). Connect the endpoints of these two perpendicular, bisecting diagonals to form the rhombus.

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• Section 6: Kite and Trapezium

A kite is a quadrilateral that can be labelled ABCD such that AB = BC (one pair of adjacent sides are equal) and CD = DA (another pair of adjacent sides are equal). Key properties of its diagonals:

The diagonal BD (connecting the vertices between the equal sides) bisects ∠ABC and ∠ADC.
The diagonal BD bisects the other diagonal AC (AO = OC), and is perpendicular to it.
A trapezium is a quadrilateral with at least one pair of parallel opposite sides.
In a trapezium, if one pair of opposite sides are parallel (e.g., PQ||SR), then the sum of the angles on the same side of a non-parallel transversal is 180°. Therefore, ∠S + ∠P = 180° and ∠R + ∠Q = 180°.
An isosceles trapezium is a trapezium where the non-parallel sides have the same lengths. Its special property is that the angles opposite to the equal sides (base angles) are equal (e.g., ∠U = ∠V in trapezium UVWX).
False. Isosceles trapeziums are not parallelograms [59(vii)]. A parallelogram requires both pairs of opposite sides to be parallel, whereas a trapezium (even isosceles) only has at least one pair of parallel opposite sides.

Venn Diagram Challenge:

The quadrilateral that is both a kite and a parallelogram is a rhombus. A rhombus has all sides equal, which satisfies the conditions of a kite (all adjacent pairs are equal) and a parallelogram (opposite sides are parallel).
Yes, there can be a quadrilateral that is both a kite and a rectangle. This quadrilateral is a square. A square is a rectangle (all angles 90°) and a kite (all adjacent sides are equal, since all sides are equal). Such a figure would have diagonals that are equal, bisect each other, and are perpendicular, which defines a square.
No, every kite is not a rhombus [55(iii)]. The correct relationship is that a rhombus is a special type of kite. While a rhombus has all four sides equal, a general kite only requires two distinct pairs of adjacent sides to be equal.

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• Section 7: General Properties and Review

A quadrilateral where the opposite sides are equal is a parallelogram. Justification: If opposite sides are equal (e.g., AB=CD and AD=BC), drawing a diagonal (say, BD) divides the quadrilateral into two triangles (∆ABD and ∆CDB). By SSS congruence, ∆ABD ≅ ∆CDB. This congruence implies that alternate interior angles are equal (e.g., ∠ABD = ∠CDB and ∠ADB = ∠CBD), which in turn proves that opposite sides are parallel (AB||CD and AD||BC). A quadrilateral with both pairs of opposite sides parallel is a parallelogram.

True/False Statements:

• (i) A quadrilateral whose diagonals are equal and bisect each other must be a square.
  • False [58(i)]. This description defines a rectangle. For it to be a square, the diagonals must also intersect at right angles.
• (ii) A quadrilateral having three right angles must be a rectangle.
  • True [58(ii)]. If three angles of a quadrilateral are 90°, then the sum of these angles is 270°. Since the total sum of angles in a quadrilateral is 360°, the fourth angle must also be 90° (360° - 270° = 90°). A quadrilateral with all four angles equal to 90° is a rectangle.
• (iii) A quadrilateral in which the opposite angles are equal must be a parallelogram.
  • True [59(v)]. In a quadrilateral, if opposite angles are equal, say ∠A=∠C and ∠B=∠D, then 2∠A + 2∠B = 360°, which means ∠A + ∠B = 180°. Since adjacent angles are supplementary, this implies that consecutive sides are parallel, thus forming a parallelogram.
• (iv) A quadrilateral in which all the angles are equal is a rectangle.
  • True [59(vi)]. If all four angles of a quadrilateral are equal and their sum is 360°, then each angle must be 360°/4 = 90°. A quadrilateral in which all angles are 90° is defined as a rectangle.

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