Class 8 Maths (Ganita Prakash) - Chapter 4: QUADRILATERALS
Study Material & Competency-Based Worksheet
1. Introduction to Quadrilaterals
A quadrilateral is a closed figure with four sides, four vertices, and four angles. The sum of the interior angles of any quadrilateral is 360°.
SUMMARY
A rectangle is a quadrilateral in which the angles are all 90°.
Properties of a rectangle —
Opposite sides of a rectangle are equal.
Opposite sides of a rectangle are parallel to each other.
Diagonals of a rectangle are of equal length and they bisect each other.
A square is a quadrilateral in which all the angles are 90°, and all the sides are of equal length. Properties of a square —
The opposite sides of a square are parallel to each other.
The diagonals of a square are of equal lengths and they bisect each other at 90°.
The diagonals of a square bisect the angles of the square.
A parallelogram is a quadrilateral in which opposite sides are parallel.
Properties of a parallelogram —
The opposite sides of a parallelogram are equal.
In a parallelogram, the adjacent angles add up to 180°, and the opposite angles are equal.
The diagonals of a parallelogram bisect each other.
A rhombus is a quadrilateral in which all the sides have the same length.
Properties of a rhombus —
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 at right angles.
The diagonals of a rhombus bisect its angles.
A kite is a quadrilateral with two non-overlapping adjacent pairs of sides having the same length.
A trapezium is a quadrilateral having at least one pair of parallel opposite sides.
The sum of the angle measures in a quadrilateral is 360°
2. Types of Quadrilaterals
Trapezium
A quadrilateral with one pair of parallel sides.
Kite
A quadrilateral with two distinct pairs of adjacent sides that are equal.
Diagonals are perpendicular to each other.
Parallelogram
A quadrilateral with both pairs of opposite sides parallel.
Key Properties:
Opposite sides are equal.
Opposite angles are equal.
Diagonals bisect each other.
Consecutive angles are supplementary (sum = 180°).
Rhombus
A parallelogram with all four sides equal.
Key Properties (inherits all parallelogram properties plus):
Diagonals are perpendicular bisectors of each other.
Diagonals bisect the interior angles.
Rectangle
A parallelogram with all angles equal to 90°.
Key Properties (inherits all parallelogram properties plus):
Each angle is 90°.
Diagonals are equal in length.
Square
A parallelogram that is both a rectangle and a rhombus.
Key Properties (inherits all properties of rectangle and rhombus):
All sides are equal.
All angles equal to 90°.
Diagonals are equal, perpendicular, and bisect each other.
3. Important Theorems & Midpoint Theorem
The line segment joining the midpoints of any two sides of a triangle is parallel to the third side and is half of it.
Question Bank
Multiple Choice Questions (20 Questions)
What is the sum of all interior angles of a quadrilateral?
(a) 180° (b) 270° (c) 360° (d) 90° [Logical Reasoning]A quadrilateral with only one pair of parallel sides is called a:
(a) Parallelogram (b) Rhombus (c) Trapezium (d) Kite [Spatial Understanding]In a parallelogram, the bisectors of two adjacent angles are:
(a) Parallel (b) Perpendicular (c) Coincident (d) Equal [Analytical Thinking]The diagonals of a square:
(a) Are equal but not perpendicular (b) Are perpendicular but not equal
(c) Are neither equal nor perpendicular (d) Are equal and perpendicular [Spatial Understanding]If two adjacent angles of a parallelogram are (2x + 25)° and (3x - 5)°, the value of x is:
(a) 60 (b) 32 (c) 100 (d) 64 [Problem-Solving]A quadrilateral whose diagonals are equal and bisect each other at right angles is a:
(a) Rectangle (b) Rhombus (c) Kite (d) Square [Logical Reasoning]The angles of a quadrilateral are in the ratio 1:2:3:4. The largest angle is:
(a) 72° (b) 144° (c) 108° (d) 180° [Problem-Solving]In a rhombus, if the length of a diagonal is equal to its side, then the angles of the rhombus are:
(a) 60° and 120° (b) 45° and 135° (c) 30° and 150° (d) 90° each [Analytical Thinking]The figure formed by joining the midpoints of the sides of a quadrilateral is a:
(a) Square (b) Rhombus (c) Rectangle (d) Parallelogram [Spatial Understanding]In a rectangle ABCD, if AB = 12 cm and BC = 5 cm, the length of diagonal AC is:
(a) 13 cm (b) 17 cm (c) 7 cm (d) 15 cm [Problem-Solving]A kite has how many pairs of equal adjacent sides?
(a) 1 (b) 2 (c) 3 (d) 0 [Spatial Understanding]In a parallelogram, if one angle is 90°, it becomes a:
(a) Kite (b) Rhombus (c) Rectangle (d) Trapezium [Logical Reasoning]The diagonals of a rectangle:
(a) Are perpendicular(b) Are equal(c) Bisect the angles(d) Are not equal [Spatial Understanding]If the diagonals of a quadrilateral bisect each other, it is a:
(a) Kite (b) Trapezium (c) Parallelogram (d) Rhombus [Logical Reasoning]In ΞABC, D and E are midpoints of AB and AC respectively. If DE = 4 cm, then BC is:
(a) 2 cm (b) 4 cm (c) 8 cm (d) 16 cm [Problem-Solving]A square is a special type of:
(a) Kite and Rectangle (b) Rhombus and Trapezium (c) Rectangle and Rhombus
(d) Parallelogram and Kite [Logical Reasoning]The number of diagonals in a convex quadrilateral is:
(a) 1 (b) 2 (c) 3 (d) 4 [Spatial Understanding]In a parallelogram PQRS, if ∠P = 70°, then ∠R is:
(a) 70° (b) 110° (c) 20° (d) 90° [Problem-Solving]Which of the following is not true for a parallelogram?
(a) Opposite sides are equal (b) Opposite angles are equal
(c) Diagonals are equal (d) Diagonals bisect each other [Analytical Thinking]A quadrilateral with all sides equal and diagonals equal is a:
(a) Rhombus (b) Square (c) Kite (d) Rectangle [Logical Reasoning]
Assertion & Reasoning Questions (20 Questions)
Directions: In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Assertion (A): A square is a rhombus.
Reason (R): A square has all four sides equal, which is a defining property of a rhombus.
[Logical Reasoning]Assertion (A): Every rectangle is a parallelogram.
Reason (R): In a rectangle, only one pair of opposite sides is parallel. [Analytical Thinking]Assertion (A): The diagonals of a kite are perpendicular to each other.
Reason (R): The diagonals of a kite bisect each other at right angles. [Analytical Thinking]Assertion (A): In a parallelogram, the diagonals are equal.
Reason (R): Only in a rectangle, the diagonals are equal. [Logical Reasoning]Assertion (A): The sum of angles of a quadrilateral is 360°.
Reason (R): A quadrilateral can be divided into two triangles, and the sum of angles in each triangle is 180°. [Spatial Understanding]Assertion (A): A trapezium is not a parallelogram.
Reason (R): In a trapezium, only one pair of opposite sides is parallel, while in a parallelogram, both pairs are parallel. [Logical Reasoning]Assertion (A): A rhombus with one right angle becomes a square.
Reason (R): All sides of a rhombus are equal. [Analytical Thinking]Assertion (A): The quadrilateral formed by joining the midpoints of a rectangle is a rhombus.
Reason (R): The sides of the new figure are equal and parallel to the diagonals of the rectangle.
[Spatial Understanding]Assertion (A): All kites are rhombuses.
Reason (R): Both kites and rhombuses have all sides equal. [Logical Reasoning]Assertion (A): In a square, the diagonals are perpendicular bisectors of each other.
Reason (R): A square inherits this property from both the rectangle and the rhombus.
[Analytical Thinking]Assertion (A): A quadrilateral can be constructed if its three sides and two included angles are given.
Reason (R): To construct a unique quadrilateral, five independent measurements are required.
[Problem-Solving]Assertion (A): In a parallelogram, the sum of consecutive angles is 180°.
Reason (R): The opposite angles of a parallelogram are equal. [Logical Reasoning]Assertion (A): The diagonals of a rhombus are always equal.
Reason (R): A square is a rhombus with equal diagonals. [Analytical Thinking]Assertion (A): If the diagonals of a quadrilateral are perpendicular, it must be a kite.
Reason (R): The diagonals of a rhombus are also perpendicular. [Logical Reasoning]Assertion (A): A rectangle is an equiangular quadrilateral.
Reason (R): All angles in a rectangle are right angles. [Spatial Understanding]Assertion (A): The Midpoint Theorem is applicable to any triangle.
Reason (R): The line segment joining midpoints of two sides is always parallel to the third side.
[Analytical Thinking]Assertion (A): All squares are kites.
Reason (R): A square has two distinct pairs of adjacent sides equal. [Logical Reasoning]Assertion (A): A parallelogram is always a cyclic quadrilateral.
Reason (R): The sum of opposite angles of a parallelogram is 180° only if it is a rectangle.
[Analytical Thinking]Assertion (A): The area of a rhombus can be calculated as half the product of its diagonals.
Reason (R): The diagonals of a rhombus divide it into four congruent right triangles.
[Problem-Solving]Assertion (A): If one pair of opposite sides of a quadrilateral are equal and parallel, it is a parallelogram.
Reason (R): This is one of the standard tests to prove a quadrilateral is a parallelogram.
[Logical Reasoning]
True/False Questions (10 Questions)
Every square is a rectangle. (True/False) [Logical Reasoning]
Every rhombus is a square. (True/False) [Logical Reasoning]
The diagonals of a parallelogram are always equal. (True/False) [Spatial Understanding]
A quadrilateral with perpendicular diagonals is always a kite. (True/False) [Analytical Thinking]
The sum of exterior angles of a quadrilateral is also 360°. (True/False) [Problem-Solving]
In a parallelogram, consecutive angles are complementary. (True/False) [Analytical Thinking]
A trapezium can have two right angles. (True/False) [Spatial Understanding]
The diagonals of a rectangle bisect the angles at the vertices. (True/False) [Analytical Thinking]
The figure formed by joining the midpoints of a rhombus is a rectangle. (True/False) [Spatial Understanding]
All rectangles are parallelograms. (True/False) [Logical Reasoning]
Short Answer Type I (2 Marks each) - 15 Questions
Two adjacent angles of a parallelogram are in the ratio 4:5. Find the measure of all its angles. [Problem-Solving]
In a parallelogram ABCD, if ∠A = (2x + 15)° and ∠B = (3x - 25)°, find the value of x. [Problem-Solving]
State the properties of the diagonals of a rhombus. [Spatial Understanding]
a kite with one diagonal drawn If AB = AD and CB = CD in kite ABCD, and ∠ABC = 50°, find ∠ADC. [Problem-Solving]
Prove that the line segment joining the midpoints of two sides of a triangle is parallel to the third side. [Analytical Thinking]
The angles of a quadrilateral are 2x, 3x, 4x, and 6x. Find the value of the smallest angle. [Problem-Solving]
Can all the angles of a quadrilateral be acute? Justify your answer. [Logical Reasoning]
In a rectangle, one diagonal is 10 cm. What is the length of the other diagonal? [Spatial Understanding]
an isosceles trapezium, if one of the base angles is 65°, what is the measure of the opposite angle? [Problem-Solving]
Define a square and list any one of its unique properties. [Spatial Understanding]
In a quadrilateral PQRS, the diagonals PR and QS bisect each other. What is the special name of this quadrilateral? [Logical Reasoning]
If the diagonals of a quadrilateral are equal and bisect each other, prove that it is a rectangle. [Analytical Thinking]
The perimeter of a parallelogram is 60 cm. If one side is 12 cm, find the length of the adjacent side. [Problem-Solving]
In ΞPQR, S and T are midpoints of sides PQ and PR respectively. If QR = 14 cm, find ST. [Problem-Solving]
A field is in the shape of a rhombus whose perimeter is 400 m and one of its diagonals is 160 m. Find the area of the field. (Hint: Use Pythagoras theorem to find the other diagonal). [Problem-Solving]
Short Answer Type II (3 Marks each) - 10 Questions
Prove that the diagonals of a rectangle are equal. [Analytical Thinking]
ABCD is a rhombus. Show that the diagonal AC bisects ∠A as well as ∠C. [Analytical Thinking]
a quadrilateral with midpoints E,F,G,H of sides AB,BC,CD,DA marked In quadrilateral ABCD, E, F, G, H are the midpoints of sides AB, BC, CD, and DA respectively. Prove that EFGH is a parallelogram. [Analytical Thinking]
In a parallelogram, show that the angle bisectors of two adjacent angles intersect at a right angle. [Analytical Thinking]
The ratio of two adjacent sides of a parallelogram is 5:4. Its perimeter is 90 cm. Find the lengths of all sides. [Problem-Solving]
Two parallel lines l and m are intersected by a transversal p. Show that the quadrilateral formed by the bisectors of the interior angles is a rectangle. [Creativity, Analytical Thinking]
In a trapezium ABCD, AB || CD. If ∠A : ∠D = 3:2 and ∠B : ∠C = 4:5, find all the angles of the trapezium. [Problem-Solving]
Prove that a diagonal of a parallelogram divides it into two congruent triangles. [Analytical Thinking]
A kite with diagonals AC and BD intersecting at O In kite ABCD, diagonals AC and BD intersect at O. If AB = 5 cm and BD = 8 cm, find the length of side AD and the area of the kite. [Problem-Solving]
In ΞABC, D, E, and F are the midpoints of sides BC, CA, and AB respectively. Show that ΞABC is divided into four congruent triangles by segments DE, EF, and FD. [Spatial Understanding, Analytical Thinking]
Long Answer Questions (5 Marks each) - 10 Questions
Construct a quadrilateral ABCD where AB = 4.5 cm, BC = 5.5 cm, CD = 4 cm, AD = 6 cm, and diagonal AC = 7 cm. Measure the length of BD. [Creativity, Problem-Solving]
Prove that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square. [Analytical Thinking]
ABCD is a rectangle in which diagonal AC bisects ∠A as well as ∠C. Show that:
(i) ABCD is a square (ii) Diagonal BD bisects ∠B as well as ∠D.[Analytical Thinking]A plot of land is in the shape of a quadrilateral. A farmer divides it equally among his two sons by building a straight boundary wall from one vertex to the midpoint of the opposite side. Which theorem did he use? Prove the theorem used.
[Creativity, Analytical Thinking]In a parallelogram, prove that the bisectors of any two consecutive angles intersect at a right angle. [Analytical Thinking]
The diagonal of a rhombus is 16 cm and 30 cm long. Find its perimeter and area. [Problem-Solving]
A complex figure made of multiple quadrilaterals The given figure shows a pattern made from tiles shaped as parallelograms and rhombuses. Using the properties of these shapes, prove that the pattern can tessellate a plane without gaps. [Creativity, Spatial Understanding]
PQRS is a trapezium with PQ || SR. A line parallel to the base PQ passes through the midpoint M of side PS and meets QR at N. Prove that N is the midpoint of QR. [Analytical Thinking]
ABC is a triangle right-angled at C. A line through the midpoint M of hypotenuse AB and parallel to BC intersects AC at D. Show that: (i) D is the midpoint of AC (ii) MD ⊥ AC
(iii) CM = MA = ½ AB [Analytical Thinking]
Construct a parallelogram whose diagonals are 6 cm and 8 cm long and the angle between them is 60°. Measure the sides of the parallelogram. [Creativity, Problem-Solving]
Case-Based Questions Case 1: The Playground
A school playground is in the shape of a quadrilateral ABCD. The coach marks points E, F, G, H as the midpoints of sides AB, BC, CD, and DA respectively for a relay race.
A quadrilateral playground with midpoints marked
The quadrilateral EFGH formed is a:
(a) Rectangle (b) Rhombus (c) Parallelogram (d) Square [Spatial Understanding]If the original quadrilateral ABCD is a rectangle, then EFGH will be a:
(a) Rectangle (b) Rhombus (c) Parallelogram (d) Square [Logical Reasoning]If the original quadrilateral ABCD is a rhombus, then EFGH will be a:
(a) Rectangle (b) Rhombus (c) Parallelogram (d) Square [Logical Reasoning]The theorem used to determine the nature of EFGH is the:
(a) Pythagoras Theorem (b) Basic Proportionality Theorem
(c) Midpoint Theorem (d) Angle Sum Property [Analytical Thinking]
Case 2: The Kite Festival
During a kite festival, Rohan made a kite ABCD. He ensures that AB = AD and CB = CD. The diagonals AC and BD intersect at O. The length of AC is 24 cm and BD is 10 cm.
A kite with diagonals intersecting at O
The diagonals of a kite are:
(a) Equal (b) Perpendicular (c) Parallel (d) None of these [Spatial Understanding]The area of Rohan's kite is:
(a) 120 cm² (b) 240 cm² (c) 60 cm² (d) 34 cm² [Problem-Solving]If ∠ABO = 35°, then what is the measure of ∠BAO?
(a) 35° (b) 55° (c) 90° (d) 70° [Problem-Solving]The shorter diagonal of the kite is:
(a) AC (b) BD (c) AB (d) AD [Spatial Understanding]
Case 3: The Picture Frame
Mohan bought a rectangular picture frame. To check if it is perfectly rectangular, he measures the diagonals and finds they are 45 cm each.
A rectangular frame with diagonals
The property Mohan used to check the frame is:
(a) Diagonals bisect each other (b) Diagonals are equal
(c) Diagonals are perpendicular d) All sides are equal [Logical Reasoning]If the length of the frame is 36 cm, what is its perimeter?
(a) 126 cm (b) 108 cm (c) 90 cm (d) 81 cm [Problem-Solving]If the frame is deformed into a parallelogram without changing side lengths, what happens to the diagonal lengths?
(a) They become equal (b) They become perpendicular (c) They become unequal (d) They bisect the angles [Analytical Thinking]A square is a special rectangle because it:
(a) Has equal diagonals (b) Has all angles of 90° (c) Has all sides equal
(d) Both (b) and (c) [Logical Reasoning]
Case 4: The Garden Plot
A garden is in the shape of a parallelogram. The gardener wants to divide it into two equal parts by planting a hedge along one of its diagonals.
A parallelogram garden with a diagonal
The diagonal divides the parallelogram into:
(a) Two similar triangles (b) Two congruent triangles
(c) Two triangles of different areas (d) A triangle and a trapezium [Spatial Understanding]If the adjacent sides of the plot are 15 m and 20 m, and one diagonal is 25 m, what is the shape of the triangular plot formed?
(a) Right-angled triangle (b) Equilateral triangle
(c) Isosceles triangle (d) Scalene triangle [Problem-Solving]If the gardener wants to divide it into four equal parts, he should plant hedges along:
(a) The diagonals (b) The medians (c) The angle bisectors (d) The lines joining midpoints
[Creativity]The diagonal of a parallelogram divides it into two congruent triangles. This property is based on which congruence rule?
(a) SSS (b) SAS (c) ASA (d) RHS [Analytical Thinking]
Case 5: The Tile Design
An interior designer uses a combination of square and rhombus-shaped tiles. All tiles have the same side length of 10 cm.
A tessellation pattern with squares and rhombuses
The main difference between the square and rhombus tiles is:
(a) Number of sides (b) Length of diagonals
(c) Measure of angles (d) Both (b) and (c) [Spatial Understanding]If the diagonals of the rhombus tile are 16 cm and 12 cm, its area is:
(a) 96 cm² b) 160 cm² (c) 192 cm² (d) 100 cm² [Problem-Solving]The designer can create a tessellation pattern without gaps because:
(a) The sum of angles around a point is 360° (b) All sides are equal
(c) Diagonals are perpendicular (d) Opposite sides are parallel [Creativity, Analytical Thinking]If the rhombus tiles are replaced with rectangular tiles of the same area, the pattern will:
(a) Have no gaps (b) Have gaps (c) Remain the same (d) Cannot be determined
[Logical Reasoning]
Figure it Out 107-109
1. Find all the sides and the angles of the quadrilateral obtained by joining two equilateral triangles with sides 4 cm.
2. Construct a kite whose diagonals are of lengths 6 cm and 8 cm.
3. Find the remaining angles in the following trapeziums — 135° 105° 100°
4. Draw a Venn diagram showing the set of parallelograms, kites, rhombuses, rectangles, and squares. Then, answer the following questions —
(i) What is the quadrilateral that is both a kite and a parallelogram?
(ii) Can there be a quadrilateral that is both a kite and a rectangle?
(iii) Is every kite a rhombus? If not, what is the correct relationship between these two types of quadrilaterals?
5. If PAIR and RODS are two rectangles, find ∠IOD.
6. Construct a square with diagonal 6 cm without using a protractor.
7. CASE is a square. The points U, V, W and X are the midpoints of the sides of the square. What type of quadrilateral is UVWX? Find this by using geometric reasoning, as well as by construction and measurement. Find other ways of constructing a square within a square such that the vertices of the inner square lie on the sides of the outer square, as shown in Figure (b).
8. If a quadrilateral has four equal sides and one angle of 90°, will it be a square? Find the answer using geometric reasoning as well as by construction and measurement.
9. What type of a quadrilateral is one in which the opposite sides are equal? Justify your answer. Hint: Draw a diagonal and check for congruent triangles.
10. Will the sum of the angles in a quadrilateral such as the following one also be 360°? Find the answer using geometric reasoning as well as by constructing this figure and measuring.
11. State whether the following statements are true or false. Justify your answers. (i) A quadrilateral whose diagonals are equal and bisect each other must be a square.
Figure it Out 102
1. Find the remaining angles in the following quadrilaterals.
2. Using the diagonal properties, construct a parallelogram whose diagonals are of lengths 7 cm and 5 cm, and intersect at an angle of 140°.
3. Using the diagonal properties, construct a rhombus whose diagonals are of lengths 4 cm and 5 cm.
Figure it Out
1. Find all the other angles inside the following rectangles.
2. Draw a quadrilateral whose diagonals have equal lengths of 8 cm that bisect each other, and intersect at an angle of (i) 30° (ii) 40° (iii) 90° (iv) 140°
3. Consider a circle with centre O. Line segments PL and AM are two perpendicular diameters of the circle. What is the figure APML? Reason and/or experiment to figure this out.
4. We have seen how to get 90° using paper folding. Now, suppose we do not have any paper but two sticks of equal length, and a thread. How do we make an exact 90° using these?
5. We saw that one of the properties of a rectangle is that its opposite sides are parallel. Can this be chosen as a definition of a rectangle? In other words, is every quadrilateral that has opposite sides parallel and equal, a rectangle?
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