Complete the Ganita Prakash Class 8 Worksheet and NCERT Class 8 Maths Chapter 4 Quadrilaterals Worksheet with Answers before your unit tests for better preparation.
Quadrilaterals Worksheet Class 8
Class 8 Maths Quadrilaterals Worksheet
Quadrilaterals Class 8 Ganita Prakash Worksheet
Rectangles and Squares
Question 1.
Riya and her father are fixing a photo frame.
Father: “The frame must be perfect. Do you know its shape?”
Riya: “It looks like a rectanqle, but how can we be sure?”
Father: “Let us check its sides, angles, and diagonals.”
Riya (measuring): “The top and bottom sides are equal and parallel; left and right sides are also equal and parallel.”
Father (with set-square): “Each corner matches the right-angle edge, so all four angles are 90°.”
Riya (stretching a thread): “Both diagonals are equal in length and when I fold the thread, the point of intersection divides both the diagonals into equal halves.
Father: Excellent! Equal and parallel opposite sides, four right angles and equal diagonals confirm that this frame is in rectangle shape.
(a) From the sides, what two clues told Riya that the frame could be a rectangle?
Answer:
The top bottom sides are equal and parallel. The left and right sides are equal and parallel.
(b) From the angles, what did the set-square check, prove?
Answer:
All 4 angles are right angles.
(c) Why is it not enough to say “opposite sides are parallel”? What extra must we check?
Answer:
All 4 angles are 90 and diagonals are equal.
(d) If you draw both diagonals of the frame, what do you notice about their lengths?
Answer:
Equal
(e) Where do the diagonals of a rectangle intersect?
Answer:
At the center of the rectangle (Bisect each other)
(f) Construct a rectangle with diagonal of length 10 cm. Verify that the diagonals bisect each other.
Answer:
Do it yourself
Question 2.
Two students each claim to have drawn a rectangle:
Student A: Diagonals are equal but do not bisect each other.
Student B: Diagonals bisect each other but are not equal.
Who, if anyone, has definitely drawn a rectangle? Give a clear reason using rectangle tests.
Answer:
Neither of the students has drawn rectangle correctly.
A special Rectangle
Question 3.
Aditi and Rohan were playing with tiles on the floor. Aditi noticed that some tiles were rectangular, while some looked perfectly equal on all sides.
Aditi: “These look like rectangles, but the smaller ones seem special.”
Rokan: “Yes, those special ones are called squares Do you know why?”
Aditi: “Maybe because all their sides are of the same length?”
Rokan: “Exactly! A square is a special type of rectangle. Like a rectangle, it has four right angles. But in addition, all four sides of a square are equal. Its diagonals are also equal in length, meet at their midpoint, and intersect each other at right angles.”
(a) How are the sides of a square different from the sides of a rectangle?
Answer:
In square, all sides are equal. In rectangle, opposite sides are equal.
(b) What do you notice about the diagonals of a square (their length and angle of intersection)?
Answer:
Diagonals are equal in length. They bisect each other at 90.
(c) What extra property makes a rectangle into a square?
Answer:
All sides are equal in length.
Question 4.
Examine each of the following statements, and determine whether it is ‘Always true’, ‘Sometimes true’, or ‘Never true’.
(a) A rectangle with perpendicular diagonals is a square.
(b) A quadrilateral with equal diagonals is a square.
Answer:
(a) Always true
(b) Sometimes true
Question 5.
Construct a square with a diagonal of length 8 cm without using a protractor. Also, write the steps.
Answer:
Do it yourself.
Question 6.
Find all the other angles inside the given rectangle:
Answer:
∠OBA = 50°, ∠DAO = 50°, ∠ADO = 40°, ∠ODC = 50°, ∠DCO = 40°, ∠OCB = 5O°, ∠OBC = 40°
Question 7.
A quadrilateral has all sides equal and diagonals perpendicular, but only one angle is a right angle. Is it a square? Explain what is missing.
Answer:
No (The other 3 angles must be equal to 90. arid diagonals must be equal.)
Question 8.
Prove that if a rectangle has perpendicular diagonals, then its adjacent sides are equal; hence, it is a square. Give a short argument using congruent right triangles formed by the diagonals.
Answer:
Do it yourself.
Question 9.
Draw a quadrilateral whose diagonals have equal lengths of 6 cm that bisect each other, and intersect at an angle of 30°. Also, write the steps of your construction.
Angles in a Quadrilateral
Question 10.
Karan was arranging four strips of paper on the floor to make a closed four-sided shape. He wondered about the angles inside the shape.
Karan: “If I tape these four strips to make any closed four-sided figure, what will be the sum of its interior angles?”
Arjun: “Let’s check. Fold a rectangle along one of its diagonals. Do you see two triangles?”
Karan: “Yes – there are two triangles.”
Arjun: “Each triangle has an angle sum of 180°. Two triangles together have 180° + 180° = 360°.
That is why the interior angles of any quadrilateral add up to 360°. The shape may be slanted or stretched, but as long as it has four sides, the sum of its interior angles is always 360°.
(a) Why does “dividing” a quadrilateral into two triangles explain the 360° angle sum?
Answer:
Each triangle has 180 as the sum of its all angle
(b) If one angle of a quadrilateral increases, what must happen to at least one of the other angles to keep the total 360°?
Answer:
Other angles must decrease
(c) Give two real-life objects that are quadrilaterals. How would you quickly check their angles?
Answer:
a book cover and a windowpane
Question 11.
Solve the following problems based on the angle sum of a quadrilateral:
(a) What should the fourth angle be for a quadrilateral if its three angles are 90°, 85° and 120°?
Answer:
65°
(b) The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. Find all four angles.
Answer:
36, 72, 108, 144
(c) One angle of a quadrilateral is 3x, the next is (x + 20)°, the third is (2x – 10)°, and the fourth is 70°. Find the value of x and all the four angles.
Answer:
x = 46.67’ Angles: 140’, 66.67’, 83.33’, 70
More Quadrilaterals with Parallel Opposite sides
Question 12.
A student was looking on the shapes drawn on the board and wondered about its shapes.
Student: “This shape looks like a rectangle, but this shape is slanted.”
Teacher: “When both pairs of opposite sides are parallel, the figure is called a parallelogram. In a parallelogram, opposite sides are equal, opposite angles are equal, adjacent angles add up to 180°, and the diagonals bisect each other.”
Student: “What if the angles were all right angles?”
Teacher: “If each interior angle is 90°, the parallelogram becomes a rectangle. If a figure has both four right angles and four equal sides, it is a square – a shape that is both a rectangle and a parallelogram.”
(a) Name two properties that differentiate a parallelogram from a rectangle.
Answer:
In rectangle, all angles are 90 and diagonals are equal.
(b) What happens to the other angles of a parallelogram when one angle increases?
Answer:
The opposite angle also increases and the two adjacent angles decrease.
(c) If one angle of a parallelogram is 40°, what are the remaining angles of the parallelogram?
Answer:
140° 40°, 140°
(d) What happens when we draw both diagonals of a parallelogram?
Answer:
Diagonals bisect each other at mid point
Question 13.
Find the remaining angles and side lengths in the adjoining quadrilateral.
Answer:
108°, 40°, 108°, BC = AD = 4 cm, CD = AB = 8 cm
Question 14.
In. Q parallelogram LMNO, diagonals intersect at X. If LX = 11 cm, MO = 30 cm, and NO = 24 cm, find XO and LN.
Answer:
XO = 11 cm, LN = 30 cm
Question 15.
Opposite angles of a parallelogram differ by 40°. Is this possible? Explain.
Answer:
No
Question 16.
A quadrilateral has opposite sides equal and parallel. One angle is 108°. Find all other angles.
Answer:
72°, 108°, 72°
Question 17.
Examine each of the follow Lag statements, and determine whether it is ‘Always true’, ‘Sometimes true’, or ‘Never true’.
(a) A quadrilateral with one pair of parallel sides is a parallelogram.
(b) A quadrilateral with diagonals that bisect each other is a parallelogram.
(c) A quadrilateral with equal diagonals is a rectangle.
(d) A parallelogram with one right angle is a rectangle.
Answer:
(a) Never true
(b) Always true
(c) Sometime’s true
(d) Always true
Quadrilaterals with Equal side Lengths
Question 18.
In rhombus PQRS, diagonals intersect at M. If PM = 9 cm and QM = 12 cm, find PR and QS.
Answer:
PR = 18 cm, QS = 24
Question 19.
If one angle of a rhombus is 68°, find all other angles.
Answer:
112°, 68°, 112°
Question 20.
Examine each of the following statements, and. determine whether it is ‘Always true’,
‘Sometimes true’, or ‘Never true’
(a) A quadrilateral with all sides equal is a rhombus.
(b) In a rhombus, opposite angles are equal
(c) In a rhombus, diagonals are equal.
(d) In a rhombus, diagonals are perpendicular.
(e) In a rhombus, each diagonal bisects the angles it touches.
Answer:
(a) Always true
(d) Always true
(b) Always true
(c) Sometimes true
(e) Always true
Kite and Trapezium
Kite
During the kite festival, Zoya was looking closely at the shape of a kite that had two pairs of equal sides placed next to each other.
Zoya: “These two shorter sides are equal, and the two longer sides are also equal.”
Arav: “That means it is a kite. In a kite, the diagonals are perpendicular. One diagonal bisects the other, and it also bisects the vertex angles at its ends.”
Zoya: “So only one pair of opposite angles is equal, not both pairs.”
Arav: “Exactly. In a kite, equal sides come in adjacent pairs, unlike a parallelogram, where opposite sides are equal.”
Question 21.
Based on the above information, answer the following:
(a) How is a kite different from a rhombus?
Answer:
Kite: Two pairs of adjacent sides equal
Rhombus: All sides are equal
(b) Which diagonal bisects the other in a kite?
Answer:
The diagonal which connects the vertices of unequal sides.
(c) Which pair of opposite angles are equal in a kite?
Answer:
Between the unequal sides
(d) If the two adjacent sides AB and AD of a kite ABCD are 6 cm and 8 cm respectively, what are the lengths of sides BC and CD?
Answer:
BC = 6 cm, CD = 8 cm
Question 22.
Construct a kite whose diagonals are of lengths 10 cm and 12 cm. Also, write the steps of your construction.
Trapezium
Arun notices a slanted panel of a roof; which has only one pair of parallel sides.
Meera: “Only the top and bottom look parallel here.”
Arun: “That makes it a trapezium. If the two non-parallel sides are equal, it becomes an isosceles trapezium.”
Meera: “Then the base angles on the same base look equal.”
Arun: “Yes – in an isosceles trapezium, the angles adjacent to each base are equal, and its diagonals are equal in length. In any trapezium, the two angles along the same leg add up to 180°.”
Question 23.
Based OR. the above Informotion, answer the following:
(a) Write one side clue that tells you a quadrilateral is a trapezium.
(b) If one diagonal of on isosceles trapezium is of length 12 cm, what is the length of the other diagonal?
(c) Which pairs of angles are supplementary in any trapezium with AB || CD?
(d) In an isosceles trapezium, what can you say about the non-parallel sides? Are the base angles equal?
Answer:
(a) Only one pair of parallel sides
(b) 12 cm
(c) ∠A + ∠D = 180 and ∠B + ∠C = 180
(d) Equal, Yes
Question 24.
Can a trapezium be a parallelogram? Why or why not?
Answer:
Yes.
Question 25.
The angles at one base of an isosceles trapezium are 70° each. Find the other two angles.
Answer:
110°, 110°
Question 26.
Compare a trapezium arid a parallelogram – give two similarities and two differences.
Answer:
Similarities: Both have 4 sides. All angles add up to 360°
Differences: Parallelogram has two pairs of parallel sides. Trapezium has only one pair of parallel side. Parallelogram has equal opposite angles, Trapezium doesn’t have equal opposite angles.
Question 27.
Find the remaining angles in the following trapezium.
Answer:
55°, 78°
Question 28.
If JUMP and DUCK are two rectangles, find ∠MCK.
Answer:
35°
Worksheet On Quadrilaterals Class 8
A. Choose the correct option.
1. The sum of all the Interior angles of a quadrilateral is:
(a) 900
(b) 1800
(c) 2700
(d) 3600
Answer:
(d) 3600
2. A square is always
(a) A rectangle only
(b) A rhombus only
(c) Both rectangle and rhombus
(d) None of these
Answer:
(c) Both rectangle and rhombus
3. In a parallelogram, opposite angles are:
(a) Always 60°
(b) Equal
(c) Supplementary
(d) Right angles
Answer:
(b) Equal
4. The diagonals of a rhombus:
(a) Are equal but not perpendicular.
(b) Bisect each other but not at right angles.
(c) Bisect each other at right angles.
(d) Never bisect each other.
Answer:
(c) Bisect each other at right angles.
5. A trapezium is a quadrilateral in which:
(a) All sides are equal.
(b) At least one pair of opposite sides are parallel.
(c) Both pairs of opposite sides are parallel.
(d) No sides are parallel.
Answer:
(b) At least one pair of opposite sides are parallel.
Directions. (For Q.6 – 10): In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option as:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
(c) Assertion (A) is true, but Reason (R) is false.
(d) Assertion (A) is false, but Reason (R) is true.
6. Assertion (A): The diagonals of a rectangle are equal and bisect each other.
Reason (R): In a rectangle, opposite sides are parallel.
Answer:
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
7. Assertion (A): Any quadrilateral whose angles are all 90° is a rectangle.
Reason (R): If all angles of a quadrilateral are right angles, then its opposite sides are equal.
Answer:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
8. Assertion (A): The diagonals of a rhombus are perpendicular bisectors of each other.
Reason (R): All the angles of a rhombus are equal.
Answer:
(c) Assertion (A) is true, but Reason (R) is false.
9. Assertion (A): In a parallelogram, each pair of adjacent angles Is supplementary.
Reason (R): A transversal cutting two parallel lines makes Interior angles on the same side supplementary.
Answer:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
10. Assertion (A): Every trapezium has exactly one pair of opposite sides parallel.
Reason (R): In a kite, one pair of opposite sides Is parallel.
Answer:
(c) Assertion (A) is true, but Reason (R) is false.
B. Fill in the blanks.
1. The diagonals of a rectangle are and ______.
Answer:
Equal, bisect each other
2. The opposite sides of a parallelogram are ______ and ______.
Answer:
Equal, parallel
3. The diagonals of a square bisect each other at ______.
Answer:
Right angles
4. A kite has two pairs of adjacent sides ______ In length.
Answer:
Equal
5. The sum of all angles of a quadrilateral Is always ______.
Answer:
360’
C. State whether the following statements are True (T) or False (F).
1. Every rhombus is a parallelogram.
Answer:
True
2. Every trapezium Is a parallelogram.
Answer:
False
3. A rectangle Is also a square.
Answer:
False
4. In an Isosceles trapezium, the base angles are equal.
Answer:
True
5. A quadrilateral having three right angles will be a rectangle.
Answer:
True
D. Solve the following.
Question 1.
Draw a quadrilateral with angles 60°, 30°, 90°, and 180°. Is It possible? Why or why not?
Answer:
Do it yourself
Question 2.
In a parallelogram, one angle Is 70°. Find the other three angles.
Answer:
110°, 70°, 110.
Question 3.
A rhombus has one angle equal to 60°. Find the remaining three angles.
Answer:
120, 60°, 120°
Question 4.
The.diagonals of a kite are 12 cm and 16 cm. Verify that the diagonals bisect each other at right angles by construction.
Answer:
Do it yourself
Question 5.
Construct a square with diagonal 14 cm. Find the length of its side.
Answer:
Do it yourself (side length = 9.89 cm)
Question 6.
Muskan draws a triangle KLM, where N and 0 are mid points of side KL and KM of ΔKLM. Now, she wants to prove NO II LM.
She constructs a ray l such that KL II MP
(a) If LM = 16 cm, what is the length of NP?
(b) If ∠NLM = 72°, then find ∠LMP.
(c) Write the figure name of NLMP.
Answer:
(a) 8 cm
(b) 72°
(c) Trapezium