Chapter 3 Understanding Quadrilaterals mcqs and important question mathematics |class 8th

MCQ Questions for Class 8th Maths: Ch 3 Quadrilaterals

1. The sum of all the angles of a quadrilateral is equal to:

(a) 180°

(b) 270°

(c) 360°

(d) 90°

► (c) 360°

2. A diagonal of a parallelogram divides it into two congruent:

(a) Square

(b) Parallelogram

(c) Triangles

(d) Rectangle

► (c) Triangles

3. The diagonals of a parallelogram:

(a) Equal

(b) Unequal

(c) Bisect each other

(d) Have no relation

► (c) Bisect each other

4. Each angle of rectangle is:

(a) More than 90°

(b) Less than 90°

(c) Equal to 90°

(d) Equal to 45°

► (c) Equal to 90°

5. If ABCD is a trapezium in which AB || CD and AD = BC, then:

(a) ∠A = ∠B

(b) ∠A > ∠B

(c) ∠A < ∠B

(d) None of the above

► (a) ∠A = ∠B

6. The diagonals AC and BD of a parallelogram ABCD intersect each other at the point O. If ∠DAC = 32°, ∠AOB = 70°, then ∠DBC is equal to:

(a) 32°

(b) 88°

(c) 24°

(d) 38°

► (d) 38°

7. In parallelogram ABCD, if ∠A = 2x + 15°, ∠B = 3x – 25°, then value of x is:

(a) 91°

(b) 89°

(c) 34°

(d) 38°

► (d) 38°

8. The opposite angles of a parallelogram are (3x – 2)° and (50 – x)° the measure of these angles is ______.​

(a) 140°, 140°

(b) 20°, 160°

(c) 37°, 143°

(d) 37°, 37°

► (d) 37°, 37°

9. In quadrilateral PQRS, if ∠P = 60° and ∠Q : ∠R : ∠S = 2 : 3 : 7, then ∠S =

(a) 175°

(b) 210°

(c) 150°

(d) 135°

► (a) 175°

10. Two angles of a quadrilateral are 50° and 80° and other two angles are in the ratio 8 : 15. Find the measure of the remaining two angles.​

(a) 100°, 130°

(b) 140°, 90°

(c) 80°, 150°

(d) 70°, 160°

► (c) 80°, 150°

11. The diagonals of rhombus are 12 cm and 16 cm. The length of the side of rhombus is:

(a) 12 cm

(b) 16 cm

(c) 8 cm

(d) 10 cm

► (d) 10 cm

12. In a parallelogram the sum of two consecutive angles is

(a) 360°

(b) 100°

(c) 180°

(d) 90°

► (c) 180°

13. The angles of a quadrilateral are (5x)°, (3x + 10)°, (6x – 20)° and (x + 25)°. Now, the measure of each angle of the quadrilateral will be

(a) 115°, 79°, 118°, 48°

(b) 100° 79°, 118°, 63°

(c) 110°, 84°, 106°, 60°

(d) 75°, 89°, 128°, 68°

► (a) 115°, 79°, 118°, 48°

14. Which of the following is not a parallelogram?

(a) Rectangle

(b) Rhombus

(c) Square

(d) Trapezium

► (d) Trapezium

15. In a Quadrilateral ABCD, AB = BC and CD = DA, then the quadrilateral is a

(a) Triangle

(b) Kite

(c) Rhombus

(d) Rectangle

► (b) Kite

16. If ABCD is a Parallelogram with 2 Adjacent angles ∠A =∠B, then the parallelogram is a

(a) Rhombus

(b) Triangle

(c) Rectangle

(d) Square

► (c) Rectangle

17. All the angles of a convex quadrilateral are congruent. However, not all its sides are congruent. What type of quadrilateral is it?

(a) Parallelogram

(b) Square

(c) Rectangle

(d) Trapezium

► (c) Rectangle

18. Perimeter of a parallelogram is 22 cm. If the longer side, measures 6.5 cm, the measure of the shorter side will be

(a) 4.5 cm

(b) 6.5 cm

(c) 2.5 cm

(d) 3.0 cm

► (a) 4.5 cm

19. If angles A, B, C and D of the quadrilateral ABCD, taken in order, are in the ratio 3:7:6:4, then ABCD is 

(a) rhombus

(b) parallelogram

(c) trapezium

(d) kite

► (c) trapezium

20. Angles of a quadrilateral are in the ratio 3 : 6 : 8: 13. The largest angle is :

(a) 178°

(b) 156°

(c) 90°​

(d) 36°​

► (b) 156°

21. The diagonals of a rectangle PQRS intersects at O. If ∠QOR = 44°, ∠OPS =?

(a) 82°

(b) 52°

(c) 68°

(d) 75°

► (c) 68°

22. In a parallelogram ABCD, if ∠A = 75°, then ∠B = ?

(a) 95°

(b) 80°

(c) 105°

(d) 15°

► (c) 105°

23. A diagonal of a Rectangle is inclines to one side of the rectangle at an angle of 25∘. The Acute Angle between the diagonals is :

(a) 115°

(b) 50°

(c) 40°

(d) 25°

► (b) 50°

24. If an angle of a parallelogram is two-third of its adjacent angle, the smallest angle of the parallelogram is:

(a) 81°

(b) 54°

(c) 108°

(d) 72°

► (d) 72°

Understanding Quadrilaterals Class 8 Extra Questions Very Short Answer Type

Question 1.
In the given figure, ABCD is a parallelogram. Find x.

Solution:
AB = DC [Opposite sides of a parallelogram]
3x + 5 = 5x – 1
⇒ 3x – 5x = -1 – 5
⇒ -2x = -6
⇒ x = 3

Question 2.
In the given figure find x + y + z.

Solution:
We know that the sum of all the exterior angles of a polygon = 360°
x + y + z = 360°

Question 3.
In the given figure, find x.

Solution:
∠A + ∠B + ∠C = 180° [Angle sum property]
(x + 10)° + (3x + 5)° + (2x + 15)° = 180°
⇒ x + 10 + 3x + 5 + 2x + 15 = 180
⇒ 6x + 30 = 180
⇒ 6x = 180 – 30
⇒ 6x = 150
⇒ x = 25

Question 4.
The angles of a quadrilateral are in the ratio of 2 : 3 : 5 : 8. Find the measure of each angle.
Solution:
Sum of all interior angles of a quadrilateral = 360°
Let the angles of the quadrilateral be 2x°, 3x°, 5x° and 8x°.
2x + 3x + 5x + 8x = 360°
⇒ 18x = 360°
⇒ x = 20°
Hence the angles are
2 × 20 = 40°,
3 × 20 = 60°,
5 × 20 = 100°
and 8 × 20 = 160°.

Question 5.
Find the measure of an interior angle of a regular polygon of 9 sides.
Solution:
Measure of an interior angle of a regular polygon

Question 6.
Length and breadth of a rectangular wire are 9 cm and 7 cm respectively. If the wire is bent into a square, find the length of its side.
Solution:
Perimeter of the rectangle = 2 [length + breadth]
= 2[9 + 7] = 2 × 16 = 32 cm.

Now perimeter of the square = Perimeter of rectangle = 32 cm.
Side of the square = 324 = 8 cm.
Hence, the length of the side of square = 8 cm.

Question 7.
In the given figure ABCD, find the value of x.

Solution:
Sum of all the exterior angles of a polygon = 360°
x + 70° + 80° + 70° = 360°
⇒ x + 220° = 360°
⇒ x = 360° – 220° = 140°

Question 8.
In the parallelogram given alongside if m∠Q = 110°, find all the other angles.

Solution:
Given m∠Q = 110°
Then m∠S = 110° (Opposite angles are equal)
Since ∠P and ∠Q are supplementary.
Then m∠P + m∠Q = 180°
⇒ m∠P + 110° = 180°
⇒ m∠P = 180° – 110° = 70°
⇒ m∠P = m∠R = 70° (Opposite angles)
Hence m∠P = 70, m∠R = 70°
and m∠S = 110°

Question 9.
In the given figure, ABCD is a rhombus. Find the values of x, y and z.

Solution:
AB = BC (Sides of a rhombus)
x = 13 cm.
Since the diagonals of a rhombus bisect each other
z = 5 and y = 12
Hence, x = 13 cm, y = 12 cm and z = 5 cm.

Question 10.
In the given figure, ABCD is a parallelogram. Find x, y and z.

Solution:
∠A + ∠D = 180° (Adjacent angles)
⇒ 125° + ∠D = 180°
⇒ ∠D = 180° – 125°
x = 55°
∠A = ∠C [Opposite angles of a parallelogram]
⇒ 125° = y + 56°
⇒ y = 125° – 56°
⇒ y = 69°
∠z + ∠y = 180° (Adjacent angles)
⇒ ∠z + 69° = 180°
⇒ ∠z = 180° – 69° = 111°
Hence the angles x = 55°, y = 69° and z = 111°

Question 11.
Find x in the following figure. (NCERT Exemplar)

Solution:
In the given figure ∠1 + 90° = 180° (linear pair)
∠1 = 90°
Now, sum of exterior angles of a polygon is 360°, therefore,
x + 60° + 90° + 90° + 40° = 360°
⇒ x + 280° = 360°
⇒ x = 80°

Understanding Quadrilaterals Class 8 Extra Questions Short Answer Type

Question 12.
In the given parallelogram ABCD, find the value of x andy.

Solution:
∠A + ∠B = 180°
3y + 2y – 5 = 180°
⇒ 5y – 5 = 180°
⇒ 5y = 180 + 5°
⇒ 5y = 185°
⇒ y = 37°
Now ∠A = ∠C [Opposite angles of a parallelogram]
3y = 3x + 3
⇒ 3 × 37 = 3x + 3
⇒ 111 = 3x + 3
⇒ 111 – 3 = 3x
⇒ 108 = 3x
⇒ x = 36°
Hence, x = 36° and y – 37°.

Question 13.
ABCD is a rhombus with ∠ABC = 126°, find the measure of ∠ACD.

Solution:
∠ABC = ∠ADC (Opposite angles of a rhombus)
∠ADC = 126°
∠ODC = 12 × ∠ADC (Diagonal of rhombus bisects the respective angles)
⇒ ∠ODC = 12 × 126° = 63°
⇒ ∠DOC = 90° (Diagonals of a rhombus bisect each other at 90°)
In ΔOCD,
∠OCD + ∠ODC + ∠DOC = 180° (Angle sum property)
⇒ ∠OCD + 63° + 90° = 180°
⇒ ∠OCD + 153° = 180°
⇒ ∠OCD = 180° – 153° = 27°
Hence ∠OCD or ∠ACD = 27°

Question 14.
Find the values of x and y in the following parallelogram.

Solution:
Since, the diagonals of a parallelogram bisect each other.
OA = OC
x + 8 = 16 – x
⇒ x + x = 16 – 8
⇒ 2x = 8
x = 4
Similarly, OB = OD
5y + 4 = 2y + 13
⇒ 3y = 9
⇒ y = 3
Hence, x = 4 and y = 3

Question 15.
Write true and false against each of the given statements.
(a) Diagonals of a rhombus are equal.
(b) Diagonals of rectangles are equal.
(c) Kite is a parallelogram.
(d) Sum of the interior angles of a triangle is 180°.
(e) A trapezium is a parallelogram.
(f) Sum of all the exterior angles of a polygon is 360°.
(g) Diagonals of a rectangle are perpendicular to each other.
(h) Triangle is possible with angles 60°, 80° and 100°.
(i) In a parallelogram, the opposite sides are equal.
Solution:
(a) False
(b) True
(c) False
(d) True
(e) False
(f) True
(g) False
(h) False
(i) True

Question 16.
The sides AB and CD of a quadrilateral ABCD are extended to points P and Q respectively. Is ∠ADQ + ∠CBP = ∠A + ∠C? Give reason.
(NCERT Exemplar)
Solution:
Join AC, then
∠CBP = ∠BCA + ∠BAC and ∠ADQ = ∠ACD + ∠DAC (Exterior angles of triangles)

Therefore,
∠CBP + ∠ADQ = ∠BCA + ∠BAC + ∠ACD + ∠DAC
= (∠BCA + ∠ACD) + (∠BAC + ∠DAC)
= ∠C + ∠A

Understanding Quadrilaterals Class 8 Extra Questions Higher Order Thinking Skills (HOTS)

Question 17.
The diagonal of a rectangle is thrice its smaller side. Find the ratio of its sides.

Solution:
Let AD = x cm
diagonal BD = 3x cm
In right-angled triangle DAB,
AD² + AB² = BD² (Using Pythagoras Theorem)
x² + AB² = (3x)²
⇒ x² + AB² = 9x²
⇒ AB² = 9x² – x²
⇒ AB² = 8x²
⇒ AB = √8x = 2√2x
Required ratio of AB : AD = 2√2x : x = 2√2 : 1

Question 18.
If AM and CN are perpendiculars on the diagonal BD of a parallelogram ABCD, Is ∆AMD = ∆CNB? Give reason. (NCERT Exemplar)
Solution:

In triangles AMD and CNB,
AD = BC (opposite sides of parallelogram)
∠AMB = ∠CNB = 90°
∠ADM = ∠NBC (AD || BC and BD is transversal.)
So, ∆AMD = ∆CNB (AAS)