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Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8

Document Description: Important Questions: Understanding Quadrilaterals for Class 8 2022 is part of Understanding Quadrilaterals for Mathematics (Maths) Class 8 preparation. The notes and questions for Important Questions: Understanding Quadrilaterals have been prepared according to the Class 8 exam syllabus. Information about Important Questions: Understanding Quadrilaterals covers topics like Practice Exercise and Important Questions: Understanding Quadrilaterals Example, for Class 8 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Important Questions: Understanding Quadrilaterals.

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Practice Exercise
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Question 1: In the adjoining figure, find x + y + z + w 

Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8
Solution: Since, the sum of the measures of interior angles of a quadrilateral is 360°.
Also, 115° + 70° + 60° = 245°
∴ 245° +∠ABC = 360°
⇒∠ABC = 360° – 245° = 115°

Now, x = ext. ∠BCD 
= 180° – ∠BCD
= 180° – 115° = 65°

Similarly, y = 180° – 70° = 110°
z = 180° – 60° = 120°
w = 180° – 115° = 65°
∴ x + y + z + w = 65° + 110° + 120° + 65°
= 360°


Question 2: In a quadrilateral ABCD, the angles A, B, C and D are in the ratio 1 : 2 : 3 : 4. Find the measure of each angle of the quadrilateral.
Solution: ∵∠A : ∠B : ∠C : ∠D = 1 : 2 : 3 : 4
∴ Let us suppose that 

A = 1x°, B = 2x° 

C = 3x°, –D = 4x°
Since, ∠A +∠B + ∠C +∠D = 360°
∴ x + 2x + 3x + 4x = 360°
⇒ 10x = 360°
⇒ x = 360/10 = 36°
∴ Angles are:
 ∠A = x° = 36° 

∠B = 2x° = 2 x 36° = 72° 
C = 3x° = 3 x 36° = 108° 
D = 4x° = 4 x 36° = 144° 

Thus, the measure of the angle of the quad. are 36°, 72°, 108° and 144°

Question 3: The interior angle of a regular is 108°. Find the number of sides of the polygon.
Solution: Let there are ‘n’ sides of the regular polygon. 

 Measure of each exterior angle =  (360/n)o
Since, the measure of each of the interior angle = 108° 
 Measure of each exterior angle = 180° – 108° = 72°
⇒ 360/n = 72   n = 360/72 = 5
Thus, the required number of sides of the regular polygon is 5.
Alternate Solution
Let there be ‘n’ sides of the regular polygon.
Since the measure of each interior angle is  Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8

Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8

⇒ (2n-4) x 90 = 108 x n

⇒ 180n - 360 = 108n

⇒ 180n - 108n = 360

⇒  72n = 360 ⇒ n= (360/72) = 5
Thus, there are 5 sides of the given regular polygon.

Question 4: Two regular polygons are such that the ratio of the measures their interior angles is 4 : 3 and the ratio between their number of sides is 2: 1. Find the number of sides of each polygon.
Solution: Let 2n and n be the number of sides of the regular polygons.
∴  Their interior angles are  

Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8
Since the ratio of the interior angles is 4 : 3
Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8
Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8
Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8
Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8
Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8   
⇒  3(n-1) = 4(n-2)

⇒  3n -3 = 4n -8

⇒  3n - 4n = -8 +3

⇒  -n = -5 ⇒ n=5

⇒   2n = 2 x 5 = 10
Thus, the number of sides of the polygons are 10 and 5 respectively.

Question 5: The exterior angle of a regular polygon is one-fifth of its interior angle. How many sides has the polygon?
Solution: Let the number of sides be ‘n’.
∴ Exterior angle of polygon =  (360/n)o

And the interior angle of the polygon = Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8
since Exterior angle = 1/5 (Interior angle)

Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8
Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8
⇒ Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8

⇒ 2n- 4 = 20 ⇒ 2n = 20 +4 = 24

⇒ n= 24/2 = 12

Thus, the polygon is having 12 sides.

Question 6: The measures of two adjacent angles of a parallelogram are in the ratio 4: 5. Find the measure of each of the angles of the parallelogram.
Solution: Let ABCD be a parallelogram such that ∠A and ∠B are 4x and 5x respectively.
Since, the adjacent angles are supplementary,

∴  ∠A+  ∠B = 1800

⇒  4x + 5x = 180o

⇒  9x+ = 1800   . ⇒   x = 180o/9 = 20o
∴ ∠A = 4x = 4 x 20° = 80° 

and ∠B = 5x = 5 x 20° = 100° 
We know that opposite angles of a parallelogram are equal. 
∴ ∠C = ∠A = 80º 

And  ∠D= ∠B = 100°
Thus,  ∠A = 80°, ∠B = 100°, ∠C = 80° and D = 100°

Question 7: In a quadrilateral ABCD, DO and CO are the bisectors of ∠D and ∠C respectively.
Prove that  Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8

 

Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8
Solution: In ΔCOD, we have 

⇒ ∠COD + ∠1 + ∠2 = 180°
⇒ ∠COD = 180° – [–1 + –2]
Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8

⇒ But ∠A+ ∠B+∠C+∠D = 360o

 ⇒ ∠C+∠D = 360o - (∠A+∠B)

 ∴Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8
Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8

 Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8

Thus,    Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8


Practice Exercise

Q 1. If an exterior angle of a regular polygon is 45°, then find the number of its sides.

Ans: 8

Q 2. If an interior angle of a regular polygon is 162°, then find the number of its sides.
Ans: 
20

Q 3. Find x in the following figure.

Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8
 Ans: 
70o


Q 4. Find the measure of an interior angle of a regular polygon having 15 sides.

Ans: 156o

Q 5. Find the measure of each of the (i) exterior angle. (ii) interior angle of an octagon.

Ans: (i) 45° (ii) 135°
 

Q 6. An angle of a parallelogram measures 70°. Find the measure of the remaining three angles.

Ans: 110°, 70°, 110°
 

Q 7. In the following figure, ABCD is a parallelogram. Find measures of x, y and z.

Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8
Ans: x = 80°, y = 100°, z = 80°

Q 8. In the following figure, find the measures of x, y, z and w.
Important Questions: Understanding Quadrilaterals Notes | Study Mathematics (Maths) Class 8 - Class 8 
Ans: x = 120°, y = 100°, z = 120°, w = 80°

Q 9. One angle of a quadrilateral is 111° and the remaining three angles are equal. Find three angles. 

Ans: 81° each

Q.10. What is the ratio of the interior angles of a pentagon and a decagon?

Note: (i) A pentagon has 5 sides.  (ii) A decagon has 10 sides.

Ans: 3: 4

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