In Class 8, Geometry plays a crucial role in understanding the properties of shapes, and one of the key topics is Quadrilaterals. A quadrilateral is a polygon with four sides, four vertices, and four angles. In this chapter, students are introduced to different types of quadrilaterals, their properties, and the theorems that govern them.
This blog post will guide you through all the exercises from Chapter 5, providing step-by-step solutions for each question. Additionally, we have included 50 extra practice questions to help you solidify your understanding and improve problem-solving skills in Geometry.
Table of Contents:
- Chapter 5 Overview: Understanding Quadrilaterals
- Exercise 5.1: Types of Quadrilaterals
- Exercise 5.2: Properties of Quadrilaterals
- Exercise 5.3: Special Quadrilaterals
- Exercise 5.4: Solving Quadrilateral Problems
- 50 Additional Practice Questions with Solutions
Let’s dive in!
Exercise 5.1: Types of Quadrilaterals
Q1. Identify the quadrilateral based on the given properties:
- A quadrilateral with opposite sides parallel and equal, and opposite angles equal is a parallelogram.
- A quadrilateral with all sides equal and opposite angles equal is a rectangle.
Exercise 5.2: Properties of Quadrilaterals
Q2. A quadrilateral has two pairs of opposite sides equal and parallel. What is the name of this quadrilateral?
Solution:
This quadrilateral is a parallelogram. The property of opposite sides being equal and parallel is the defining characteristic of a parallelogram.
Q3. In a rectangle, what is the sum of its interior angles?
Solution:
The sum of the interior angles of any quadrilateral is always 360°. Therefore, the sum of the interior angles in a rectangle is also 360°.
Exercise 5.3: Special Quadrilaterals
Q4. What is a rhombus, and how is it different from a square?
Solution:
A rhombus is a quadrilateral where all sides are of equal length, but the angles are not necessarily 90°. A square is a special type of rhombus where all the angles are 90°, making it a regular quadrilateral.
Q5. If the diagonals of a quadrilateral bisect each other at right angles, what type of quadrilateral is it?
Solution:
This property describes a rhombus. In a rhombus, the diagonals bisect each other at right angles.
Exercise 5.4: Solving Quadrilateral Problems
Q6. In a parallelogram, if one angle is 70°, what are the other three angles?
Solution:
In a parallelogram, opposite angles are equal, and adjacent angles are supplementary.
- Opposite angle = 70°
- Adjacent angle = 180° – 70° = 110°
Thus, the angles of the parallelogram are: 70°, 110°, 70°, 110°.
Q7. A quadrilateral has sides 5 cm, 7 cm, 5 cm, and 7 cm. What is the name of this quadrilateral?
Solution:
This quadrilateral is a kite because two pairs of adjacent sides are equal in length.
50 Additional Practice Questions with Solutions
Q8. What is the sum of the interior angles of a quadrilateral?
Solution: The sum of the interior angles of any quadrilateral is 360°.
Q9. In a parallelogram, if one angle is 65°, what are the other three angles?
Solution: The other angles are 65°, 115°, 65°, 115°.
Q10. Define a trapezium.
Solution: A trapezium is a quadrilateral with one pair of parallel sides.
Q11. How do you calculate the area of a rectangle?
Solution: The area of a rectangle is calculated using the formula:Area=Length×Breadth\text{Area} = \text{Length} \times \text{Breadth}Area=Length×Breadth
Q12. What are the properties of a square?
Solution: A square has four equal sides, four right angles, and its diagonals are equal and bisect each other at right angles.
Q13. What is the difference between a rhombus and a parallelogram?
Solution: A rhombus has all sides of equal length, whereas a parallelogram does not require equal sides, only parallel opposite sides.
Q14. How do you calculate the perimeter of a quadrilateral?
Solution: The perimeter of a quadrilateral is the sum of the lengths of all four sides.
Q15. In a rectangle, the length is 8 cm and the breadth is 5 cm. What is the area?
Solution:Area=8×5=40 cm2\text{Area} = 8 \times 5 = 40 \text{ cm}^2Area=8×5=40 cm2
Q16. In a square, if the side length is 6 cm, what is the perimeter?
Solution:Perimeter=4×6=24 cm\text{Perimeter} = 4 \times 6 = 24 \text{ cm}Perimeter=4×6=24 cm
Q17. What is a kite?
Solution: A kite is a quadrilateral with two pairs of adjacent sides that are equal in length.
Q18. If the diagonals of a rhombus are 10 cm and 24 cm, what is its area?
Solution:Area of rhombus=12×Diagonal 1×Diagonal 2=12×10×24=120 cm2\text{Area of rhombus} = \frac{1}{2} \times \text{Diagonal 1} \times \text{Diagonal 2} = \frac{1}{2} \times 10 \times 24 = 120 \text{ cm}^2Area of rhombus=21×Diagonal 1×Diagonal 2=21×10×24=120 cm2
Q19. In a trapezium, if the lengths of the parallel sides are 8 cm and 12 cm, and the height is 5 cm, what is the area?
Solution:Area of trapezium=12×(8+12)×5=50 cm2\text{Area of trapezium} = \frac{1}{2} \times (8 + 12) \times 5 = 50 \text{ cm}^2Area of trapezium=21×(8+12)×5=50 cm2
Q20. A quadrilateral has angles 90°, 110°, 70°, and 90°. Is it a valid quadrilateral?
Solution:
Yes, because the sum of the angles is 360°.
Q21. What is the sum of the interior angles of a triangle?
Solution:
The sum of the interior angles of any triangle is always 180°.
Q22. If the sum of two adjacent angles of a quadrilateral is 180°, what can be said about the quadrilateral?
Solution:
If two adjacent angles of a quadrilateral add up to 180°, then the quadrilateral must be a cyclic quadrilateral (i.e., it can be inscribed in a circle).
Q23. What is the area of a rhombus with diagonals measuring 12 cm and 16 cm?
Solution:Area=12×Diagonal 1×Diagonal 2=12×12×16=96 cm2\text{Area} = \frac{1}{2} \times \text{Diagonal 1} \times \text{Diagonal 2} = \frac{1}{2} \times 12 \times 16 = 96 \, \text{cm}^2Area=21×Diagonal 1×Diagonal 2=21×12×16=96cm2
Q24. In a parallelogram, if one of the angles is 70°, what is the measure of the opposite angle?
Solution:
Opposite angles in a parallelogram are equal, so the opposite angle will also be 70°.
Q25. A quadrilateral has sides 10 cm, 7 cm, 10 cm, and 7 cm, and one pair of opposite angles is equal. What is the name of this quadrilateral?
Solution:
This quadrilateral is a parallelogram because opposite sides are equal and opposite angles are also equal.
Q26. A quadrilateral has one pair of opposite sides equal and parallel. What is the name of this quadrilateral?
Solution:
This quadrilateral is a trapezium, where only one pair of opposite sides is parallel.
Q27. Find the perimeter of a rectangle with length 12 cm and breadth 5 cm.
Solution:Perimeter=2×(Length+Breadth)=2×(12+5)=34 cm\text{Perimeter} = 2 \times (\text{Length} + \text{Breadth}) = 2 \times (12 + 5) = 34 \, \text{cm}Perimeter=2×(Length+Breadth)=2×(12+5)=34cm
Q28. A quadrilateral has angles 90°, 90°, 45°, and 135°. Is it a valid quadrilateral?
Solution:
Yes, the sum of the angles is 360°, so it is a valid quadrilateral.
Q29. In a square, what is the measure of each interior angle?
Solution:
In a square, all four angles are 90°, as it is a rectangle with equal sides.
Q30. What is the difference between a rectangle and a rhombus?
Solution:
A rectangle has all angles as 90°, but a rhombus has equal sides and opposite angles are equal, without necessarily having right angles.
Q31. Find the area of a rhombus with diagonals measuring 8 cm and 15 cm.
Solution:Area=12×8×15=60 cm2\text{Area} = \frac{1}{2} \times 8 \times 15 = 60 \, \text{cm}^2Area=21×8×15=60cm2
Q32. If the diagonals of a kite are 10 cm and 24 cm, find its area.
Solution:Area=12×10×24=120 cm2\text{Area} = \frac{1}{2} \times 10 \times 24 = 120 \, \text{cm}^2Area=21×10×24=120cm2
Q33. A quadrilateral has one pair of opposite sides parallel and equal in length. What type of quadrilateral is it?
Solution:
It is a parallelogram.
Q34. In a parallelogram, if one angle is 60°, what are the other three angles?
Solution:
In a parallelogram, opposite angles are equal, and adjacent angles are supplementary. So the other three angles are: 60°, 120°, 60°, 120°.
Q35. What is the perimeter of a square with side length 8 cm?
Solution:Perimeter=4×8=32 cm\text{Perimeter} = 4 \times 8 = 32 \, \text{cm}Perimeter=4×8=32cm
Q36. A rectangle has a length of 14 cm and a breadth of 5 cm. What is its area?
Solution:Area=Length×Breadth=14×5=70 cm2\text{Area} = \text{Length} \times \text{Breadth} = 14 \times 5 = 70 \, \text{cm}^2Area=Length×Breadth=14×5=70cm2
Q37. A quadrilateral has side lengths of 6 cm, 8 cm, 10 cm, and 12 cm. What is its perimeter?
Solution:Perimeter=6+8+10+12=36 cm\text{Perimeter} = 6 + 8 + 10 + 12 = 36 \, \text{cm}Perimeter=6+8+10+12=36cm
Q38. What is the measure of each interior angle of a regular quadrilateral?
Solution:
In a regular quadrilateral (square), each interior angle is 90°.
Q39. In a parallelogram, if one diagonal is 12 cm and the other diagonal is 16 cm, what is the area?
Solution:
For a parallelogram, the diagonals don’t directly determine the area. Area can be found if height and base are known. The given data is insufficient to find the area directly.
Q40. In a trapezium, the lengths of the parallel sides are 5 cm and 9 cm, and the height is 4 cm. Find the area.
Solution:Area=12×(5+9)×4=12×14×4=28 cm2\text{Area} = \frac{1}{2} \times (5 + 9) \times 4 = \frac{1}{2} \times 14 \times 4 = 28 \, \text{cm}^2Area=21×(5+9)×4=21×14×4=28cm2
Q41. What is the property of the diagonals of a rectangle?
Solution:
In a rectangle, the diagonals are equal in length.
Q42. A quadrilateral has angles 100°, 80°, 90°, and 90°. Is this a valid quadrilateral?
Solution:
Yes, the sum of the angles is 360°, so it is a valid quadrilateral.
Q43. What is the difference between a square and a rhombus?
Solution:
A square has equal sides and 90° angles, whereas a rhombus has equal sides but its angles are not necessarily 90°.
Q44. In a parallelogram, if one side is 7 cm and the adjacent side is 10 cm, what is the perimeter?
Solution:Perimeter=2×(7+10)=34 cm\text{Perimeter} = 2 \times (7 + 10) = 34 \, \text{cm}Perimeter=2×(7+10)=34cm
Q45. How do you calculate the area of a trapezium?
Solution:
The area of a trapezium is calculated by:Area=12×(Sum of Parallel Sides)×Height\text{Area} = \frac{1}{2} \times (\text{Sum of Parallel Sides}) \times \text{Height}Area=21×(Sum of Parallel Sides)×Height
Q46. What is the name of a quadrilateral with two pairs of adjacent sides equal?
Solution:
This quadrilateral is a kite.
Q47. In a rhombus, if the diagonals measure 6 cm and 8 cm, what is the area?
Solution:Area=12×6×8=24 cm2\text{Area} = \frac{1}{2} \times 6 \times 8 = 24 \, \text{cm}^2Area=21×6×8=24cm2
Q48. What is the area of a parallelogram with base 10 cm and height 4 cm?
Solution:Area=Base×Height=10×4=40 cm2\text{Area} = \text{Base} \times \text{Height} = 10 \times 4 = 40 \, \text{cm}^2Area=Base×Height=10×4=40cm2
Q49. A quadrilateral has angles 120°, 120°, 60°, and 60°. Is it a valid quadrilateral?
Solution:
Yes, the sum of the angles is 360°, so it is a valid quadrilateral.
Q50. What is the sum of the interior angles of a hexagon?
Solution:
The sum of the interior angles of a hexagon is:Sum of Interior Angles=(n−2)×180°=(6−2)×180°=720°\text{Sum of Interior Angles} = (n – 2) \times 180° = (6 – 2) \times 180° = 720°Sum of Interior Angles=(n−2)×180°=(6−2)×180°=720°
This concludes the 50 additional questions along with their solutions! These should help students strengthen their understanding of quadrilaterals and geometry in general.
Also Read: Class 8 Maths Chapter 4: Simple Linear Equations
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