Important Questions CBSE Class 9 Maths Chapter 8 Quadrilaterals

Quadrilaterals are a vital topic in Class 9 Mathematics, covered in Chapter 8 of the CBSE curriculum. Mastering this topic requires understanding key concepts, theorems, and problem-solving techniques. This article presents important questions for Class 9 Maths Chapter 8 (Quadrilaterals) to help students practice and excel in their exams.

Key Concepts in Quadrilaterals

Before diving into the questions, ensure you understand these fundamental concepts:

• Types of quadrilaterals: parallelograms, rectangles, squares, rhombuses, and trapeziums.
• Properties of quadrilaterals, such as opposite sides, angles, and diagonals.
• Important theorems like the Midpoint Theorem and their applications.

Also Check: CBSE Syllabus for Class 9

Quadrilateral Class 9 Extra Questions: Very Short Answer Type Questions

1. Question: What is a quadrilateral?Solution: A quadrilateral is a polygon with four sides and four angles.
2. Question: Name a quadrilateral whose diagonals are equal and bisect each other at right angles.Solution: Square.
3. Question: If one angle of a parallelogram is 70°, what are the measures of the other three angles?Solution: The opposite angle is also 70°, and the adjacent angles are 110° each (since adjacent angles in a parallelogram are supplementary).
4. Question: In a rhombus, if one angle is 60°, what are the measures of the other three angles?Solution: The opposite angle is also 60°, and the adjacent angles are 120° each.
5. Question: True or False: All rectangles are squares.Solution: False. While all squares are rectangles (having equal angles and opposite sides equal), not all rectangles are squares (since squares require all four sides to be equal).
6. Question: What is the sum of the interior angles of a quadrilateral?Solution: 360°.
7. Question: In a parallelogram, if one angle is twice its adjacent angle, find the measures of all angles.Solution: Let the smaller angle be x. Then, the adjacent angle is 2x. Since adjacent angles are supplementary: x + 2x = 180° ⇒ 3x = 180° ⇒ x = 60°. Thus, the angles are 60°, 120°, 60°, and 120°.
8. Question: Name a quadrilateral whose diagonals bisect each other but are not equal.Solution: Parallelogram.
9. Question: True or False: The diagonals of a rhombus are equal.Solution: False. The diagonals of a rhombus bisect each other at right angles but are not necessarily equal.
10. Question: If the diagonals of a quadrilateral bisect each other at right angles, what type of quadrilateral is it?Solution: Rhombus.

    Also Check: NCERT Solutions for Class 9 Science

1. If the diagonals of a quadrilateral bisect each other, what type of quadrilateral is it?

Solution:
A parallelogram is a quadrilateral whose diagonals bisect each other.

2. In a parallelogram, one angle is 75°. Find the other three angles.

Solution:
In a parallelogram, opposite angles are equal, and adjacent angles are supplementary.
Given: One angle = 75°

• Opposite angle = 75°
• Adjacent angles = 180° – 75° = 105°Thus, the angles are 75°, 105°, 75°, and 105°.

3. Prove that the sum of all angles in a quadrilateral is 360°.

Solution:
A quadrilateral can be divided into two triangles, and the sum of interior angles of a triangle is 180°.
Since there are two triangles, the total sum of the angles is:

 

$$180° + 180° = 360°$$

180°+180°=360°

Thus, the sum of all angles in a quadrilateral is 360°.

4. If the diagonals of a rhombus are 12 cm and 16 cm, find its side length.

Solution:
In a rhombus, the diagonals bisect each other perpendicularly.
Each half-diagonal measures:

 

$$\frac{12}{2} = 6 \text{ cm}, \quad \frac{16}{2} = 8 \text{ cm}$$

212​=6 cm,216​=8 cm

Using Pythagoras’ theorem:

 

$$\text{Side}^2 = 6^2 + 8^2$$

Side2=62+82

$$\text{Side}^2 = 36 + 64 = 100$$

Side2=36+64=100

$$\text{Side} = \sqrt{100} = 10 \text{ cm}$$

Side=100​=10 cm

Thus, each side of the rhombus is 10 cm.

5. In a rectangle, one diagonal is inclined to a side at 30°. Find the acute angle between the diagonals.

Solution:
In a rectangle, the diagonals bisect each other and are equal in length.
Given: One diagonal makes 30° with a side.
Since diagonals form congruent triangles, the acute angle between the diagonals is:

 

$$2 \times 30° = 60°$$

2×30°=60°

Thus, the acute angle between the diagonals is 60°.

6. A quadrilateral has three angles as 90°, 70°, and 85°. Find the fourth angle.

Solution:
Sum of all angles in a quadrilateral = 360°.
Let the fourth angle be x.

 

$$90° + 70° + 85° + x = 360°$$

90°+70°+85°+x=360°

$$x = 360° – (90° + 70° + 85°)$$

x=360°−(90°+70°+85°)

$$x = 360° – 245° = 115°$$

x=360°−245°=115°

Thus, the fourth angle is 115°.

7. A parallelogram has one angle as three times its adjacent angle. Find all angles.

Solution:
Let one angle be x. Then, the adjacent angle is 3x.
Since adjacent angles in a parallelogram are supplementary:

 

$$x + 3x = 180°$$

x+3x=180°

$$4x = 180°$$

4x=180°

$$x = 45°$$

x=45°

Thus, the angles are 45°, 135°, 45°, and 135°.

8. The angles of a quadrilateral are in the ratio 2:3:4:5. Find the measure of each angle.

Solution:
Let the angles be 2x, 3x, 4x, and 5x.
Since the sum of all angles in a quadrilateral is 360°:

 

$$2x + 3x + 4x + 5x = 360°$$

2x+3x+4x+5x=360°

$$14x = 360°$$

14x=360°

$$x = 25°$$

x=25°

Thus, the angles are:

 

$$2x = 50°, \quad 3x = 75°, \quad 4x = 100°, \quad 5x = 125°$$

2x=50°,3x=75°,4x=100°,5x=125°

9. Prove that the diagonals of a rectangle are equal.

Solution:
Let ABCD be a rectangle with diagonals AC and BD.
We prove AC = BD using congruence of triangles.
In ΔABC and ΔBAD:

• AB = AD (opposite sides of a rectangle are equal).
• BC = CD (opposite sides of a rectangle are equal).
• ∠ABC = ∠BAD = 90° (rectangle has right angles).Thus, by SAS Congruence,

 

$$\triangle ABC \cong \triangle BAD$$

△ABC≅△BAD

So, AC = BD (corresponding parts of congruent triangles are equal).

Hence, the diagonals of a rectangle are equal.

10. If one diagonal of a parallelogram bisects one of its angles, show that it is a rhombus.

Solution:
Let ABCD be a parallelogram where diagonal AC bisects ∠A.
Since diagonal bisects the angle, we get:

 

$$\angle 1 = \angle 2$$

∠1=∠2

In ΔABC and ΔADC:

• AB = AD (opposite sides of parallelogram).
• AC = AC (common side).
• ∠1 = ∠2 (given).Thus, ΔABC ≅ ΔADC (SAS congruence).So, BC = CD (corresponding parts of congruent triangles).

  Since all sides are equal, ABCD is a rhombus.

Quadrilateral Class 9 Extra Questions

1. Define a quadrilateral. List six types of quadrilaterals.

Solution:

A quadrilateral is a polygon with four sides (edges) and four vertices (corners).

Six types of quadrilaterals are:

1. Parallelogram
2. Rectangle
3. Square
4. Rhombus
5. Trapezium (or Trapezoid)
6. Kite

2. In which quadrilateral are the diagonals equal and bisect each other at right angles?

Solution:

In a square, the diagonals are equal in length and bisect each other at right angles (90°).

3. Identify the type of quadrilateral formed by joining the midpoints of consecutive sides of a quadrilateral whose diagonals are:

a) Perpendicular

b) Equal

Solution:

a) If the diagonals of a quadrilateral are perpendicular, joining the midpoints of its consecutive sides forms a rectangle.

b) If the diagonals of a quadrilateral are equal, joining the midpoints of its consecutive sides forms a rhombus.

4. In a parallelogram, if one angle measures 80°, find all the angles.

Solution:

In a parallelogram, opposite angles are equal, and adjacent angles are supplementary (sum to 180°).

Given one angle = 80°.

Therefore, the opposite angle is also 80°.

The adjacent angles = 180° – 80° = 100°.

Thus, the angles are 80°, 100°, 80°, and 100°.

5. In a rectangle, one diagonal is inclined to one of its sides at 25°. Determine the acute angle between the two diagonals.

Solution:

In a rectangle, diagonals are equal and bisect each other.

Given: One diagonal makes a 25° angle with a side.

Since diagonals bisect each other, they form two congruent triangles.

The acute angle between the diagonals = 2 × 25° = 50°.

6. Is it possible to have a quadrilateral with all angles obtuse? Explain.

Solution:

No, it’s not possible.

The sum of all interior angles of a quadrilateral is 360°.

An obtuse angle is greater than 90°.

If all four angles were obtuse, their sum would exceed 360°, which contradicts the angle sum property of quadrilaterals.

7. Prove that the angle bisectors of a parallelogram form a rectangle.

Solution:

In a parallelogram, adjacent angles are supplementary.

The bisectors of adjacent angles intersect to form angles of 90°.

Thus, the quadrilateral formed by the angle bisectors is a rectangle.

8. In a trapezium, the angles adjacent to the non-parallel sides are 55° and 70°. Find the other two angles.

Solution:

In a trapezium, the sum of angles adjacent to each non-parallel side is 180°.

Let the trapezium be ABCD with AB || CD.

Given: ∠A = 55°, ∠B = 70°.

∠D = 180° – ∠A = 180° – 55° = 125°.

∠C = 180° – ∠B = 180° – 70° = 110°.

Thus, the angles are 55°, 70°, 125°, and 110°.

9. Calculate all the angles of a parallelogram if one of its angles is twice its adjacent angle.

Solution:

Let one angle be x.

Its adjacent angle = 2x.

Since adjacent angles in a parallelogram are supplementary:

x + 2x = 180°

3x = 180°

x = 60°

Therefore, the angles are 60°, 120°, 60°, and 120°.

10. The angles of a quadrilateral are in the ratio 2:5:4:1. Find the measure of each angle.

Solution:

Let the common ratio be x.

Then, the angles are 2x, 5x, 4x, and 1x.

Sum of angles in a quadrilateral = 360°

2x + 5x + 4x + x = 360°

12x = 360°

x = 30°

Therefore, the angles are:

2x = 60°

5x = 150°

4x = 120°

1x = 30°

FAQs on CBSE Class 9 Maths Chapter 8

What are the important points of Chapter Quadrilateral Class 9?

Properties of quadrilaterals (parallelogram, rectangle, rhombus, square, trapezium), mid-point theorem, and diagonal properties.

<>How do you get 100% in Maths Class 9?

Practice daily, understand concepts, solve NCERT exercises, and revise formulas regularly.

<>Is Quadrilateral Chapter Class 9 hard?

No, it is easy if you understand properties and practice problems regularly.

<>What is the hardest math in Class 9?

Geometry (Circles, Constructions), Algebra (Polynomials), and Trigonometry can be tricky for some students.

<>What is important in Quadrilateral Class 9?

Theorems, properties of different quadrilaterals, and proving questions are important.

<>How to get full marks in Maths Class 9 CBSE?

Focus on concept clarity, practice NCERT thoroughly, solve sample papers, and avoid silly mistakes in calculations.