Quadrilaterals — Class 8 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 8 Mathematics chapter "Quadrilaterals" — 7 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 8 Mathematics chapter "Quadrilaterals" — 7 important questions with detailed answers for CBSE board exam p…
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NCERT Solutions for Class 8 Mathematics — Quadrilaterals. Step-by-step answers to all 7 textbook questions from this chapter, written for CBSE board preparation and free to use.
Quadrilaterals — All 7 Questions Solved
Q1. Find the fourth angle of a quadrilateral if three angles are 70°, 85°, and 95°.
We use the angle sum property of quadrilaterals.
Step 1: Recall the angle sum property
The sum of all interior angles of a quadrilateral = 360°
Step 2: Let the fourth angle be x
70° + 85° + 95° + x = 360°
Step 3: Add the three known angles
70 + 85 + 95 = 250
250 + x = 360°
Step 4: Solve for x
x = 360° - 250°
x = 110°
Step 5: Verify
70 + 85 + 95 + 110 = 360 ✓
Answer: The fourth angle is 110°
Q2. Identify the quadrilateral with all sides equal and all angles equal.
We need to identify which quadrilateral has these properties.
Step 1: Analyze the property "all sides equal"
Quadrilaterals with all sides equal: rhombus, square
Step 2: Analyze the property "all angles equal"
If all angles of a quadrilateral are equal:
Each angle = 360°/4 = 90°
Quadrilaterals with all angles = 90°: rectangle, square
Step 3: Find the quadrilateral with both properties
Rhombus: all sides equal, opposite angles equal (but not all angles equal)
Square: all sides equal AND all angles equal (each 90°)
Rectangle: all angles equal but not all sides equal
Step 4: Verification
A square has:
- All four sides equal
- All four angles equal to 90°
- Diagonals equal and bisect each other at right angles
Answer: Square is the quadrilateral with all sides equal and all angles equal
Q3. In a parallelogram ABCD, angle A = 60°. Find angles B, C, and D.
We use the properties of angles in a parallelogram.
Step 1: Recall properties of a parallelogram
- Opposite angles are equal
- Adjacent angles are supplementary (sum = 180°)
Step 2: Find angle B
Angles A and B are adjacent in the parallelogram
A + B = 180°
60° + B = 180°
B = 120°
Step 3: Find angle C
Angles A and C are opposite in the parallelogram
C = A
C = 60°
Step 4: Find angle D
Angles B and D are opposite in the parallelogram
D = B
D = 120°
Alternatively, angles A and D are adjacent:
A + D = 180°
60° + D = 180°
D = 120° ✓
Step 5: Verify angle sum
60 + 120 + 60 + 120 = 360 ✓
Answer: B = 120°, C = 60°, D = 120°
Q4. What is the sum of an interior angle and an exterior angle of a quadrilateral?
We need to find the relationship between an interior angle and its adjacent exterior angle.
Step 1: Define exterior angle
An exterior angle is formed by extending one side of the quadrilateral.
The exterior angle and its adjacent interior angle are supplementary.
Step 2: At any vertex of a quadrilateral
Interior angle + Exterior angle = 180°
(They form a linear pair)
Step 3: Example
If an interior angle is 80°, the exterior angle at that vertex is:
180° - 80° = 100°
Sum = 80° + 100° = 180°
Step 4: This applies to all quadrilaterals
Any interior angle + its adjacent exterior angle = 180°
Answer: The sum of an interior angle and an exterior angle at the same vertex is 180°
Q5. Classify the quadrilateral with one pair of parallel sides.
We need to identify the quadrilateral based on its side properties.
Step 1: Recall quadrilateral types
- Parallelogram: both pairs of opposite sides parallel
- Trapezium (or Trapezoid): exactly one pair of parallel sides
- Rhombus: all sides equal, opposite sides parallel
- Rectangle: all angles 90°, opposite sides parallel
- Square: all sides equal, all angles 90°, opposite sides parallel
- Kite: two pairs of consecutive sides equal, but no parallel sides
Step 2: Identify by "one pair of parallel sides"
A quadrilateral with exactly one pair of parallel sides is a TRAPEZIUM.
Step 3: Properties of trapezium
- One pair of parallel sides (called bases)
- One pair of non-parallel sides (called legs)
- Can be isosceles (legs equal) or right-angled
Step 4: Example
If AB || CD, but AD and BC are not parallel, then ABCD is a trapezium.
Answer: A quadrilateral with one pair of parallel sides is called a Trapezium (or Trapezoid)
Q6. In rectangle PQRS, diagonals PR and QS intersect at O. If PO = 5 cm, find QS.
We use the property of diagonals in a rectangle.
Step 1: Recall properties of rectangle diagonals
- Diagonals of a rectangle are equal in length
- Diagonals bisect each other
- Each diagonal divides the rectangle into two congruent triangles
Step 2: Since diagonals bisect each other
O is the midpoint of both diagonals PR and QS
Therefore: PO = OR and QO = OS
Step 3: Given information
PO = 5 cm
Step 4: Find PR (the full diagonal)
Since O is the midpoint of PR:
PR = 2 × PO = 2 × 5 = 10 cm
Step 5: Use the property that diagonals are equal
QS = PR = 10 cm
Step 6: Verify
Since QS = 10 cm and O is its midpoint:
QO = OS = 10/2 = 5 cm ✓
Answer: QS = 10 cm
Q7. Why is a square also a rhombus? Explain with properties.
We need to show how a square satisfies all properties of a rhombus.
Step 1: List properties of a rhombus
- All four sides are equal
- Opposite angles are equal
- Diagonals bisect each other at right angles
- Diagonals bisect the angles
Step 2: List properties of a square
- All four sides are equal
- All four angles are 90°
- Opposite sides are parallel
- Diagonals are equal
- Diagonals bisect each other at right angles
- Diagonals bisect the angles
Step 3: Compare
Property 1: All sides equal
✓ Square has all sides equal (like rhombus)
Property 2: Opposite angles equal
✓ Square has all angles 90°, so opposite angles are 90° = 90° (satisfied)
Property 3: Diagonals bisect at right angles
✓ Square's diagonals bisect at 90° (verified geometrically)
Property 4: Diagonals bisect angles
✓ In a square, diagonals bisect the 90° angles into 45° each
Step 4: Conclusion
Since a square satisfies all the properties of a rhombus, a square IS a special type of rhombus.
Specifically: a square is a rhombus with all angles equal to 90°.
Answer: A square is a rhombus because it has all four sides equal and diagonals that bisect each other at right angles
Quadrilaterals — Practice MCQs with Answers
Attempt these 20 multiple-choice questions after working through the solutions above. Each carries the correct option and the reasoning behind it, so a wrong answer tells you which idea to revisit.
MCQ 1. What is a polygon with four sides called?
- A. Triangle
- B. Pentagon
- C. Hexagon
- D. Quadrilateral
Answer: D. Quadrilateral — A quadrilateral is defined as a polygon with exactly four sides and four vertices.
MCQ 2. The sum of the interior angles of any quadrilateral is:
- A. 180°
- B. 360°
- C. 540°
- D. 720°
Answer: B. 360° — A quadrilateral can be divided into two triangles, and since each triangle's angle sum is 180°, the total for a quadrilateral is 2 * 180° = 360°.
MCQ 3. In a parallelogram ABCD, if angle A is 70°, what is angle C?
- A. 70°
- B. 110°
- C. 20°
- D. 180°
Answer: A. 70° — In a parallelogram, opposite angles are equal. Therefore, angle C is equal to angle A.
MCQ 4. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a:
- A. Trapezium
- B. Kite
- C. Parallelogram
- D. Rhombus
Answer: C. Parallelogram — A fundamental property of parallelograms is that their diagonals bisect each other.
MCQ 5. The adjacent angles of a parallelogram are in the ratio 2:3. Find the measure of the smaller angle.
- A. 72°
- B. 108°
- C. 36°
- D. 144°
Answer: A. 72° — Adjacent angles of a parallelogram are supplementary. Let the angles be 2x and 3x. Then 2x + 3x = 180°, so 5x = 180°, which means x = 36°. The smaller angle is 2x = 2 * 36° = 72°.
MCQ 6. Which of the following quadrilaterals has all four angles equal to 90 degrees?
- A. Rhombus
- B. Parallelogram
- C. Trapezium
- D. Rectangle
Answer: D. Rectangle — A rectangle is defined as a parallelogram with all its angles equal to 90 degrees.
MCQ 7. A rhombus has all sides equal. What else is true about its diagonals?
- A. They are equal in length
- B. They bisect each other at right angles
- C. They are parallel
- D. They bisect opposite angles but not perpendicularly
Answer: B. They bisect each other at right angles — In a rhombus, the diagonals not only bisect each other but also intersect at right angles and bisect the vertex angles.
MCQ 8. Which quadrilateral has equal diagonals, and they also bisect each other at right angles?
- A. Rhombus
- B. Rectangle
- C. Square
- D. Kite
Answer: C. Square — A square combines properties of a rectangle (equal diagonals) and a rhombus (diagonals bisect at right angles).
MCQ 9. A quadrilateral has diagonals that are equal in length and bisect each other. What kind of quadrilateral is it?
- A. Rhombus
- B. Rectangle
- C. Square
- D. Parallelogram (general)
Answer: B. Rectangle — If diagonals bisect each other, it implies it's a parallelogram. If they are also equal in length, then the parallelogram must be a rectangle.
MCQ 10. A quadrilateral with exactly one pair of parallel sides is called a:
- A. Parallelogram
- B. Rhombus
- C. Trapezium
- D. Kite
Answer: C. Trapezium — A trapezium is defined as a quadrilateral with exactly one pair of parallel sides.
MCQ 11. In a kite, which of the following is true?
- A. All sides are equal
- B. Diagonals bisect each other
- C. Exactly one diagonal is the perpendicular bisector of the other
- D. All angles are equal
Answer: C. Exactly one diagonal is the perpendicular bisector of the other — In a kite, one of the diagonals (the one between non-equal sides) is the perpendicular bisector of the other diagonal.
MCQ 12. The exterior angles of a quadrilateral are x°, (x+10)°, (x+20)° and (x+30)°. Find the value of x.
- A. 75°
- B. 90°
- C. 100°
- D. 30°
Answer: A. 75° — The sum of the exterior angles of any polygon is 360°. So, x + (x+10) + (x+20) + (x+30) = 360°. This simplifies to 4x + 60 = 360°, so 4x = 300°, and x = 75°.
MCQ 13. A parallelogram has a perimeter of 50 cm. If one side is 10 cm, what is the length of the adjacent side?
- A. 10 cm
- B. 15 cm
- C. 20 cm
- D. 25 cm
Answer: B. 15 cm — In a parallelogram, opposite sides are equal. Let the adjacent sides be 'a' and 'b'. Perimeter = 2(a+b). Given a=10 cm and Perimeter=50 cm. So, 2(10+b) = 50 => 10+b = 25 => b = 15 cm.
MCQ 14. Three angles of a quadrilateral are 75°, 90°, and 100°. What is the measure of the fourth angle?
- A. 85°
- B. 95°
- C. 105°
- D. 115°
Answer: B. 95° — The sum of the interior angles of a quadrilateral is 360°. So, 75° + 90° + 100° + x = 360°. This means 265° + x = 360°, so x = 360° - 265° = 95°.
MCQ 15. Which of the following statements is FALSE for a square?
- A. All sides are equal
- B. All angles are right angles
- C. Diagonals bisect each other perpendicularly
- D. Diagonals are unequal in length
Answer: D. Diagonals are unequal in length — A square is a type of rectangle, and rectangles have equal diagonals. Therefore, diagonals of a square are equal, making the statement false.
MCQ 16. A quadrilateral whose opposite sides are parallel is called a:
- A. Trapezium
- B. Kite
- C. Parallelogram
- D. Rhombus
Answer: C. Parallelogram — A parallelogram is defined as a quadrilateral with two pairs of parallel sides.
MCQ 17. The diagonals of a rhombus are 16 cm and 12 cm. What is the length of each side of the rhombus?
- A. 8 cm
- B. 10 cm
- C. 14 cm
- D. 20 cm
Answer: B. 10 cm — Diagonals of a rhombus bisect each other at right angles. So, we have a right-angled triangle with legs 16/2 = 8 cm and 12/2 = 6 cm. Using Pythagoras theorem, side² = 8² + 6² = 100. So, side = √100 = 10 cm.
MCQ 18. In a parallelogram ABCD, angle DAB = 60°. If the diagonal AC bisects angle A, find angle BAC.
- A. 30°
- B. 60°
- C. 120°
- D. 90°
Answer: A. 30° — If AC bisects angle A (DAB), then angle BAC is half of angle DAB. So, angle BAC = 60° / 2 = 30°.
MCQ 19. A quadrilateral in which at least one interior angle is greater than 180° is called a:
- A. Convex quadrilateral
- B. Regular quadrilateral
- C. Concave quadrilateral
- D. Irregular quadrilateral
Answer: C. Concave quadrilateral — A concave quadrilateral has at least one interior angle greater than 180°, which makes one of its diagonals lie partly or entirely outside the figure.
MCQ 20. Which of the following statements is ALWAYS true?
- A. All rhombuses are squares
- B. All parallelograms are rectangles
- C. All squares are rhombuses
- D. All trapeziums are parallelograms
Answer: C. All squares are rhombuses — A square has all four sides equal, which is the defining property of a rhombus. Since a square satisfies this, all squares are rhombuses.
How to Use These Solutions
Attempt each question yourself first and only then compare with the worked answer. Marks in the CBSE board exam are awarded for the METHOD as much as the final result, so reproduce the steps rather than memorising the last line. Where a solution states a law, a formula or a definition, learn that wording — examiners look for it.
Related practice for this chapter: chapter MCQs, previous-year questions, revision notes and sample papers.