Chapter 5 (Balbharati) · Class 9 Maharashtra SSC · MCQ Test
Quadrilaterals (चतुर्भुज) MCQ Test — Class 9 Maharashtra SSC
Practice 10 multiple-choice questions with instant answer reveal and explanations.
Quadrilaterals (चतुर्भुज) — MCQ Questions
1Which of the following statements about the angle sum property of a quadrilateral is TRUE?
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Answer: The sum of interior angles of a convex quadrilateral is 360°.
Hint: Recall how you can divide any quadrilateral into two triangles to find the sum of its angles.
Solution:
Any quadrilateral, whether convex or concave, can be divided into two triangles by drawing one of its diagonals.
Since the sum of angles in each triangle is 180°, the sum of angles in the quadrilateral will be 2 × 180° = 360°.
This property holds true for all quadrilaterals, convex or concave.
2Ravi was given a quadrilateral ABCD. He concluded that if AB = CD and BC = DA, then ABCD must be a parallelogram. Is Ravi's reasoning correct?
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Answer: Yes, because a quadrilateral with opposite sides equal is always a parallelogram.
Hint: Consider the defining properties of a parallelogram and the conditions that make a quadrilateral a parallelogram.
Solution:
One of the conditions for a quadrilateral to be a parallelogram is that its opposite sides are equal in length.
If AB = CD and BC = DA, this condition is met.
Therefore, Ravi's reasoning is correct; the quadrilateral ABCD must be a parallelogram.
3Consider a quadrilateral PQRS where the diagonals PR and QS intersect at point O. If PO = OR and QO = OS, which of the following statements is definitely TRUE?
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Answer: PQRS is a parallelogram.
Hint: Think about the unique property of diagonals in a parallelogram.
Solution:
The given information states that the diagonals PR and QS bisect each other at point O (PO=OR and QO=OS).
A fundamental property of a parallelogram is that its diagonals bisect each other.
Therefore, if the diagonals of a quadrilateral bisect each other, it must be a parallelogram. It doesn't necessarily have to be a rhombus or a rectangle unless additional conditions (like perpendicular diagonals or equal diagonals) are met.
4In a parallelogram ABCD, ∠A = 70°. What are the measures of ∠B, ∠C, and ∠D respectively?
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Answer: ∠B = 110°, ∠C = 70°, ∠D = 110°
Hint: Remember the properties of opposite angles and consecutive angles in a parallelogram.
Solution:
In a parallelogram, opposite angles are equal. So, ∠C = ∠A = 70°.
Also, consecutive angles are supplementary (sum to 180°). So, ∠A + ∠B = 180°.
Substituting ∠A = 70°, we get 70° + ∠B = 180°, which means ∠B = 110°.
Since opposite angles are equal, ∠D = ∠B = 110°.
5Which of the following conditions is NOT sufficient to prove that a quadrilateral is a parallelogram?
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Answer: One pair of opposite sides is equal.
Hint: Consider specific examples of quadrilaterals where only one pair of opposite sides is equal but it's not a parallelogram.
Solution:
The conditions for a quadrilateral to be a parallelogram include: (1) both pairs of opposite sides are equal, (2) both pairs of opposite angles are equal, (3) diagonals bisect each other, and (4) one pair of opposite sides is equal and parallel.
If only one pair of opposite sides is equal (e.g., AB = CD), it does not guarantee that the quadrilateral is a parallelogram. It could be an isosceles trapezium or another non-parallelogram figure.
Therefore, 'One pair of opposite sides is equal' is not a sufficient condition.
6A student claims that every rectangle is a square. Is this statement true or false, and why?
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Answer: False, because a rectangle only requires opposite sides to be equal, while a square requires all sides to be equal.
Hint: Think about the specific properties that define a square versus a rectangle. Is one a more general case of the other?
Solution:
A rectangle is a parallelogram with all angles equal to 90°. Its opposite sides are equal.
A square is a rectangle with all sides equal. It is also a rhombus.
Therefore, while every square is a rectangle (because it satisfies all properties of a rectangle), not every rectangle is a square (unless all its sides are equal). The statement is false because a rectangle can have unequal adjacent sides, unlike a square.
7Consider a triangle ABC. P and Q are the mid-points of sides AB and AC respectively. Which of the following statements is a direct consequence of the Mid-point Theorem?
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Answer: The line segment PQ is half the length of BC and parallel to BC.
Hint: Recall the exact statement of the Mid-point Theorem.
Solution:
The Mid-point Theorem states that the line segment connecting the mid-points of two sides of a triangle is parallel to the third side and is half its length.
Given P is the mid-point of AB and Q is the mid-point of AC, the line segment PQ connects these mid-points.
Therefore, according to the theorem, PQ || BC and PQ = 1/2 BC.
8A student is asked to prove that if the diagonals of a quadrilateral bisect each other, then it is a parallelogram. Which pair of triangles should the student primarily focus on proving congruent to establish that opposite sides are equal?
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Answer: △AOB and △COD (where O is the intersection of diagonals)
Hint: To prove opposite sides are equal using congruence, you usually need to use the given information about the diagonals bisecting each other.
Solution:
Let the quadrilateral be ABCD and its diagonals AC and BD intersect at O.
Given that the diagonals bisect each other, we have AO = OC and BO = OD.
To prove AB = CD, we can consider △AOB and △COD. We have AO = OC, BO = OD, and ∠AOB = ∠COD (vertically opposite angles).
By SAS congruence criterion, △AOB ≅ △COD, which implies AB = CD (CPCTC). Similarly, we can prove AD = BC using △AOD and △COB.
Proving both pairs of opposite sides equal is a direct way to show it's a parallelogram.
9In a quadrilateral ABCD, it is known that AB || CD. Which additional piece of information is required to conclude that ABCD is a parallelogram?
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Answer: AD || BC
Hint: Recall the definition of a parallelogram in terms of its parallel sides.
Solution:
A parallelogram is defined as a quadrilateral in which both pairs of opposite sides are parallel.
We are given that AB || CD (one pair of opposite sides is parallel).
To conclude that ABCD is a parallelogram, we need the other pair of opposite sides to also be parallel, i.e., AD || BC.
Options like AD = BC (isosceles trapezium possible) or ∠A = 90° (trapezium with right angle) or AC = BD (rectangle or isosceles trapezium) are not sufficient on their own.
10A farmer wants to fence a rectangular plot of land. He measures two adjacent sides as 20 meters and 30 meters. What is the total length of fencing required?
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Answer: 100 meters
Hint: A rectangular plot is a parallelogram where all angles are 90°. Think about the lengths of all its sides.
Solution:
A rectangular plot has two pairs of equal sides. If two adjacent sides are 20 meters and 30 meters, then the other two sides will also be 20 meters and 30 meters respectively.
The total length of fencing required is the perimeter of the rectangle.
Perimeter = 2 × (length + width) = 2 × (30 m + 20 m).
Perimeter = 2 × 50 m = 100 meters.
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Tips for Quadrilaterals (चतुर्भुज) MCQs
- 1Read each question carefully and identify what is being asked before looking at the options.
- 2Try to solve the problem mentally or on paper first, then match your answer with the options.
- 3Use elimination — rule out clearly wrong options to improve your chances even when unsure.
- 4Check units, signs, and edge cases — these are common traps in Quadrilaterals (चतुर्भुज) MCQs.
- 5Review your mistakes after completing the test to build lasting understanding.
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