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Special Parallelograms Quiz - CBSE Class 8 Mathematics

Free practice quiz: Special Parallelograms (CBSE Class 8 Mathematics). 18 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This CBSE Class 8 Mathematics practice quiz has 18 multiple-choice questions on Special Parallelograms, and 18 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

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Parallelogram and trapezium basics

  • A parallelogram has both pairs of opposite sides parallel. Its opposite sides and opposite angles are equal.
  • Adjacent angles add to 180°, and the diagonals bisect each other. A trapezium has exactly one pair of parallel sides.
  • If diagonals of a quadrilateral bisect each other, or one pair of opposite sides is equal and parallel, it is a parallelogram.
Adjacent angles
\[\angle A + \angle B = 180^\circ\]
Opposite angles
\[\angle A = \angle C,\ \angle B = \angle D\]
Diagonals bisect (O is the meeting point)
\[OA = OC,\ OB = OD\]

Common mistake: Calling every trapezium a parallelogram. A trapezium has exactly one parallel pair, so it is never a parallelogram.

Rectangle: equal diagonals

  • A rectangle is a parallelogram with every angle 90°. Its diagonals are equal and bisect each other.
  • A parallelogram with equal diagonals is a rectangle. Prove it using congruent triangles and adjacent angles adding to 180°.
  • If adjacent sides a, b and a diagonal d satisfy a² + b² = d², the angle between a and b is 90°.
Equal diagonals
\[AC = BD\]
Diagonal from sides
\[d = \sqrt{a^2 + b^2}\]
Converse of Pythagoras
\[a^2 + b^2 = d^2 \Rightarrow \text{angle between } a, b = 90^\circ\]

Common mistake: Thinking a rectangle's diagonals are perpendicular. They are equal and bisect each other, but usually do not meet at 90°.

Rhombus: perpendicular bisecting diagonals

  • A rhombus is a parallelogram with all four sides equal.
  • Its diagonals bisect each other at right angles, but they need not be equal in length.
  • Half-diagonals and a side form a right triangle, so the side length comes from Pythagoras.
Side from diagonals
\[s = \sqrt{\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2}\]
Area of rhombus
\[A = \frac{1}{2} d_1 d_2\]
Perimeter
\[P = 4s\]

Common mistake: Using full diagonals in Pythagoras. Use half of each diagonal, because the diagonals bisect each other.

Square: rectangle and rhombus together

  • A square has all sides equal and all angles 90°. It is both a rectangle and a rhombus.
  • Its diagonals are equal, bisect each other, and meet at right angles. Only the square has all three properties.
Diagonal of square with side a
\[d = a\sqrt{2}\]
Area of square
\[A = a^2 = \frac{1}{2} d^2\]

Common mistake: Saying a square is not a rhombus or rectangle. A square is a special case of both.

Exam tips

Beta: made with AI and checked by a second AI. A teacher has not reviewed it yet.

Sample questions

  1. Which of the following is NOT a property of a rectangle?

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    Answer: D. All sides are equal

    Rectangles have opposite sides equal and parallel, all angles are 90 degrees, and diagonals bisect each other. However, only squares have all sides equal.

  2. If the diagonals of a parallelogram are equal and bisect each other at right angles, what type of parallelogram is it?

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    Answer: C. Square

    Equal diagonals that bisect each other at right angles are properties of a square. A rectangle has equal diagonals but they don't bisect at right angles. A rhombus has perpendicular bisecting diagonals but they are not equal.

  3. Prove that the diagonals of a rhombus bisect each other.

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    Answer: B. This is a property of all parallelograms, including rhombuses.

    All parallelograms have diagonals that bisect each other. Since a rhombus is a type of parallelogram, this property holds true.

  4. If the diagonals of a quadrilateral bisect each other at right angles, what can be said about the quadrilateral?

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    Answer: D. It is either a rhombus or a square

    While it is a parallelogram, the right angle condition narrows it down to either a rhombus or a square.

  5. A quadrilateral ABCD has AB || CD and AB = CD. What additional condition would ensure ABCD is a parallelogram?

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    Answer: D. All of the above

    Each condition individually ensures that ABCD is a parallelogram.

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More questions in this quiz

Answers and explanations are in the full quiz.

  1. A parallelogram with all sides equal is called a:
  2. What is the unique property of a square compared to other parallelograms?
  3. The diagonals of a rectangle are 10cm each. What can be said about the diagonals?
  4. A quadrilateral with exactly one pair of parallel sides is called a:
  5. If the diagonals of a parallelogram are perpendicular bisectors of each other, but not necessarily equal, what is the parallelogram?
  6. Given a parallelogram with angles 70°, 110°, 70°, and 110°, what type of parallelogram is it?
  7. Construct a rhombus with diagonals of length 6cm and 8cm. What is the length of each side?
  8. Prove that the diagonals of a rectangle are equal in length.
  9. A parallelogram has adjacent sides of lengths 5cm and 12cm, and one diagonal is 13cm. What type of parallelogram is it?
  10. What is the area of a rhombus with diagonals of length 10cm and 24cm?
  11. Can a trapezium be a parallelogram?
  12. Prove that if the diagonals of a parallelogram are equal, then the parallelogram is a rectangle.
  13. A quadrilateral has its diagonals bisecting each other. What can you conclude about the quadrilateral?

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