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Understanding Elementary Shapes - Types of Quadrilaterals

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A Quadrilateral is a closed polygon with four sides, four vertices, and four angles. The sum of all interior angles is always 360∘360^{\circ}.

General quadrilateral ABCD with four vertices and four sides.
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A Parallelogram is a quadrilateral where both pairs of opposite sides are parallel and equal. Opposite angles are also equal.

Parallelogram showing parallel opposite sides.
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A Rectangle is a special parallelogram where every interior angle is a right angle (90∘90^{\circ}). Opposite sides are equal.

Rectangle with all 90 degree angles and labeled length and breadth.
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A Square is a quadrilateral with four equal sides and four right angles. It is both a rectangle and a rhombus.

Square with all sides equal to s and right angles.
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A Trapezium is a quadrilateral with at least one pair of parallel sides.

Trapezium showing one pair of parallel horizontal sides.

📐Formulae

Sum of interior angles: ∠A+∠B+∠C+∠D=360∘\angle A + \angle B + \angle C + \angle D = 360^{\circ}

Perimeter of a general Quadrilateral: P=side1+side2+side3+side4P = side_1 + side_2 + side_3 + side_4

Perimeter of a Rectangle: P=2×(l+b)P = 2 \times (l + b) where ll is length and bb is breadth

Perimeter of a Square: P=4×sP = 4 \times s where ss is the length of a side

Perimeter of a Rhombus: P=4×sP = 4 \times s where ss is the length of a side

💡Examples

Problem 1:

In a quadrilateral ABCDABCD, three angles are measured as 75∘75^{\circ}, 105∘105^{\circ}, and 100∘100^{\circ}. Find the measure of the fourth angle.

Solution:

  1. Let the fourth angle be xx.
  2. According to the angle sum property of a quadrilateral: ∠A+∠B+∠C+∠D=360∘\angle A + \angle B + \angle C + \angle D = 360^{\circ}.
  3. Substitute the known values: 75∘+105∘+100∘+x=360∘75^{\circ} + 105^{\circ} + 100^{\circ} + x = 360^{\circ}.
  4. Add the known angles: 280∘+x=360∘280^{\circ} + x = 360^{\circ}.
  5. Subtract 280∘280^{\circ} from both sides: x=360∘−280∘=80∘x = 360^{\circ} - 280^{\circ} = 80^{\circ}.
  6. Therefore, the fourth angle is 80∘80^{\circ}.

Explanation:

We use the fundamental property that all interior angles of any quadrilateral must add up to exactly 360∘360^{\circ} to find the unknown value.

Problem 2:

Identify the quadrilateral which has all sides equal in length but does not necessarily have right angles at its vertices. If one side is 55 cm, what is its perimeter?

Solution:

  1. A quadrilateral with all sides equal is either a square or a rhombus.
  2. Since the problem states it does not necessarily have right angles, the most accurate name is a Rhombus.
  3. All sides of a rhombus are equal, so s=5s = 5 cm.
  4. Perimeter P=4×s=4×5 cm=20 cmP = 4 \times s = 4 \times 5\text{ cm} = 20\text{ cm}.

Explanation:

This problem tests the property-based identification of shapes. Since all sides are equal, we use the perimeter formula 4×side4 \times side.

Problem 3:

In the given rectangle PQRSPQRS, if PQ=8PQ = 8 cm and QR=6QR = 6 cm, find the lengths of RSRS and SPSP.

Rectangle PQRS with dimensions 8cm and 6cm.

Solution:

In a rectangle, opposite sides are equal. Therefore, RS=PQ=8RS = PQ = 8 cm and SP=QR=6SP = QR = 6 cm.

Explanation:

Since PQRSPQRS is a rectangle, the side opposite to PQPQ is RSRS, making them equal. Similarly, the side opposite to QRQR is SPSP, making them equal.

Problem 4:

Identify the type of quadrilateral ABCDABCD where AB∥CDAB \parallel CD and AD∥BCAD \parallel BC, and all sides are equal to 77 cm, but angles are not 90∘90^{\circ}.

A rhombus ABCD with all sides 7 cm.

Solution:

  1. Opposite sides are parallel (AB∥CDAB \parallel CD, AD∥BCAD \parallel BC), so it is a parallelogram.
  2. All sides are equal (77 cm).
  3. Since all sides are equal and angles are not 90∘90^{\circ}, the quadrilateral is a Rhombus.

Explanation:

A quadrilateral with two pairs of parallel sides and all four sides of equal length is a Rhombus. If the angles were 90∘90^{\circ}, it would be a square.