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Understanding Quadrilaterals - Types of Quadrilaterals

Grade 8CBSE

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 the interior angles of any quadrilateral is always 360∘360^\circ. Visually, it can be any shape formed by four straight line segments joined end-to-end, such as a kite, a box, or a slanted frame.

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A Parallelogram is a special quadrilateral where both pairs of opposite sides are parallel and equal in length. Visually, imagine a rectangle being pushed from one corner so it leans to the side; the top and bottom remain horizontal and parallel, while the left and right sides remain slanted and parallel. Key properties include: opposite angles are equal, and diagonals bisect each other (cut each other into two equal parts).

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A Rhombus is a parallelogram where all four sides are of equal length. Visually, it looks like a diamond shape. A unique property is that its diagonals are perpendicular bisectors of each other, meaning they meet at a right angle (90∘90^\circ) and cut each other in half.

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A Rectangle is a parallelogram where every interior angle is a right angle (90∘90^\circ). Visually, it is a perfectly 'square' cornered box where opposite sides are equal. A defining characteristic is that its diagonals are equal in length.

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A Square is a regular quadrilateral, meaning it is both a rectangle (all angles are 90∘90^\circ) and a rhombus (all sides are equal). Visually, it is perfectly symmetrical. It inherits all properties: diagonals are equal, they bisect at 90∘90^\circ, and all sides are congruent.

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A Trapezium is a quadrilateral with at least one pair of parallel sides. Visually, it often resembles a triangle with the top point cut off by a horizontal line. If the non-parallel sides are equal in length, it is called an 'Isosceles Trapezium', where the base angles are also equal.

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A Kite is a quadrilateral with two distinct pairs of equal adjacent sides. Visually, it looks like a traditional flying kite with two shorter sides at the top and two longer sides at the bottom. Its diagonals intersect at 90∘90^\circ, and the longer diagonal bisects the shorter one.

📐Formulae

Sum of interior angles of a quadrilateral = 360∘360^\circ

Sum of exterior angles of any convex quadrilateral = 360∘360^\circ

Number of diagonals in a quadrilateral = n(n−3)2=4(4−3)2=2\frac{n(n-3)}{2} = \frac{4(4-3)}{2} = 2

Area of a Parallelogram = base×height\text{base} \times \text{height}

Area of a Rhombus = 12×d1×d2\frac{1}{2} \times d_1 \times d_2 (where d1,d2d_1, d_2 are diagonals)

Area of a Trapezium = 12×(a+b)×h\frac{1}{2} \times (a + b) \times h (where a,ba, b are parallel sides and hh is the height)

Perimeter of a Quadrilateral = Sum of all four sides\text{Sum of all four sides}

💡Examples

Problem 1:

In a parallelogram ABCDABCD, the measure of ∠A\angle A is 70∘70^\circ. Find the measures of the remaining angles ∠B\angle B, ∠C\angle C, and ∠D\angle D.

Solution:

  1. In a parallelogram, adjacent angles are supplementary. Therefore, ∠A+∠B=180∘\angle A + \angle B = 180^\circ.
  2. Substitute the given value: 70∘+∠B=180∘  ⟹  ∠B=180∘−70∘=110∘70^\circ + \angle B = 180^\circ \implies \angle B = 180^\circ - 70^\circ = 110^\circ.
  3. Opposite angles of a parallelogram are equal. Therefore, ∠C=∠A=70∘\angle C = \angle A = 70^\circ and ∠D=∠B=110∘\angle D = \angle B = 110^\circ.

Explanation:

This solution uses the property that consecutive angles in a parallelogram sum to 180∘180^\circ and opposite angles are congruent.

Problem 2:

The diagonals of a rhombus are 1616 cm and 1212 cm. Find the length of each side of the rhombus.

Solution:

  1. Let the diagonals be d1=16d_1 = 16 cm and d2=12d_2 = 12 cm. They bisect each other at 90∘90^\circ.
  2. The half-lengths of the diagonals are 162=8\frac{16}{2} = 8 cm and 122=6\frac{12}{2} = 6 cm.
  3. These half-lengths form the legs of a right-angled triangle where the side of the rhombus (ss) is the hypotenuse.
  4. Using Pythagoras theorem: s2=82+62=64+36=100s^2 = 8^2 + 6^2 = 64 + 36 = 100.
  5. s=100=10s = \sqrt{100} = 10 cm.

Explanation:

Since diagonals of a rhombus are perpendicular bisectors, we can use the Pythagorean theorem on one of the four internal right-angled triangles to find the side length.