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Understanding Quadrilaterals - Types of Quadrilaterals: Trapezium, Kite, Parallelogram

Grade 8ICSE

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

🔑Concepts

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A Quadrilateral is a closed two-dimensional shape with four straight sides, four vertices, and four interior angles. The Angle Sum Property states that the sum of all interior angles in any quadrilateral is exactly 360∘360^{\circ}. Visually, if you draw a diagonal connecting two opposite vertices, the quadrilateral is split into two triangles, each contributing 180∘180^{\circ}.

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A Trapezium is a quadrilateral with at least one pair of parallel sides. In a diagram, these parallel sides are often marked with matching arrowheads. An 'Isosceles Trapezium' is a special case where the non-parallel sides are equal in length and the base angles are equal, appearing perfectly symmetrical across a central vertical axis.

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A Kite is a quadrilateral that has two distinct pairs of equal-length sides that are adjacent to each other. Visually, it looks like a traditional diamond shape where the top two sides are shorter and equal, and the bottom two sides are longer and equal. Its diagonals intersect at right angles (90∘90^{\circ}), and the longer diagonal bisects the shorter one.

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A Parallelogram is a quadrilateral where both pairs of opposite sides are parallel and equal in length. Visually, it looks like a rectangle that has been tilted. Its key properties include: opposite angles are equal, opposite sides are equal, and diagonals bisect each other (meaning they cut each other exactly in half).

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Adjacent angles in a parallelogram are supplementary, meaning they add up to 180∘180^{\circ}. If you look at the parallel lines of a parallelogram, the adjacent angles are 'interior angles on the same side of the transversal,' forming a 'C' or 'U' shape visually between the parallel lines.

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The diagonals of a Kite have a unique property: the main diagonal (the one between the vertices where equal sides meet) acts as a line of symmetry. It bisects the interior angles at those vertices and perpendicularly bisects the other diagonal, forming four right-angled triangles inside the kite.

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Properties of diagonals differ across types: In a Parallelogram, diagonals bisect each other but are not necessarily equal. In a Kite, diagonals are perpendicular. In an Isosceles Trapezium, the diagonals are actually equal in length.

📐Formulae

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

Area of a Parallelogram: Area=Base×HeightArea = Base \times Height

Perimeter of a Parallelogram: P=2(a+b)P = 2(a + b), where aa and bb are lengths of adjacent sides

Area of a Trapezium: Area=12×(sum of parallel sides)×height=12×(a+b)×hArea = \frac{1}{2} \times (sum\ of\ parallel\ sides) \times height = \frac{1}{2} \times (a + b) \times h

Area of a Kite: Area=12×d1×d2Area = \frac{1}{2} \times d_1 \times d_2, where d1d_1 and d2d_2 are the lengths of the diagonals

In a Parallelogram: Adjacent angles ∠A+∠B=180∘\angle A + \angle B = 180^{\circ}

💡Examples

Problem 1:

In a parallelogram ABCDABCD, if ∠A=(2x+10)∘\angle A = (2x + 10)^{\circ} and ∠B=(3x−40)∘\angle B = (3x - 40)^{\circ}, find the measure of all the angles of the parallelogram.

Solution:

  1. In a parallelogram, adjacent angles are supplementary. Therefore, ∠A+∠B=180∘\angle A + \angle B = 180^{\circ}.
  2. Substitute the expressions: (2x+10)+(3x−40)=180(2x + 10) + (3x - 40) = 180.
  3. Simplify: 5x−30=1805x - 30 = 180.
  4. Add 3030 to both sides: 5x=2105x = 210.
  5. Divide by 55: x=42x = 42.
  6. Calculate ∠A\angle A: 2(42)+10=84+10=94∘2(42) + 10 = 84 + 10 = 94^{\circ}.
  7. Calculate ∠B\angle B: 3(42)−40=126−40=86∘3(42) - 40 = 126 - 40 = 86^{\circ}.
  8. Since opposite angles are equal: ∠C=∠A=94∘\angle C = \angle A = 94^{\circ} and ∠D=∠B=86∘\angle D = \angle B = 86^{\circ}.

Explanation:

This problem uses the property that consecutive (adjacent) angles in a parallelogram add up to 180∘180^{\circ} and opposite angles are equal.

Problem 2:

The diagonals of a kite are 12 cm12\ cm and 18 cm18\ cm long. Find the area of the kite. Also, if one of the interior angles formed by the intersection of diagonals is given, what is its value?

Solution:

  1. The formula for the area of a kite is Area=12×d1×d2Area = \frac{1}{2} \times d_1 \times d_2.
  2. Substitute the given values: Area=12×12×18Area = \frac{1}{2} \times 12 \times 18.
  3. Calculate: Area=6×18=108 cm2Area = 6 \times 18 = 108\ cm^2.
  4. By property, the diagonals of a kite always intersect at right angles.
  5. Therefore, the angle formed by the intersection of diagonals is 90∘90^{\circ}.

Explanation:

The solution applies the specific area formula for kites using diagonals and utilizes the geometric property that kite diagonals are perpendicular.