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Shape and Space - Properties and Classification of Quadrilaterals

Grade 6IB

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

🔑Concepts

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A quadrilateral is a polygon with 44 sides, 44 vertices, and 44 interior angles. Visualize it as any closed shape made of four straight line segments joined end-to-end.

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Parallelogram: A quadrilateral where opposite sides are parallel and equal in length. Opposite angles are equal, and diagonals bisect each other. Visualize this as a rectangle that has been tilted or 'pushed' over so its corners are no longer square.

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Rectangle: A special type of parallelogram that contains four right angles (90∘90^{\circ}). The diagonals of a rectangle are equal in length and bisect each other. It looks like a box where the opposite sides are perfectly parallel and all corners are 'L' shaped.

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Rhombus: A parallelogram where all four sides are of equal length. Its diagonals are perpendicular to each other, meaning they cross at a 90∘90^{\circ} angle. It is often visualized as a diamond shape.

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Square: A regular quadrilateral that possesses the properties of both a rectangle and a rhombus. It has four equal sides and four 90∘90^{\circ} angles. Visualize this as the most 'balanced' four-sided shape where every side and every corner is identical.

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Trapezium: A quadrilateral with at least one pair of parallel sides. In an isosceles trapezium, the non-parallel sides are equal in length. Visualize this as a shape with a flat top and bottom of different lengths, with sides that may slant inward.

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Kite: A quadrilateral with two pairs of equal-length sides that are adjacent to each other (not opposite). One pair of opposite angles is equal, and the diagonals intersect at a right angle (90∘90^{\circ}). It looks like the traditional diamond-shaped kites used for flying.

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Interior Angle Sum: The sum of the interior angles of any quadrilateral is always 360∘360^{\circ}. This is because any quadrilateral can be divided into two triangles by drawing a single diagonal, and each triangle contributes 180∘180^{\circ} (2×180∘=360∘2 \times 180^{\circ} = 360^{\circ}).

📐Formulae

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

Perimeter of any quadrilateral: P=side1+side2+side3+side4P = side_1 + side_2 + side_3 + side_4

Area of a Rectangle: Area=length×widthArea = length \times width

Area of a Square: Area=side2Area = side^2

Area of a Parallelogram: Area=base×heightperpendicularArea = base \times height_{perpendicular}

Perimeter of a Rhombus or Square: P=4×sideP = 4 \times side

💡Examples

Problem 1:

In a quadrilateral ABCDABCD, three of the interior angles are 110∘110^{\circ}, 85∘85^{\circ}, and 75∘75^{\circ}. Find the measure of the fourth angle, xx.

Solution:

  1. Recall the Angle Sum Property: The sum of all angles in a quadrilateral is 360∘360^{\circ}.
  2. Set up the equation: 110∘+85∘+75∘+x=360∘110^{\circ} + 85^{\circ} + 75^{\circ} + x = 360^{\circ}.
  3. Add the known angles: 270∘+x=360∘270^{\circ} + x = 360^{\circ}.
  4. Subtract the sum from 360∘360^{\circ}: x=360∘−270∘x = 360^{\circ} - 270^{\circ}.
  5. Result: x=90∘x = 90^{\circ}.

Explanation:

Since any quadrilateral can be split into two triangles, its total internal degrees must equal 360360. We subtract the sum of the known angles from this total to find the missing value.

Problem 2:

Calculate the perimeter of a rhombus where one side measures 7.57.5 cm.

Solution:

  1. Identify the property: In a rhombus, all four sides are equal in length.
  2. Use the formula: Perimeter=4×sidePerimeter = 4 \times side.
  3. Substitute the value: P=4×7.5 cmP = 4 \times 7.5 \text{ cm}.
  4. Calculate: P=30 cmP = 30 \text{ cm}.

Explanation:

Because a rhombus is equilateral (all sides equal), you simply multiply the length of one side by four to find the total distance around the shape.