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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 four sides, four vertices, and four interior angles. The sum of the interior angles of any quadrilateral is always 360∘360^{\circ}.

A general quadrilateral ABCD illustrating four sides and four vertices.
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A Parallelogram is a quadrilateral where opposite sides are parallel and equal in length. Opposite angles are also equal.

A parallelogram showing base and perpendicular height.
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A Trapezium (or Trapezoid) is a quadrilateral with at least one pair of parallel sides. If the non-parallel sides are equal, it is called an isosceles trapezium.

A trapezium with a pair of parallel top and bottom sides.
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A Kite is a quadrilateral with two pairs of adjacent sides that are equal in length. Its diagonals intersect at right angles (90∘90^{\circ}).

A kite showing two pairs of equal adjacent sides and intersecting diagonals.

📐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.

Problem 3:

Given a rectangle with a length of 1212 cm and a diagonal of 1313 cm, find the width of the rectangle.

A rectangle with length 12cm, diagonal 13cm, and unknown width w.

Solution:

  1. Let the width be ww. In a rectangle, the diagonal forms a right-angled triangle with the length and width.
  2. Using the Pythagorean theorem: 122+w2=13212^2 + w^2 = 13^2
  3. 144+w2=169144 + w^2 = 169
  4. w2=169−144=25w^2 = 169 - 144 = 25
  5. w=25=5w = \sqrt{25} = 5 cm.

Explanation:

Because a rectangle has four right angles, we can use the Pythagorean theorem on the triangle formed by two sides and the diagonal.

Problem 4:

In the parallelogram PQRSPQRS, ∠P=70∘\angle P = 70^{\circ}. Find the measures of the remaining three angles.

Parallelogram PQRS with angle P labeled as 70 degrees.

Solution:

  1. In a parallelogram, opposite angles are equal: ∠R=∠P=70∘\angle R = \angle P = 70^{\circ}.
  2. Adjacent angles are supplementary (sum to 180∘180^{\circ}): ∠Q=180∘−70∘=110∘\angle Q = 180^{\circ} - 70^{\circ} = 110^{\circ}.
  3. Since ∠S\angle S is opposite to ∠Q\angle Q, ∠S=110∘\angle S = 110^{\circ}.
  4. The angles are 70∘70^{\circ}, 110∘110^{\circ}, 70∘70^{\circ}, and 110∘110^{\circ}.

Explanation:

Properties of parallelograms state that opposite angles are equal and consecutive (adjacent) angles add up to 180 degrees.