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Shape and Space - Classification of Quadrilaterals and Triangles

Grade 5IB

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

🔑Concepts

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Triangles can be classified by their sides: Scalene (no equal sides), Isosceles (at least two equal sides), and Equilateral (all three sides equal). By angles, they are Acute (all angles <90∘< 90^{\circ}), Right (one angle =90∘= 90^{\circ}), or Obtuse (one angle >90∘> 90^{\circ}).

Comparison of an equilateral triangle and a right-angled isosceles triangle.
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Quadrilaterals are four-sided polygons. Special types include Parallelograms (opposite sides parallel and equal), Rectangles (parallelograms with four right angles), Rhombuses (parallelograms with four equal sides), and Squares (four equal sides and four right angles).

Diagram showing a parallelogram and a rhombus to illustrate side and angle properties.
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A Trapezium (or Trapezoid) is a quadrilateral with at least one pair of parallel sides. A Kite has two pairs of adjacent sides that are equal in length.

Diagram of a trapezium showing one pair of parallel horizontal bases.
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The sum of interior angles in any triangle is always 180∘180^{\circ}, while the sum of interior angles in any quadrilateral is 360∘360^{\circ}.

📐Formulae

Sum of angles in a triangle: a+b+c=180∘a + b + c = 180^{\circ}

Sum of angles in a quadrilateral: a+b+c+d=360∘a + b + c + d = 360^{\circ}

Perimeter of a Square: P=4×sP = 4 \times s

Perimeter of a Rectangle: P=2×(l+w)P = 2 \times (l + w)

Perimeter of any Triangle: P=side1+side2+side3P = side_1 + side_2 + side_3

💡Examples

Problem 1:

A triangle has two angles that measure 35∘35^{\circ} and 110∘110^{\circ}. Calculate the size of the third angle and classify the triangle by its angles and sides.

Solution:

Step 1: Use the triangle angle sum formula: 35∘+110∘+x=180∘35^{\circ} + 110^{\circ} + x = 180^{\circ}. Step 2: Add the known angles: 145∘+x=180∘145^{\circ} + x = 180^{\circ}. Step 3: Subtract from 180∘180^{\circ} to find xx: x=180∘−145∘=35∘x = 180^{\circ} - 145^{\circ} = 35^{\circ}. Step 4: Identify the properties. The angles are 35∘35^{\circ}, 110∘110^{\circ}, and 35∘35^{\circ}.

Explanation:

Because one angle is 110∘110^{\circ} (greater than 90∘90^{\circ}), it is an obtuse triangle. Because two angles are equal (35∘35^{\circ}), two sides must also be equal, making it an isosceles triangle. Result: Obtuse Isosceles Triangle.

Problem 2:

A quadrilateral has four equal sides. Its opposite angles are equal, but it does not have any right angles. Identify the shape and find the sum of its interior angles.

Solution:

Step 1: Analyze the properties. Four equal sides mean the shape is either a square or a rhombus. Step 2: Check the angles. Since there are no right angles (90∘90^{\circ}), it cannot be a square. Step 3: Conclude the shape is a Rhombus. Step 4: Use the quadrilateral angle sum rule.

Explanation:

A rhombus is a member of the parallelogram family. Like all quadrilaterals, the sum of its interior angles is always 360∘360^{\circ}.

Problem 3:

In the triangle shown, the angles at the base are 70∘70^{\circ} each. Find the value of the third angle xx and classify the triangle by its sides.

An isosceles triangle with base angles of 70 degrees and top angle marked x.

Solution:

  1. Use the angle sum property: 70∘+70∘+x=180∘70^{\circ} + 70^{\circ} + x = 180^{\circ}.
  2. 140∘+x=180∘140^{\circ} + x = 180^{\circ}.
  3. x=180∘−140∘=40∘x = 180^{\circ} - 140^{\circ} = 40^{\circ}.
  4. Since two angles are equal (70∘70^{\circ}), the triangle is an Isosceles triangle.

Explanation:

Because two angles are equal, the sides opposite those angles must also be equal, which is the defining property of an isosceles triangle.

Problem 4:

A rectangle has a length of 12 cm12 \text{ cm} and a width of 5 cm5 \text{ cm}. Calculate its perimeter and state the measure of each of its interior angles.

A rectangle with length 12 cm and width 5 cm, showing a right angle symbol.

Solution:

  1. Perimeter P=2×(l+w)=2×(12+5)=2×17=34 cmP = 2 \times (l + w) = 2 \times (12 + 5) = 2 \times 17 = 34 \text{ cm}.
  2. By definition, a rectangle has four right angles, so each interior angle is 90∘90^{\circ}.

Explanation:

A rectangle is a special quadrilateral where opposite sides are equal and all internal angles are right angles (90∘90^{\circ}).