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Geometry - Properties of 2D shapes (triangles, quadrilaterals, polygons)

Grade 5Cambridge (IGCSE)

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

🔑Concepts

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Triangles are classified by their sides and angles. An Equilateral triangle has three equal sides and three equal angles of 60∘60^{\circ}, while an Isosceles triangle has at least two equal sides and two equal base angles.

Equilateral triangle with all angles equal to 60 degrees.
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Quadrilaterals are 4-sided polygons where the sum of interior angles is always 360∘360^{\circ}. Common types include Parallelograms (opposite sides parallel and equal), Rhombuses (all sides equal), and Trapeziums (one pair of parallel sides).

Parallelogram showing equal opposite sides labeled a and b.
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A Regular Polygon has all sides of equal length and all interior angles of equal size. The exterior angle and the interior angle at any vertex always lie on a straight line and sum to 180∘180^{\circ}.

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The sum of exterior angles of any convex polygon is always 360∘360^{\circ}, regardless of the number of sides.

Pentagon showing an exterior angle produced by extending one side.

📐Formulae

Sum of interior angles in any triangle = 180∘180^{\circ}

Sum of interior angles in any quadrilateral = 360∘360^{\circ}

Sum of interior angles in an nn-sided polygon = (n−2)×180∘(n - 2) \times 180^{\circ}

Interior angle of a regular nn-sided polygon = (n−2)×180∘n\frac{(n - 2) \times 180^{\circ}}{n}

💡Examples

Problem 1:

An isosceles triangle has one angle of 70∘70^{\circ} at its apex (the angle between the two equal sides). Calculate the size of the other two angles.

Solution:

55∘55^{\circ} each

Explanation:

Since the triangle is isosceles, the two base angles are equal. The sum of angles is 180∘180^{\circ}. Subtract the apex angle: 180∘−70∘=110∘180^{\circ} - 70^{\circ} = 110^{\circ}. Divide by 2 for the two equal angles: 110∘÷2=55∘110^{\circ} \div 2 = 55^{\circ}.

Problem 2:

A quadrilateral has three angles measuring 90∘90^{\circ}, 85∘85^{\circ}, and 105∘105^{\circ}. Find the value of the fourth angle.

Solution:

80∘80^{\circ}

Explanation:

The sum of angles in a quadrilateral is 360∘360^{\circ}. Add the known angles: 90+85+105=280∘90 + 85 + 105 = 280^{\circ}. Subtract from the total: 360∘−280∘=80∘360^{\circ} - 280^{\circ} = 80^{\circ}.

Problem 3:

What is the sum of the interior angles of a regular Hexagon?

Solution:

720∘720^{\circ}

Explanation:

A hexagon has n=6n = 6 sides. Using the formula (n−2)×180∘(n - 2) \times 180^{\circ}, we get (6−2)×180∘=4×180∘=720∘(6 - 2) \times 180^{\circ} = 4 \times 180^{\circ} = 720^{\circ}.

Problem 4:

Find the size of the missing angle xx in the right-angled triangle shown below.

Right-angled triangle with one angle labeled 35 degrees and another x.

Solution:

  1. We know the sum of angles in a triangle is 180∘180^{\circ}.
  2. The square symbol indicates a right angle, which is 90∘90^{\circ}.
  3. The given angle is 35∘35^{\circ}.
  4. x=180∘−(90∘+35∘)x = 180^{\circ} - (90^{\circ} + 35^{\circ})
  5. x=180∘−125∘=55∘x = 180^{\circ} - 125^{\circ} = 55^{\circ} The missing angle xx is 55∘55^{\circ}.

Explanation:

In any right-angled triangle, the two non-right angles are complementary, meaning they must add up to 90∘90^{\circ}.

Problem 5:

Calculate the size of one interior angle of a regular octagon (an 8-sided polygon).

Regular octagon with 8 equal sides.

Solution:

  1. Use the formula for the sum of interior angles: (n−2)×180∘(n - 2) \times 180^{\circ}.
  2. For an octagon, n=8n = 8.
  3. Sum=(8−2)×180∘=6×180∘=1080∘\text{Sum} = (8 - 2) \times 180^{\circ} = 6 \times 180^{\circ} = 1080^{\circ}.
  4. Since it is a regular polygon, divide the sum by the number of angles:
  5. Interior angle=1080∘8=135∘\text{Interior angle} = \frac{1080^{\circ}}{8} = 135^{\circ}. Each interior angle is 135∘135^{\circ}.

Explanation:

A regular polygon has all equal angles. Alternatively, you can find the exterior angle first (360/8=45∘360/8 = 45^{\circ}) and subtract from 180∘180^{\circ}.