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Angles in polygonsQuadrilaterals

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Here we will learn about *polygons*, including regular polygons, angles in polygons, and complex polygons.

There are also *polygons *worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

An irregular polygon is a 2D shape that has straight sides that are not equal to each other and angles that are not equal to each other. You need to be able to classify geometric shapes based on their properties and sizes and find unknown angles in any polygon. To do this, we need to look closely at the properties of these shapes.

E.g.

Trapezium | Irregular Pentagon | Irregular dodecagon |

To do this, we need to look closely at the properties of these shapes.

**Irregular polygons have sides that are not equal**to each other or**angles that are not equal**to each other or both.- The table below gives the name of several irregular polygon.

Polygon | Number of sides | Image (irregular polygon) |

Scalene triangle | 3 | |

Irregular quadrilateral | 4 | |

Irregular pentagon | 5 | |

Irregular hexagon | 6 | |

Irregular heptagon | 7 | |

Irregular octagon | 8 | |

Irregular nonagon | 9 | |

Irregular decagon | 10 | |

Irregular hendecagon | 11 | |

Irregular dodecagon | 12 |

- The sum of interior angles can be calculated using the formula:

**Sum of interior angles = (n-2) × 180^{\circ} **

where n represents the number of sides.

As the size of each angle is not equal, we can determine the size of each angle by adding together the interior angles that we know and putting it equal to the sum of the interior angles, then solving the equation**.**

**Exterior angles**are supplementary to the interior angle.

**Sum of exterior angles of a polygon = 360° **

We can use this property to find either the interior angle, or exterior angle at a vertex.

As the size of each angle is not equal, we can determine the size of an exterior angle by adding together the exterior angles that we know and subtracting it from 360° .

- Irregular polygons can be either
**convex polygons**or**concave polygons**.

A **convex polygon** is a polygon with all the interior angles less than 180° . A **concave polygon** has at least one angle that is greater than 180° (or a reflex angle).

E.g.

Convex hexagon | Concave hexagon |

All interior angles are either acute or obtuse. | At least one angle is greater than 180° |

In order to classify an irregular polygon:

**State/calculate the number of sides of the polygon.****Determine the size of the angles and side lengths within the polygon.****Recognise the other properties of the polygon**.

Get your free Irregular polygons worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONGet your free Irregular polygons worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOONGiven that ABC is an isosceles triangle and BCD is a straight line, classify the polygon ABD .

**State/calculate the number of sides of the polygon**.

ABC has 3 sides and so it is a triangle.

2**Determine the size of the angles and/or side lengths within the polygon**.

As BCA = 116° , angle ACD = 180 - 116 = 64° .

As ABC is an isosceles triangle, ABC = BAC = (180 - 116)\div 2 = 32° .

As angles in a triangle total 180° , angle CAD = 180 - (74 + 64) = 42° .

Angle BAD = 42 + 32 = 74° .

3**Recognise the other properties of the polygon**.

As the three angles in the triangle ABC are 74°, 74° and 32° . The triangle ABD has two equal angles and so it is isosceles.

The polygon is an **isosceles triangle**, which is an irregular polygon.

Classify the following polygon:

**State/calculate the number of sides of the polygon**.

The polygon has 4 sides so this is a type of quadrilateral.

**Determine the size of the angles and/or side lengths within the polygon**.

Angle ACD is alternate to angle BAC and so angle ACD = 48° .

The angle COD = 180 - (41 + 48) = 91° .

**Recognise the other properties of the polygon**.

As the angle at the centre is not equal to 90 degrees and the four sides are two pairs of parallel lines, the polygon ABCD is a parallelogram, which is an irregular quadrilateral.

The polygon below is a quadrilateral. By using angle properties, determine the type of quadrilateral.

**State/calculate the number of sides of the polygon**.

It is stated in the question that we have a quadrilateral.

**Determine the size of the angles and/or side lengths within the polygon**.

We do not have any angles to use to classify this polygon. This in turn discounts many quadrilaterals that have specific angle properties. We can however use the information provided to find the length of AD .

As the vertical height is perpendicular to the base, we can use Pythagoras’ Theorem to calculate the length of AD .

3^{2}+4^{2}=AD^{2}

9+16=AD^{2}

AD^{2}=25

AD=5cm

**Recognise the other properties of the polygon**.

As AD does not equal BC but AB and CD are parallel, the quadrilateral is a **trapezium **(which is an irregular polygon).

A pentagon is constructed by joining an isosceles triangle with a rectangle. The line AM is a line of symmetry. Show that the polygon is irregular.

**State/calculate the number of sides of the polygon**.

The pentagon has 5 sides.

**Determine the size of the angles and/or side lengths within the polygon**.

As angles in a triangle total 180° , angle AEB = 180 - (90+53) = 37° .

As BCDE is a rectangle, angle BED = 90° and so angle AED = 90 + 37 = 127° .

As the polygon is symmetrical, angle BAE = 53 × 2 = 106° .

**Recognise the other properties of the polygon**.

The interior angles of the polygon are equal to 106°, 120°, 90°, 106° and 120° so, as the angles are not the same, the **pentagon **is **irregular**.

Below is a description of a polygon. From the description, determine the classification of the polygon.

The polygon has 10 edges.

The polygon has a rotational symmetry of order 5 .

The interior angles are equal to 42° or 246° .

The polygon can be constructed from 5 congruent kites.

**State/calculate the number of sides of the polygon**.

As the polygon has 10 edges, this polygon is a type of decagon.

**Determine the size of the angles and/or side lengths within the polygon**.

As the polygon can be constructed from 5 congruent kites, each side length of the decagon is the same. As the polygon has a rotational symmetry of order 5 and the interior angles are 42° or 246° , the polygon looks like a five pointed star.

**Recognise the other properties of the polygon**.

As the angles in the polygon are not equal, the shape is an irregular decagon.

The interior angles of a polygon are: 4x+16, 6x+8 and 2x+12 . Classify the polygon.

**State/calculate the number of sides of the polygon**.

As there are 3 interior angles, the polygon is a type of triangle.

**Determine the size of the angles and/or side lengths within the polygon**.

As the sum of angles in a triangle total 180°

4x+16+6x+8+2x+12=180

12x+36=180

12x=144

x=12

As x=12

4x+15=4\times 12+16=64°

6x+8=6\times 12+8=80°

2x+12=2\times 12+12=36°

**Recognise the other properties of the polygon**.

As all the angels in the polygon are different, the triangle is a **scalene triangle** (an irregular polygon).

**Angles in polygons**

Make sure you know your angle properties. Getting these confused causes quite a few misconceptions.

- Angles in a triangle total 180°
- Angles in a quadrilateral total 360°
- Angles on a straight line total 180°

**Incorrect quadrilateral classification**

There are many quadrilaterals and it is common to confuse the properties, especially for a rhombus, parallelogram, or trapezium.

As well as this, stating that the polygon is a quadrilateral is not enough information for a classification.

**Incorrect assumptions for triangles**

Assuming a triangle is isosceles or equilateral can have an impact on the size of angles within the rest of the polygon, so make sure you can explain why you have chosen a specific type of triangle.

As well as this, stating that the polygon is a triangle is not enough information for a classification. You must state whether it is isosceles/ equilateral etc.

1. BD is parallel to AE in the triangle ACE . Classify the polygon BCD

Equilateral triangle

Right Angle triangle

Isosceles triangle

Scalene triangle

The angle BDC = 180 – 127 = 53^{\circ} (angles on a straight line total 180^{\circ} ).

Angle CBD = 180 – (65 + 53) = 62^{\circ} (angles in a triangle total 180^{\circ} ).

All angles of BCD are different.

2. A quadrilateral is inscribed in a circle with centre O . The diagonals meet perpendicularly at the point E where AC = 2AE . Use circle theorems to determine the classification of the quadrilateral ABCD .

Diamond

Arrowhead

Isosceles triangle

Kite

As AC = 2AE , and BD is perpendicular to AC , triangles ABC and ADC are isosceles triangles. This means that the two angles at A and C are equal and the side lengths AB and BC , and AD and CD are equal. This is a kite.

3. The polygon below has an order of rotational symmetry of 4 . Classify the polygon.

Octagon

Quadrilateral

Regular octagon

Regular pentagon

As the order of rotational symmetry is 4 , the angles at A, C, E and G are all equal to 40^{\circ} , and the angles at B, D, F, and H are equal to 360 – 130 = 230^{\circ} . The polygon has 8 sides but the interior angles are not equal.

4. 6 congruent triangles are joined together to form a 6 sided polygon. Each line segment within the polygon is the same length. The new polygon has two acute angles, 3 obtuse angles and one reflex angle. The polygon has one line of symmetry and no rotational symmetry. Classify the polygon.

Arrowhead

Regular hexagon

Hexagon

Parallelogram

As the polygon has 6 sides and different interior angles, the polygon is a hexagon.

5. The polygon below is incomplete. Given that the polygon is made up of 10 congruent kites, classify the polygon.

Regular decagon

Regular dodecagon

Regular icosagon

Regular octagon

As the polygon is made of 10 congruent kites, each side length is the same and the interior angles are also equal due to the kite also being symmetrical.

6. The interior angles of a polygon are: 12x-30, 8x+40, 7x+30, 8x, and 5x+20 . The sum of the angles is equal to 540^{\circ} . Classify the polygon.

Pentagon

Hexagon

Regular pentagon

Regular hexagon

12x-30+8x+40+7x+30+8x+5x+20=540\\
40x+60=540\\
40x=480\\
x=12

As x=12\\ 12x-30=12\times12-30=114^{\circ}\\ 8x+40=8\times12+40=136^{\circ}\\ 7x+30=7\times12+30=114^{\circ}\\ 8x=8\times12=96^{\circ}\\ 5x+20=5\times12+20=80^{\circ}

The internal angles are not equal and so the polygon is a pentagon.

1. (a) An isosceles trapezium is made when you join two triangles together along an edge with the same length. ABC is an isosceles triangle. Work out the size of the angle ACD .

(b) What type of triangle is ACD ?

**(5 marks)**

Show answer

(a)

BCA = (180 – 116)\div2 = 32^{\circ}

**(1)**

**(1)**

**(1)**

**(1)**

(b)

Scalene

**(1)**

2. (a) Show that the triangle is a right angle triangle.

(b)The area of both polygons are equal. Calculate the length x .

**(5 marks)**

Show answer

(a)

5^{2}+12^{2}=25+144=169

**(1)**

**(1)**

The sum of the square of the two shorter sides is equal to the square of the hypotenuse so the triangle contains a right angle

**(1)**

(b)

Triangle area = 5\times12\div2=30cm^{2}

**(1)**

**(1)**

3. (a) The polygon ABC is half of a kite where AC is the longest length of the kite. Write an expression for the area of the kite.

(b) Given that the area of the smaller triangle ABO = 40cm^{2} , calculate the area of the kite.

(c) Calculate the size of angle OAB .

**(5 marks)**

Show answer

(a)

8x^{2}

**(1)**

(b)

x^{2}=40

**(1)**

**(1)**

(c)

OAB=tan^{-1}(\frac{2x}{x})

**(1)**

**(1)**

You have now learned how to:

- Plot specified points and draw sides to complete a given polygon
- Distinguish between regular and irregular polygons based on reasoning about equal sides and angles
- Compare and classify geometric shapes based on their properties and sizes and find unknown angles in any triangles, quadrilaterals, and regular polygons
- Describe, sketch and draw using conventional terms and notations: points, lines, parallel lines, perpendicular lines, right angles, regular polygons, and other polygons that are reflectively and rotationally symmetric
- Derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon, and to derive properties of regular polygons

- Regular polygons
- Types of quadrilaterals
- Types of triangle

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