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Understanding Angles and Polygons: Exterior and Interior Angles Explained

In geometry, angles play a crucial role in understanding polygons. This guide focuses on exterior and interior angles, using a pentagon as an example. We learn that the sum of the exterior angles of any polygon is always 360 degrees, and how to calculate the number of sides based on the exterior angle. Additionally, the relationship between interior and exterior angles is explored, helping to derive the sum of interior angles using the number of sides. This resource is valuable for anyone looking to deepen their understanding of polygon properties.

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Understanding Angles and Polygons: Exterior and Interior Angles Explained

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  1. Angles and Polygons By Mr Robertson

  2. In and Out Exterior angle • In this pentagon there are: • 5 Interior angles • 5 exterior angles • 5 sides Interior angle

  3. Exterior Angles The sum of the exterior angles = 360 Another way to express this is to say that the number of exterior angles × the size of each exterior angle = 360 And since the number of sides is the same as the number of angles we can write: number of sides x exterior angle = 360 se = 360

  4. Exterior Angles And if we know the size of the exterior angle we can find the number of sides: number of sides= 360 ÷ exterior angle s = 360 e

  5. Exterior Angles We can rearrange this so if we know the number of sides we can find the exterior angle: exterior angle= 360 ÷ number of sides e = 360 s

  6. In and Out Look at the diagram and you will see that the exterior and the interior angle combine to make a straight line. This means that: interior angle = 180 – exterior angle exterior angle = 180 – interior angle i = 180 – ee = 180 – i

  7. Summing Up Sometimes you have to calculate the sum of the interior angles. This is going to be equal to the number of sides × the size of the interior angle. This can be written like this: Sum of interior angles= si But the interior angle is just 180 – the exterior angle, so we can write it like this instead: si = s(180 – e) But the exterior angle is just 360 ÷ the number of sides, so we can write it like this instead: si = s(180 – 360 ÷ s) And if we multiply out this bracketed expression and simplify we get: si = 180s – 360 So it turns out we can work out the sum of the interior angles, just by knowing the sides!

  8. In and Out • So it turns out that we can work out everything else just by knowing one of: • Number of sides • Size of exterior angle • Size of interior angle • Sum of interior angles • Let’s see an example

  9. Example • The sum of the interior angles of a regular polygon is 9000°. How many sides has it got? • Start with:si = 9000 • Then replace i with 180 – e to get:s(180 – e) = 9000 • Then replace e with 360/s to get:s(180 – 360/s) = 9000 • Multiply out the bracket and solve:180s – 360 = 9000 180s = 9360 s = 9360 ÷ 180 s = 52 • So the polygon has 52 sides interior angle = 180 – exterior angle exterior angle = 360 ÷ number of sides

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