Question

1) Show that the exterior angle of a regular dodecagon (12 sides) is 30 degrees
2) The size of an interior angle in a regular polygon is 175 degrees. How many sides has the polygon?

I've tried and I am still confused as to how to answer these questions!

Answer 1

Let's do it again, OK?

Answer 2

Ok.

Answer 3

Sorry!

Answer 4

Imagine your dodecahedron is a real building (a castle) and you are alking along the walls. You are walking around ccw. So at each vertex you turn left by X degrees. OK?

Answer 5

Yes

Answer 6

The critical thing is, after you go all the way around, you are facing the same direction. Like, if you had a compass and on the first segment you faced north, after all 12 turns you would face north again. OK?

Answer 7

Oh ok

Answer 8

So you have 12 turns that total 360, that is 30 degrees per turn. These are the "exterior" angles.

Answer 9

Ok

Answer 10

But geometrically, how would I prove that? Since we are doing geometry in class

Answer 11

In particular, deductive geometry

Answer 12

Dunno what deductive geometry is. The standard proof is about the internal angles and uses induction.

Answer 13

Can I show an example?

Answer 14

We never learnt this 'induction' thing though.

Answer 15

Hmm.. so deductive means starting from axioms and such..

Answer 16

It might be better to leave it for now then. Shall we move on to question 2 please?

Answer 17

I'll give you my proof anyway. See if it makes any sense. Draw a square. Total of interior angles is 360. Now add another point outside and draw a triangle using that point and the vertices of the nearest side.

Answer 18

Need a picture?

Answer 19

No, that actually makes sense :) We are done with question 1 then. Shall we start with question 2?

Answer 20

If you're happy with the answer.. But what axiom (from Euclid, say) allows us to get that answer. I'd have to think about that...

Answer 21

OK, question 2. The key point is that at each vertex, the interior angle and the exterior angle add up to 180 degrees. OK?

Answer 22

So that will always happen?

Answer 23

Yes

Answer 24

Can you show a diagram please?

Answer 25

Hang on..