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No-Numbers Geometry

November 30th, 2009 Leave a comment Go to comments

Can you work out the size of angle xº?

int-ext-angles problem

That’s a regular hexagon and a regular pentagon.  Good luck!

(Hint: Consider the size of the interior angles in a polygon, the sum of angles round a point, and the sum of angles on a line)

(This is my first post in quite a while – apologies for the delay. I’ve been trying to become a qualified teacher!).

Solutions

When I designed this problem I didn’t realise initially that it could be solved without using the hexagon, so here’s my original reasoning (method 1) along with a more efficient method (method 2).

The key starting point is to think, “what would I need to know to be able to calculate angle x?” then to work backwards until you stumble across angles you do know.  We need to know the little angle next to x, indicated in yellow on the diagrams that follow, because this will lead us directly to angle x.  There are (at least) two ways of going about finding the yellow angle….


Method 1: Using the hexagon

Firstly, you need to know the interior angles in a pentagon add up to 540º, which means that one interior angle in a regular pentagon is 108º.

Next, similarly, you need to know the interior angles in a hexagon add up to 720º, which means that one interior angle in a regular hexagon is 120º.

Then you can use the facts that angles on a straight line add up to 180º and angles around a point add up to 360º, to find five of the six angles in the blue no-mans-land shape in the middle.

Now, once again, use the fact that the interior angles in a hexagon add up to 720º (the blue shape is an irregular hexagon) to find the sixth angle (indicated in yellow) which works out at 36º.

Finally, since angles on a straight line add up to 180º, we can deduce that angle x = 144º.

method1

Method 2:  Without using the hexagon.

You need four geometrical tools.  Firstly, once again, you need to know the interior angles in a pentagon add up to 540º, which means that one interior angle in a regular pentagon is 108º.

Next, you need to know that consecutive interior angles in parallel line situations are supplementary (i.e. add up to 180º), which means the pair of angles drawn in red below add up to 180º.  So the smaller of the two angles drawn in red is 72º.  We can use this fact to determine the size of the yellow angle drawn next to it: they add up to 108º so the yellow angle must be 36º.

Next, you need to know that alternate angles in parallel line situations are equal, which means the two yellow angles are equal.

Finally, using the fact that angles on a straight line add up to 180º, we can deduce that angle x = 144º.

method2

  1. rich
    December 13th, 2009 at 22:25 | #1

    didnt use the hexagon.

    just slammed some right angles into the pentagon. bang.

  2. Micky
    December 13th, 2009 at 21:30 | #2

    well done Rich. Now, did you use the hexagon or not? there are two ways to the solution. one ignores the hexagon completely

  3. rich
    December 13th, 2009 at 20:47 | #3

    bla bla bla bla your comment is too short

    144

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