Determine the size of the angles X and W (given in the image)

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First realise that each arm of the five-pointed star in the center forms a straight line with the next but one arm (so that we could inscribe a regular pentagon in the center of the star). Looking at an individual arm as a triangle with base one of the sides of the inscribed pentagon we see that that triangle is isoceles - the two angles `a` at the base are equal and the third is ` ``X =180 - 2a` degrees (since angles in a triangle add up to 180 degrees).

Now, since angles on a straight line add up to 180 degrees also, we see that the angles `a` at the base are equal to 180 - 110 = 70 degrees each so that the third is equal to 180 - 2 x 70 = 180 - 140= 40 degrees. Therefore `X=40` degrees.

The green shape is a regular ten-sided polygon (a *decagon*). Given the result about internal angles of a polygon (they sum to *` `*`(n-2) times 180` degrees), the sum of the internal angles of the decagon here is 8 x 180 = 1440. Since there are 10 internal angles, each is equal to 144 degrees.

Looking at the diagram, the angle `W` plus this internal angle of the decagon of size 144 degrees is equal to 180 (since the side of the outermost white pentagon - five-sided polygon - is a straight line and angles on a straight line add up to 180 degrees). Therefore `W = 180 - 144 = 36` degrees.

**Answer X = 40 degrees and W = 36 degrees (using results about angles on a straight line and the sum of internal angles of an n-sided polygon).**

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