Finding angle created by polygon within polygon

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Simonsky

Junior Member
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Jul 4, 2017
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Here's the question (number 14 on the page):

number 14.jpg

And here's as far as I got making two triangles from BGC and using interior and exterior angle information-but I still got stuck:

fourteen.jpg
 
Dr.Peterson said:
Start by making a more accurate picture, so you will be more aware that GBC is isosceles.
Click to expand...


Dr. Peterson, I thought about that but didn't know how to prove it was. However, I now realise that the sides of the pentagon are the same length as the sides of the hexagon. I should have noticed that:eek: I think the original diagram (as well as mine) is deceptive because the line BG should be a rotation of the line BC. I assume the original is drawn that way to make it less obvious. Now it is straight forward to work out:

The angle GBC is 12 (interior angle of 6gon - interior angle 5gon) and the base angles of the isosceles are 84. So angle HGC will be 360 -( interior angle 5gon + 84) = 168.


Very deceptive original diagram!

many thanks as always!
 
Simonsky said:
Dr. Peterson, I thought about that but didn't know how to prove it was. However, I now realise that the sides of the pentagon are the same length as the sides of the hexagon. I should have noticed that:eek: I think the original diagram (as well as mine) is deceptive because the line BG should be a rotation of the line BC. I assume the original is drawn that way to make it less obvious. Now it is straight forward to work out:

The angle GBC is 12 (interior angle of 6gon - interior angle 5gon) and the base angles of the isosceles are 84. So angle HGC will be 360 -( interior angle 5gon + 84) = 168.

Very deceptive original diagram!
Click to expand...

It is standard for geometrical diagrams in problems to be "not drawn to scale", in part to avoid giving away an answer.

I agree with your result.
 
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