MHB Find the angle X inside a pentagon with a square

[mathlearn]

If the the question is too small , Please be kind enough to read it from here Question

The interior angles of a pentagon add up to 540 degrees. So thinking that this is a regular pentagon with all 5 sides equal an interior would be 108 degrees.

And speaking of the square All four sides are equal and all 4 angles are 0 degrees.

Could have easily found X if the the side with the angle X (shortest side) was a straight line.It could be said using angles on a straight line add up to 180.

But the line seems to be slanted.

I included the information in a diagram,

Any Ideas on How to begin ?

Many Thanks :)

 

[I like Serena]

Hey mathlearn! ;)

Can we already find one of the other angles in the small triangle that contains x?
What kind of triangle is it anyway?

 

[mathlearn]

It looks like that side of the pentagon and the side of the square are equal at a glance , But don't know whether it's correct. If I'm correct then it would be an isosceles triangle

Then this will be the result

Many Thanks :)

 

[I like Serena]

It looks like that side of the pentagon and the side of the square are equal at a glance , But don't know whether it's correct. If I'm correct then it would be an isosceles triangle

Then this will be the result

Many Thanks :)

The sides of the square have the same length.
The sides of the pentagon have the same length.
One side of the square has the same length as the corresponding side of the pentagon.
So yes, those sides are indeed equal. :)

Can we say anything about the small angle in the triangle?

 

[mathlearn]

As the interior angles of the pentagon add up to 540 degrees. So One angle should be equal to 108 degrees. Correct I guess ?

The sides of the square have the same length.

Can we say anything about the small angle in the triangle?

:) Yes the small angle of the triangle would be 108(int. angle of the pentagon)=90(int. angle of the square)-(the small angle)

108-90 =small angle
18 degrees = small angle

So 180 (interior angles of a triangle add up to 180 ) = 2x + 18
180 -18 = 2x
162 = 2x
81 = x

Many Thanks :)