Find the angle X inside a pentagon with a square

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SUMMARY

The discussion centers on calculating the angle X within a pentagon that shares a side with a square. The interior angles of a pentagon total 540 degrees, leading to each angle measuring 108 degrees in a regular pentagon. The relationship between the angles of the square and the triangle formed by the pentagon and square reveals that angle X is 81 degrees, derived from the equation 180 - 18 = 2x, where 18 degrees is the small angle adjacent to angle X.

PREREQUISITES
  • Understanding of basic geometry concepts, including angles and triangles.
  • Familiarity with the properties of polygons, specifically pentagons and squares.
  • Knowledge of the triangle angle sum theorem.
  • Ability to interpret geometric diagrams and relationships between shapes.
NEXT STEPS
  • Study the properties of regular polygons, focusing on pentagons and their angles.
  • Learn about the triangle angle sum theorem and its applications in geometry.
  • Explore isosceles triangles and their properties in relation to equal sides and angles.
  • Practice solving geometric problems involving multiple shapes and their relationships.
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Students, educators, and anyone interested in geometry, particularly those looking to enhance their problem-solving skills in polygon angle calculations.

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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,

Untitledpen.png


Any Ideas on How to begin ?

Many Thanks :)
 
Last edited:
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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?
 
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

2qcju68.png


Many Thanks :)
 
Last edited:
mathlearn said:
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?
 
As the interior angles of the pentagon add up to 540 degrees. So One angle should be equal to 108 degrees. Correct I guess ?

I like Serena said:
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

2ef8epl.jpg


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

Many Thanks :)
 

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