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The two green lines are parallel. The two red segments are equal and the two purple segments are equal too. The blue angle is 90°.
 

How big is the black angle:

 

 Jan 29, 2021
 #1
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Let  the  legs of the isosceles  right triangle = 1

Then  the   purple  segmemt on the base = sqrt (2)

The  area of the right triangle =  1/2 (1)^2   = 1/2

And  1/2 =   (1/2) base (height

base * height  =  1

sqrt (2)  height   =1

height =1/sqrt (2)

 

So the  height  of this triangle  will    be 1/ sqrt (2)

 

And the  perpendicular distance  from  the point  where the  purple segment meets the  green line at the top left  to the base = the height of the triangle =  1/sqrt (2)

 

So   

 

Let A  be  the  angle  unshaded  part  of   the 45°  base angle

 

sin A  =   1/sqrt (2)  / sqrt (2)  = 1/2

A = 30°

 

So....the shaded angle = 45  - 30  = 15°

 

 

cool cool cool

 Jan 29, 2021

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