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problem

I know that the answer is 180 from using the arctangent (arctan(1)+arctan(2)+arctan(3)), but thats boring. Is there any way to solve this problem without the help of trig at all?

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  • $\begingroup$ Just thought of constructing a 5 sided polygon with the colored lines as a few of the sides, and using the fact that any convex 5-sided polygons external angles add to 450 (many of them will be right angles, if properly constructed) but any better ideas? $\endgroup$ – D. W. Nov 4 '16 at 9:09
  • $\begingroup$ See the third answer here. $\endgroup$ – David Mitra Nov 4 '16 at 9:31
  • $\begingroup$ Not always the third answer @David. Answer orders change as votes come in. You can only reder to the question which, true, duplicates this one (and which I did not see before posting my answer). $\endgroup$ – Oscar Lanzi Nov 4 '16 at 9:37
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    $\begingroup$ math.hmc.edu/funfacts/ffiles/20005.2.shtml $\endgroup$ – Hans Lundmark Nov 4 '16 at 10:10
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enter image description here

I think the picture is self-explanatory, but let me justify some details.

The segment in the middle, the one that's across two vertical squares on the top picture, is reflected with respect to the vertical axis of symmetry of the two vertical squares to obtain segment $AB$ on the bottom picture.

Clearly $x = 45^{\circ}$ because it is the angle between an edge and a diagonal of a square (see first segment on top picture).

Triangle $ABC$ is isosceles with right engle $\angle\, BAC = 90^{\circ}$, as its base angle. Simply perform a $90^{\circ}$ counterclockwise rotation around point $A$ and you see that $AB$ is mapped to $AC$.

Alternatively, look at this picture, where clearly $BACD$ is a square:

enter image description here

From all these pictures it follows that $x + y + z = 45^{\circ} + y + z = 180^{\circ}$

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  • $\begingroup$ According to the first picture : X = 90 - Y + 90 - Z ; X + Y + Z = 90 + 90; X+Y+Z=180 $\endgroup$ – Natali Nov 4 '16 at 13:58
  • $\begingroup$ Your fourth paragraph should not say "equilateral". Your third paragraph could also be something like "Clearly $x=45^\circ$ because it is an angle of an isosceles right-angled triangle." $\endgroup$ – Henry Nov 4 '16 at 14:23
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In the diagram below, green and red angles in triangle $ABC$ are equal by construction to your $x$ and $z$ angles. If we can show that blue angle is equal to angle $y$ then we are done, because the angles in a triangle sum up to 180°.

Drawing line $AD$ perpendicular to $AB$, we can see that $AD=2AB$, because $AD={1\over2}AE={1\over2}AF$ and $AB={1\over4}AF$. It follows that triangle $ABD$ is similar to your $y$ triangle and $\angle B=y$.

enter image description here

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Define the following points:

$O=(0,0)$

$A=(1,0)$

$B=(1,1)$

$C=(0,2)$

$D=(-1,3)$

$E=(-1,0)$

Points $B,C,D$ are collinear, and that line is perpendicular to $OB$ (slopes are $+1, -1$). Also $BD=BC+CD=2BC=2OB$, with triangle $OBC$ being isosceles. Then triangles $OAB, OBD, ODE$ are similar to the ones given and the angles at $O$ add up to $180°$.

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