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Three squares are drawn next to each other. Three lines are drawn from a corner as illustrated. Determine the sum of the three angles exposed (the exact number of degrees or radians):

enter image description here

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  • $\begingroup$ Do you know about trigonometry, like tangents and stuff? $\endgroup$ – Justpassingby Dec 13 '15 at 22:09
  • $\begingroup$ Yes. Try do derive me to the answer. $\endgroup$ – algebra1 Dec 13 '15 at 22:19
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Note that: $$\tan \alpha_1=1 \qquad \tan \alpha_2=\frac{1}{2} \qquad \tan \alpha_3=\frac{1}{3}$$

and $$ \tan (\alpha_1+\alpha_2)=\frac{\tan \alpha_1+\tan \alpha_2}{1-\tan \alpha_2\tan \alpha_2}=\frac{1+1/2}{1-1/2}=3 $$ so: $$\tan (\alpha_1+\alpha_2)=\frac{1}{\tan \alpha_3}$$ and this means that $\alpha_3$ is the complement of $\alpha_1+\alpha_2$, i.e. $\alpha_1+\alpha_2+\alpha_3=90°$

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Hint. Draw another one-by-two domino, but aligned at $45$ degrees to this one, whose lower-left and upper-right corners are the upper-left and lower-right corners of the original diagram. You will find that the proposition becomes self evident.

(When I get to a console, I'll draw it out if that helps.) Martin Gardner fans should recognize it.

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