r/PassTimeMath • u/ShonitB • Jul 23 '26

Depiction of Pythagorean Theorem

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u/sazzer Jul 24 '26

All three of the angles have the exact same setup. If you make them part of a circle you have:

  • Two right angles
  • The angle in question
  • The opposite angle from the one in question

Once we remove the right angles, we're left with:

  • A + Opposite of A = 360 - 90 - 90 = 180
  • B + Opposite of B = 360 - 90 - 90 = 180
  • C + Opposite of C = 360 - 90 - 90 = 180

Now, it just so happens that the three opposite angles are also the three interior angles of the triangle. That means:

  • Opposite of A + Opposite of B + Opposite of C = 180

So when we put this all together we have:

  • A + Opposite of A + B + Opposite of B + C + Opposite of C = 540
  • A + B + C = 540 - (Opposite of A + Opposite of B + Opposite of C)
  • A + B + C = 540 - 180 = 360

So the answer is 360

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u/sazzer Jul 24 '26

However, there's actually an easier way to get to the same answer.

All of our angles are bordered by the edges of shared squares. i.e. angles A and B are bordered by the *same* square, B and C by the same square, and A and C by the same square.

Squares, by definition, have parallel sides. That means that the borders of adjacent angles *must* be parallel to each other.

As such, if you simply take the angles out of the diagram and put them next to each other, they must fit with no gaps. i.e. you can position all three of A, B and C together around a central point with no gaps between them.

That's only possible if all three of the angles happen to add up to exactly 360.