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How do we know that the 4-sided shape in the middle of this picture, with sides of length c, is actually a square? Does it have any credentials or some sort of "square ID" badge?

For starters, the angles of a triangle must add to 180 degrees, that much we know for sure. In a right triangle, the measures of the other two angles must add to 90 degrees:

Any straight line, such as the straight line across the bottom of the big square, is also 180 degrees. Adding the measures of the blue angle and red angle yields 90 degrees, so we have 90 degrees left for the angle between the blue and red line segments; in other words, between the two sides of length c. The figure with 4 sides of length c is a square.

It was nice that our shape passed the initial eyeball test, but we feel even better knowing it can be proved. In fact, we are now on such a proving high that we'll take on the existence of Bigfoot next. Wish us luck.

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