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I'm looking at this link:

http://www.mathsisfun.com/geometry/trapezoid.html

..and it says "Called an Isosceles trapezoid when the sides that aren't parallel are equal in length and both angles coming from a parallel side are equal."

Isn't the second part, namely "both angles coming from a parallel side are equal" redundant? I mean, if the two sides that aren't parallel are equal in length, then doesn't it follow that the angles will be equal as well? I can't prove this, but my intuition tells me this is how it is. Am I correct in making this assumption?

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Yes, you are correct. Given any trapezoid, we can determine whether or not it is also isosceles by using any number of necessary and sufficient characterizations, including "equal legs" and "equal base angles".

To see why "equal legs" implies "equal base angles", imagine extending the legs (that is, extending the sides that aren't parallel) until they form a triangle. Notice that, because trapezoids have one pair of parallel sides, there are now two similar triangles. Since the trapezoid has two equal legs, the larger triangle must also have two equal legs. Hence, the larger triangle is isosceles and therefore has two equal base angles, and thus so too does the trapezoid.

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There is some disagreement as to whether a trapezooid must only have one pair of parallel sides. The source you cite above doesn't seem to hold this restrictive view, so a parallelogram would also be technically considered to be a trapezoid. In this case the condition on the angles is necessary to differentiate from a parallelogram.

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