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Math Help - Euclid's Exterior Angle Theorem (EEAT)

  1. #1
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    Euclid's Exterior Angle Theorem (EEAT)

    A.) Any exterior angle of a triangle is greater than each of the opposite interior angles. Warning: Euclid's EAT is not the same as the Exterior Angle Theorem usually studied in high school.

    B.) When is EEAT true on the plane, on a sphere, and on a hyperbolic plane?

    Here are some suggestions:

    Draw a line from the vertex of a to the midpoint, M, of the opposite side, BC. Extend that lien beyond M to a point A' in such a way that AM is congruent to MA". Join A' to C. Be cautious when tranferring this to a sphere, it will probably help to draw Euclid's hint directly to a physical sphere.

    As a final note, remember you do not have to look at figures using only one orientation, rotations and reflections of a figure do not change its properties, so if you have trouble "seeing" somethig, check to see if it's something you're familiar with by orienting it differently on the page.

    EEAT is not always true on a sphere, even for small triangles.

    This was just background stuff....any help would be appreciated
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    The Euclidean EEAT says that an exterior angle is equal to the sum of the two opposite angles. That is, of course, because the exterior angle is 180 degrees minus the adjacent angle in the triangle and, since the angles in a Euclidean triangle sum to 180 degrees, that is the same as sum of the two opposite angles. On a sphere, the angle in a triangle sum to more than 180 degrees so the exterior angle is less than the sum of the two opposite angles. In hyperbolic geometry, the angles in a triangle sum to [b]less[b] than 180 degrees so the exterior angle is greater than the sum of the two opposite angles.
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    This is very helpful thank you
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