Ideas in Geometry/Instructive examples/Proving Parallel Bisectors in Quadrilaterals

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Proving Parallel Bisectors in Quadrilaterals:

Problem 9 :Explain how the following picture "proves" that if a quadrilateral has two opposite angles that are equal, then the bisectors of the other two angles are parallel or top of each other. In order for this to be true within our picture we first found that each shape has two right angles in them. This means that these angles have equal measures of 90 degrees. This satisfies the first part of the statement ( two angles that are equal) Our next step was to prove that the lines drawn in the picture are parallel. To do this we wanted to work with interior/exterior angles. We need to prove that two angles are equal in order to show that the lines within the picture are indeed parallel.

We looked at different shapes to see how this statement would work. In a square, and within a parallelagram, the lines are on top of one another. In a rectangle the lines are parallel. - we haven't exactly figured out how to do this, we have a number of different ideas of how to go about finding our solution. We are assuming that the line is bisecting the small angle within the dark grey shape and the white grey shape. This angles total measure is 45 degrees. We figured this out because when we bisected the top right angle into 45 degrees. We knew that the two angles added up to 180, and we had the measures of 90 degrees and 45 degrees leaving us with an angle measure of 45 degrees for the missing angle. We assumed that the line is a bisector, leaving us bisecting this 45 degree angle bisected into two 22.5 degree angles. This is the measure of the exterior angle that we are using to prove the lines are parallel. We repeated this process on the lower quadrilaterals, proving that the other exterior angle also equals 22.5 degrees. These two angles have the same measure, and they are exterior angles. If two exterior angles are equal, then the lines creating them are parallel. We proved that these two lines are parallel. Because these lines are parallel, and our quadrilaterals have two opposite angles of equal measure, this original statement holds true. The bisectors of the remaining two angles within the quadrilateral are parallel ( in this case) or on top of one another.