How can I prove that the angle bisectors of a rectangle form a square?

Expert Solution

The angle bisectors will always be 45 degrees from both sides of each corner of the rectangle. Angles A ,B, C, and D will all be 90 degrees, because the other two angles in the triangles are 45 degrees. So now you know ABCD has 4 right angles, so it must be a rectangle. To show ABCD is a square show AB = BC, and since it is a rectangle already we know AB = CD and BC = AD. By drawing and extending AC, Ac will pass through the midpoints of the sides of the rectangle and thus be parallel to to top and bottom of the rectangle, so Angle BAC is congruent to the top part of the angle bisector, or 45 degrees. This shows that triangle ABC is isosceles, because the base angles are both 45 degrees, so the sides must be congruent. So AB = BC and ABCD is a square.

Given below

Nipun Maity answered 2 years ago

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