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which of the following statements are correct?

A. If one interior angle of a parallelogram is a right angle, then the parallelogram must be a rectangle.

B. If two diagonals of a rectangle are perpendicular, then the rectangle must be a square.

C. If two diagonals of a rhombus are equal, then the rhombus must be a square.

D. If one interior angle of a rhombus is a right angle, then the rhombus must be a square.
E. If two diagonals of a parallelogram are equal, then the parallelogram must be a rectangle

 Jan 3, 2021
edited by Guest  Jan 3, 2021
edited by Guest  Jan 3, 2021
 #1
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Keep in mind that we only need 1 example to disprove something.

 

A. As one angle determines all the other ones in a parallelogram, A is true, because all the other angles are 180-90 = 90, so it is a rectangle.

 

B. True, same reason as E

 

C. True, I think same reason as E

 

D. True, opposite angles are equal, so all the angles are auto-completed, making the equal-sided rhombus a square.

 

E.  True, as the diagonals must bisect each other (definition of a parallelogram) and so all the small triangles formed by the diagonals are congruent by SSS and have largest angle 360/4=90˚. Because a parallelogram's angle bisectors are diagonals, and all the triangles are isosceles right, each angle is 45*2 = 90, and as at least two sides of a parallelogram are equal, and each angle is 90 from part A, we have that all the sides and angles are equal, so it is a square.

 Jan 3, 2021

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