Geometry Question Bank

[FunmiO404]

Hello,

So I just finished going through the Geometry 6th edition and tried the question bank. I need clarification on 2 question.

Ques 1: Is quadrilateral ABCD a rectangle?

(1) Line segment AC and BD bisect one another.
(2) Angle ABC is a right angle

Answer :
(1) INSUFFICIENT: The diagonals of a parallelogram bisect one another. Knowing that the diagonals of quadrilateral ABCD (i.e. AC and BD) bisect one another establishes that ABCD is a parallelogram, but not necessarily a rectangle. (I agree with this statement)

(2) INSUFFICIENT: Having one right right angle is not enough to establish a quadrilateral as a rectangle (i agree with this statement)

(1) AND (2) SUFFICIENT: According to statement (1), quadrilateral ABCD is a parallelogram. If a parallelogram has one right angle, all of its angles are right angles (in a parallelogram opposite angles are equal and adjacent angles add up to 180), therefore the parallelogram is a rectangle. (Is it possible that the parallelogram is also a square?)

Ques 2 : Is quadrilateral ABCD a rhombus?

(1) Line segments AC and BD are perpendicular bisectors of each other.

(2) AB = BC = CD = AD

Ans: (1) SUFFICIENT: The diagonals of a rhombus are perpendicular bisectors of one another. This is in fact enough information to prove that a quadrilateral is a rhombus ( I agree with this statement)

(2) SUFFICIENT: A quadrilateral with four equal sides is by definition a rhombus. (Again, why wasn't a square considered since a square has 4 equal sides)

Thank you.

[StaceyKoprince]

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Some quick answers to your questions:
All squares are rectangles (but not all rectangles are squares)—so yes, it's possible that a rectangle could be a square. Even if it is a square, though, it's still a rectangle.

All squares are rhombi (the plural of rhombus), but not all rhombi are squares. Again, it's possible that a rhombus could be a square. Even if it is a square, it's still a rhombus.