gottabwise wrote:
Is quadrilateral ABCD a rectangle?
(1) Line segments AC and BD bisect one another.
(2) Angle ABC is a right angle.
Is quadrilateral ABCD a rectangle?
Rephrase: does ABCD have 1)opposite sides that are parallel and equal, 2) bisecting diagonals and 3) opposite and equal angles of 90; adjacent angles that sum to 180.
(1) Line segments AC and BD bisect one another.
Bisecting diagonals make quadrilateral ABCD a parallelogram but not necessarily a rectangle.
(2) Angle ABC is a right angle.
One right angle isn't enough to establish that quadrilateral ABCD is a rectangle.
Answer C: A paralleogram with one right angle has all right angles because opposite angles are equal. Therefore ABCD is a rectangle
Source:
MGMAT question bank
Recall how we go from general quadrilateral to specific:
Quadrilateral
- trapezoid (one set of opposite sides parallel)
- parallelogram (both sets of opposite sides parallel, diagonals bisect each other)
- rhombus (all sides equal and opposite sides parallel, diagonal bisect and perpendicular)
- rectangle (Opposite sides parallel and all 90 degree angles, diagonals bisect)
- square (all sides equal, all 90 degree angles, diagonals bisect and perpendicular)
(1) Line segments AC and BD bisect one another.
All we can say is that we have a parallelogram at least. Is it a rectangle, we don't know.
(2) Angle ABC is a right angle.
We have one right angle. It's possible that all other 3 angles are not 90 degrees. Not sufficient.
Using both, we have a parallelogram with a 90 degree angle. So the angle opposite to 90 degree angle will also be 90 degrees. This means the other two angles with have a sum of 180 and will be equal so then they both will be 90 too. Hence all angles will be 90.
A parallelogram with all angles 90 degrees is a rectangle.
Sufficient
Answer (C)
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