Open Middle®
Directions: Use the terms square, rhombus, kite, parallelogram, trapezoid, rectangle, irregular quadrilateral at most one time each to complete two sentences
Hint
How many degrees are in a straight angle? What do you know about the measure of acute and obtuse angles?
Answer
Sample answers:
A square is a rectangle but not a kite.
A rhombus is a parallelogram but not a trapezoid.
A square is a rectangle but not a kite.
A rhombus is a parallelogram but not a trapezoid.
Source: Alice Keeler and Miguel Ruiz
I’m not 100% sure how it is defined in English-speaking countries, but in German, a kite is a quadrilateral with two pairs of adjacent and equally long sides. This is also the case for a square, which makes the square a special case of the kite.
Additionally, a trapezoid in German is defined as a quadrilateral with (at least) two opposing parallel sides, which makes the rhombus a special case of a trapezoid.
Also, with the article “a” being given, filling in “irregular quadrilateral” in any of the blanks results in an agrammatical sentence.
And lastly: I’d say “but not always a …” instead of “but not a …”, because – for example – a rectangle *can* be a square if a = b and a parallelogramm *can* be a rhombus.
With these definitions, the four solutions excluding the irregular quadrilateral are:
#1:
– A rhombus is a kite, but not (always) a square.
– A parallelogramm is a trapezoid, but not (always) a rectangle.
#2:
– A rhombus is a kite, but not (always) a rectangle.
– A parallelogramm is a trapezoid, but not (always) a square.
#3:
– A rectangle is a parallelogramm, but not (always) a kite.
– A rhombus is a trapezoid, but not (always) a square.
#4:
– A rectangle is a trapezoid, but not (always) a kite.
– A rhombus is a parallelogramm, but not (always) a square.
Including the irregular quadrilateral, there are another 26 possibilities, for a total of 30:
#5:
– A parallelogramm is a trapezoid, but not a kite.
– A square is a rhombus, but not an irregular quadrilateral.
#6:
– A parallelogramm is a trapezoid, but not a rhombus.
– A square is a kite, but not an irregular quadrilateral.
#7:
– A parallelogramm is a trapezoid, but not a kite.
– A square is a rectangle, but not an irregular quadrilateral.
#8:
– A parallelogramm is a trapezoid, but not a rectangle.
– A square is a kite, but not an irregular quadrilateral.
#9:
– A parallelogramm is a trapezoid, but not a rhombus.
– A square is a rectangle, but not an irregular quadrilateral.
#10:
– A parallelogramm is a trapezoid, but not a rectangle.
– A square is a rhombus, but not an irregular quadrilateral.
#11:
– A parallelogramm is a trapezoid, but not a square.
– A rhombus is a kite, but not an irregular quadrilateral.
#12:
– A parallelogramm is a trapezoid, but not an irregular quadrilateral.
– A rhombus is a kite, but not a square.
#13:
– A parallelogramm is a trapezoid, but not an irregular quadrilateral.
– A rhombus is a kite, but not a rectangle.
#14:
– A parallelogramm is a trapezoid, but not a rectangle.
– A rhombus is a kite, but not an irregular quadrilateral.
#15:
– A square is a kite, but not an irregular quadrilateral.
– A rhombus is a parallelogramm, but not a rectangle.
#16:
– A square is a parallelogramm, but not an irregular quadrilateral.
– A rhombus is a kite, but not a rectangle.
#17:
– A square is a kite, but not an irregular quadrilateral.
– A rhombus is a trapezoid, but not a rectangle.
#18:
– A square is a trapezoid, but not an irregular quadrilateral.
– A rhombus is a kite, but not a rectangle.
#19:
– A square is a parallelogramm, but not an irregular quadrilateral.
– A rhombus is a trapezoid, but not a rectangle.
#20:
– A square is a trapezoid, but not an irregular quadrilateral.
– A rhombus is a parallelogramm, but not a rectangle.
#21:
– A rhombus is a kite, but not a square.
– A rectangle is a parallelogramm, but not an irregular quadrilateral
#22:
– A rhombus is a kite, but not a square.
– A rectangle is a trapezoid, but not an irregular quadrilateral
#23:
– A rhombus is a parallelogramm, but not a square.
– A rectangle is a trapezoid, but not an irregular quadrilateral
#24:
– A rhombus is a trapezoid, but not a square.
– A rectangle is a parallelogramm, but not an irregular quadrilateral
#25:
– A rhombus is a kite, but not an irregular quadrilateral.
– A rectangle is a parallelogramm, but not a square.
#26:
– A rhombus is a kite, but not an irregular quadrilateral.
– A rectangle is a trapezoid, but not a square.
#27:
– A rhombus is a parallelogramm, but not an irregular quadrilateral.
– A rectangle is a trapezoid, but not a square.
#28:
– A rhombus is a trapezoid, but not an irregular quadrilateral.
– A rectangle is a parallelogramm, but not a square.
#29:
– A rhombus is a parallellogramm, but not an irregular quadrilateral.
– A rectangle is a trapezoid, but not a kite.
#30:
– A rhombus is a trapezoid, but not an irregular quadrilateral.
– A rectangle is a parallelogramm, but not a kite.
____
To get all these, I searched for all the valid solutions to these:
– A parallelogramm is a trapezoid, but not a _______(square/kite/rhombus/irregular quadrilateral/rectangle).
– A square is a ___(kite/rectangle/parallelogramm/rhombus/trapezoid), but not an irregular quadrilateral.
– A rhombus is a ____(kite/parallogramm/trapezoid), but not a ____(square/irregular quadrilateral/rectangle).
– A rectangle is a _____(parallelogramm/trapezoid), but not a ______(square/kite/rhombus/irregular quadrilateral).
The sentence can never begin with “A trapezoid is a”, because there’s no quadrilateral that fits in the blank after that. Same with the kite and the irregular quadrilateral.
To clarify: I think the solutions given by the authors are wrong.