Which of the following statements are true?
(A) If a kite is also a parallelogram, then it must be a rhombus.
(B) If a trapezoid is also a parallelogram, then it must be a rhombus.
(C) If a trapezoid is also a kite, then it must be a parallelogram.
(D) If a rhombus is also a trapezoid, then it must be a kite.
A and C are both true
(I haven't formally proved it which I should do for absolute certainty)
I think for C it has to be a rhombus as well.
I think what you call a trapezoid, I would call a trapezium and all rhombuses are trapeziums.
Just as all rhombuses are paralellograms.
Here is one problem with this question.
According to several sources on the net a trapexium is the same thing as a trapezoid BUT
according to Wolfram|Alpha, and I quote
"There are two common definitions of the trapezium. The American definition is a quadrilateral with no parallel sides; the British (AND AUSTRALIAN) definition is a quadrilateral with two sides parallel (e.g., Bronshtein and Semendyayev 1977, p. 174)--which Americans call a trapezoid."
So before all these answers can be accepted or rejected we may have to agree upon a defintion of a trapezium or trapezoid!
However, A is still definitely correct :)
I suppose I should at least give reasons
(A) If a kite is also a parallelogram, then it must be a rhombus.
With Kites and rhombuses the diagonals MUST intersect at right angles.
Parallelograms have opposite sides parallel.
A parallelogram where the diagonals interect at right angles is a rhombus
So A is DEFINITELY correct
(B) If a trapezoid is also a parallelogram, then it must be a rhombus.
All parallelograms are trapeziums (they have at least one pair of parallel sides)
so If a trapezoid is also a parallelogram it can be ANY parallelogram
so B is false
(C) If a trapezoid is also a kite, then it must be a parallelogram.
This one depends on the definition of a trapezoid. Using the American definition it is definitely false.
Using the English (and Australia) definition it is true but it is harder to prove.
(D) If a rhombus is also a trapezoid, then it must be a kite.
All rhombuses are trapeziums. end of story. false