Quadrilateral relationships answers

Note: you should make sure you understand the reasoning and examples, not just the yes/no answer.

Squares:
square

1. Is every square a rhombus?
Yes, a rhombus is a quadrilateral with 4 equal sides.  Every square has 4 equal length sides, so every square is a rhombus.
2. Is every square a rectangle?
Yes.  A rectangle is a quadrilateral with 4 right angles.  Every square has 4 right angles, so every square is a rectangle.
3. Is every square a parallelogram?
Yes.  A parallelogram is a quadrilateral with 2 pairs of parallel sides.  The opposite sides on every square are parallel, so every square is a parallelogram.
4. Is every square a quadrilateral?
Yes.  Every square is a closed figure, and every square has 4 straight sides, so every square is a quadrilateral.


Rectangles:

5. Is every rectangle a square?
No--Some rectangles are not squares. This one is not, because it doesn't have 4 equal length sides:
A non square rectangle
6. Is every rectangle a rhombus?
No--Some rectangles are not rhombi. This one is not, because it doesn't have 4 equal length sides:
A non square rectangle

7. is every rectangle a parallelogram?
Yes.  A parallelogram is a quadrilateral with 2 pairs of parallel sides.  The opposite sides on every rectangle are parallel, so every rectangle is a parallelogram.
rectangle with opposite paralles marked
8. Are there any rectangles that are squares?
Yes.   This shape has 4 right angles, so it is a rectangle.  It also has 4 equal sides, so it is also a square.
square

9. Are there any rectangles that are rhombi?
Yes.   This shape has 4 right angles, so it is a rectangle.  It also has 4 equal sides, so it is also a rhombus.
square

Rhombi

10. Is every rhombus a square?
   No.  This rhombus does not have right angles, so it isn't a square:
rhombus
11. Is every rhombus a rectangle?
   No.  This rhombus does not have right angles, so it isn't a rectangle:
rhombus

12. Is every rhombus a parallelogram?
Yes.  A parallelogram is a quadrilateral with 2 pairs of parallel sides.  The opposite sides on every rhombus are parallel, so every rhombus is a parallelogram.
rhombus with parallel sides marked
13. Are there any rhombi that are squares?
Yes.  This shape has 4 equal sides so it is a rhombus.  It also has 4 right angles, so it is also a square:
square
14. Are there any rhombi that are rectangles?
Yes.  This shape has 4 equal sides so it is a rhombus.  It also has 4 right angles, so it is also a rectangle:
square
15. Are there any rhombi that are parallelograms?
Yes.  All of them are
rhombus with parallel sides marked

Parallelograms.

16. Is every parallelogram a square?
No.  This quadrilateral has 2 pairs of parallel sides, so it is a parallelogram, but it does not have equal sides or right angles so it is not a square
parallelogram

17. Is every parallelogram a rectangle?
No.  This quadrilateral has 2 pairs of parallel sides, so it is a parallelogram, but it does not have right angles so it is not a rectangle
parallelogram
18. Is every parallelogram a rhombus?
No.  This quadrilateral has 2 pairs of parallel sides, so it is a parallelogram, but it does not have equal sides so it is not a rhombus
parallelogram

19. Are any parallelograms squares?
Yes.  This quadrilateral has parallel opposite sides so it is a parallelogram.   It also has equal sides and right angles, so it is a square
square

20. Are any parallelograms rectangles?
Yes.  This quadrilateral has parallel opposite sides, so it is a parallelogram.  It also has right angles, so it is a rectangle
rectangle
21. Are any parallelograms rhombi?
Yes.  This quadrilateral has parallel opposite sides, so it is a parallelogram.  it also has equal length sides, so it is a rhombus
rhombus