Draw a Venn diagram showing the set of parallelograms, kites, rhombuses, rectangles, and squares. Then, answer the following questions—
What is the quadrilateral that is both a kite and a parallelogram?
A rhombus is both a kite and a parallelogram. A rhombus has all sides equal (making it a kite with two pairs of adjacent equal sides) and opposite sides parallel (making it a parallelogram).
Can there be a quadrilateral that is both a kite and a rectangle?
Yes, a square is both a kite and a rectangle. A square has all sides equal (satisfying the kite condition) and all angles 90° (satisfying the rectangle condition).
Is every kite a rhombus? If not, what is the correct relationship between these two types of quadrilaterals?
No, not every kite is a rhombus. A kite has two pairs of adjacent sides equal, but a rhombus has all four sides equal. Every rhombus is a kite (since all sides equal means adjacent sides are equal), but not every kite is a rhombus.
Explanation
(i) A kite has two pairs of adjacent sides equal. A parallelogram has opposite sides parallel and equal. A rhombus satisfies both conditions: all sides are equal (so adjacent sides are equal), and opposite sides are parallel. (ii) A square has all sides equal, so it has two pairs of adjacent equal sides (kite property). A square also has all angles 90° and opposite sides parallel (rectangle property). Thus, a square belongs to both categories. (iii) The chapter defines a kite as having two pairs of adjacent sides equal (AB = BC and CD = DA), while a rhombus has all four sides equal. A rhombus satisfies the kite condition, but a general kite may have different pairs of equal adjacent sides (like sides of 5 cm, 5 cm, 3 cm, 3 cm), which is not a rhombus.