Consider the following two statements about quadrilaterals

Consider the following two statements about quadrilaterals: \" If a quadrilateral is a square, then it is a rectangle and a rhombus \" \" If a quadrilateral is a rectangle and a rhombus, then it is a square \" a ) Are these two statements equivalent ? Why or why not ? How could you convinve someone whether they are or not ? b) Under what conditions are pairs of statements like this both going to be true together? Describe the conditions on the sets decribed in the two parts of the statment that would make both statements true together.

Solution

Let us first look at following definitions:

A quadrilateral is a four-sided polygon, like a square, rectangle, or rhombus. Quadrilaterals are also called quadrangles and tetragons.

A rhombus is a quadrilateral with four equal sides and that opposite angles are equal. All the properties of a parallelogram apply to rhombus.

A rectangle is any shape with four sides and four right angles. All squares are rectangles but not all rectangles are squares (all the sides in a square have to be the same length), and

The square is a quadrilateral, defined as having all sides equal, and its interior angles all right angles (90°). From this it follows that the opposite sides are also parallel.

From above definitions, we see that \" If a quadrilateral is a square, then it is a rectangle and a rhombus\".

Further, from definition of rhombus all four sides must be equal and from definition of rectangle all interior angles must be right angle, therefore we have

\" If a quadrilateral is a rectangle and a rhombus, then it is a square \"

Hence, the two statements are equivalent.