To further illustrate the need for careful definition consider the following pos

Question

To further illustrate the need for careful definition consider the following possible definitions of rectangle: (i) A quadrilateral with four right angles (ii) A quadrilateral with all angles congruent to one another (iii) A parallelogram with at least one right angle Prove that these three definitions are equivalent

Explanation / Answer

i) => ii) quadrilateral with 4 right angles = quadrilateral with 4 congruent angles. ii) => iii) quadrilateral with all congruent angles implies that atleast one (infact all) angles are right angles. also since opposite sides make the same angle with the sides adjacent to them, the opposite sides are infact parallel. hence it is a parallelogram with atleast one right angle. iii) => i) a parallelogram will have opposite sides equal as a consequence of the parallel postulate. one right angle will mean that the opposite angle is also a right angle. the other angles are equal and must sum to 180 which mean again that both are right angles. hence it is a quadrilateral with all angles = 90. Thus these 3 definitions of rectangle are logically equivalent.

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