To further illustrate the need for careful definition, consider the following po
ID: 2899611 • Letter: T
Question
To further illustrate the need for careful definition, consider the following possible definitions of rectangle: A quadrilateral with four right angles. A quadrilateral with all angles congruent to one another. A parallelogram with at least one right angle. In this book we will take (i) as our definition. Your experience with Euclidean geometry may lead you to believe that these three definitions are equivalent; sketch informally how you might prove that and notice carefully which theorems you are tacitly assuming. In hyperbolic geometry, these definitions give rise to three different sets of quadrilaterals (see Chapter 6).Explanation / Answer
i) When a quadrilateral is having 4 right angles, then opposite sides parallel as interior angles add to 180 for every pair.
Hence it is a parallelogram with one angle =90 degrees and hence rectangle
ii) Sum of 4 angles in a quadrilateral = 360
Hence when 4 angles are congruent each = 90 and hence a rectangle
iii)
In a parallelogram adjacent angles add to 180. Hence if one angle is 90, adjacent angles to it would be 90 each.
The fourth angle = 360-3(90) = 90