To prove that ABCD is a rectangle, we can use the following steps:
1. Let l, p, s, and r be the lengths of the sides of ABCD, where l and p are opposite sides and s and r are opposite sides.
2. Since ABCD is not a square, we know that l is not equal to p.
3. By definition, a rectangle is a quadrilateral with opposite sides that are parallel and congruent. Therefore, to prove that ABCD is a rectangle, we need to show that l is parallel and congruent to p, and that s is parallel and congruent to r.
4. We can use the fact that ABCD is a plane figure to draw a diagram of the quadrilateral, with l and p as the horizontal sides and s and r as the vertical sides.
5. By definition, opposite sides of a parallelogram are parallel and congruent. Therefore, if we can show that ABCD is a parallelogram, we can conclude that it is also a rectangle.
6. To show that ABCD is a parallelogram, we can use the fact that opposite sides of a parallelogram are parallel. Since l and p are opposite sides of ABCD, we know that they are parallel.
7. To show that ABCD is a parallelogram, we can also use the fact that opposite sides of a parallelogram are congruent. Since l and p are not equal, we need to show that s and r are not equal. We can do this by assuming that s is equal to r and showing that this leads to a contradiction.
8. Suppose s is equal to r. Then, by definition, ABCD is a trapezoid with bases l and s. However, since l is not equal to p, we know that ABCD is not an isosceles trapezoid. Therefore, the diagonals of ABCD do not bisect each other, which contradicts the fact that ABCD is a parallelogram.
9. Since assuming that s is equal to r leads to a contradiction, we can conclude that s is not equal to r. Therefore, ABCD is a parallelogram with opposite sides that are parallel and congruent, which means that it is also a rectangle.
In summary, we can prove that ABCD is a rectangle by showing that it is a parallelogram with opposite sides that are parallel and congruent. This can be done by assuming that s is equal to r and showing that this leads to a contradiction. Therefore, we can conclude that ABCD is a rectangle..