I know that a rectangle has 2 opposite equal and parallel sides but is that enough to say that a rectangle is a parallelogram , i mean how can someone prove it in geometry, any help appreciated
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$\begingroup$Ultimately, it boils down to definitions. Any level of rigor can only be established if you can define the object you are talking about. Essentially, what I am asking is what do you mean by this thing, parallelogram? Before we work, we need to pinpoint that what it is exactly. So using Wikipedia's definition of parallelograms:
- In Euclidean geometry, a parallelogram is a simple (non self-intersecting) quadrilateral with two pairs of parallel sides.
And, for rectangle:
- In Euclidean geometry, a rectangle is any quadrilateral with four right angles.
The above two definitions pinpoint what exactly a parallelogram and rectangle, so we can start working without any ambiguity or problems. So what does it mean that a rectangle is a parallelogram? We can say that it means the definition of rectangle fits in the definition of parallelogram. So, the question is, that is any quadrilateral with four right angles [in short, rectangles] has two pairs of parallel sides and is simple (non self-intersecting)?
So what are parallel sides, then? Obviously, they can be defined as any two lines in a plane that never meet. But in Euclidean geometry, the one we are working in, we have an axiom (they are statements that cannot really be proven but have to assumed to move ahead) to work with, called the parallel postulate (Google if you want to know more). An equivalent version states:
- Lines m and l are both intersected by a third straight line (a transversal) in the same plane, and the corresponding angles of intersection with the transversal are congruent, then m and l are parallel.
We can see that what we want is an immediate corollary of this. Let the lines m and l be the two [extended] non adjacent sides of the rectangle and we see that the corresponding angles of intersection are $90 ^{\circ}$ and hence congruent, the exact thing can be done for the other pair of sides. So, we have proved that the rectangle has two pairs of parallel sides. Now, since the adjacent sides are parallel, they do not intersect [remember the definition], and hence is simple. Thus, every condition is met with, and hence a rectangle is a parallelogram.
If you want to make it feel more rigorous, you could rewrite the stuff in formal logic, use stricter definitions or whatever, but I think it is useless to do so, because it does not help us anyway.. I have introduced and specifically wrote out the definitions [and axioms, another important concept, both of which have to be assumed], because I feel that is where you are going into problems. You wrote the properties of the rectangle, that would easily fit within the parallelogram definition, but you did not know when or how to use it. So, I started with definitions and then manipulated to get from one end to the other, so you get a better feel of what is happening.
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