# Parallelogram

A **parallelogram** is a quadrilateral whose opposite sides are parallel.
A parallelogram, in its most general form, looks something like this:

#### Contents

## Special Cases

## Basic Properties

## Area

## Some Cooler Properties

## Properties of Parallelograms

A **parallelogram** is a quadrilateral with two pairs of parallel sides.

## Properties

The fundamental definition of a **parallelogram** is as follows:

A

parallelogramis a quadrilateral whose opposite sides are parallel.

In the diagram of a general parallelogram above, \( AB || DC \) and \( AD || BC \). Several important properties then follow. First, it is clear that the opposite angles must be equal \(( \angle A = \angle C \) and \( \angle B = \angle D) \) since they make corresponding angles with the two sets of parallel lines. Meanwhile, consecutive angles are supplementary.

Property 1.The opposite angles of a parallelogram are equal.

Property 2.Consecutive angles of a parallelogram are supplementary.

One can also show that the opposite sides are equal \(( AB = DC \) and \( AD = BC): \) the two triangles formed by drawing in a diagonal of the parallelogram \((\)i.e. either the segment \( AC \) or \( BD \) above\()\) must be congruent by angle-side-angle, so the corresponding sides of the two triangles must be congruent as well.

Property 3.The lengths of the opposite sides of a parallelogram are equal.

Now that the lengths of all of the sides are known, it is easy to compute the perimeter of a parallelogram.

Property 4.A parallelogram whose side lengths are \( a \) and \( b \) has perimeter \( 2a + 2 b \).

Meanwhile, the area of the parallelogram can be found by computing the sum of the area of the two triangles formed.

Property 5.The area of a parallelogram with side lengths \( a \) and \( b \), with the acute angle formed between them \( \theta \), is given by \( ab \sin{\theta} \).

Drawing in both diagonals simultaneously produces four congruent triangles. Therefore, the intersection of the two diagonals must be the midpoint of each diagonal.

Property 6.The diagonals of a parallelogram bisect each other.

One last result is left as an exercise for the reader.

Property 7.Lines that connect midpoints of opposite sides with opposite vertices trisect the diagonal.

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Given that \(Q\) has base \(12\) and perpendicular height \(4\) and that the area of \(ABCD\) is \(200\), find the sum of all possible values of the base of \(P\).

## Special Parallelograms

Commonly encountered special cases of parallelograms include

- rectangles, all of whose angles are equal;
- rhombuses, all of whose sides are equal;
- squares, all of whose sides and angles are equal.

Because opposite angles of a cyclic quadrilateral are supplementary, all cyclic parallelograms are rectangles. Furthermore, the only rectangles with an incircle are squares, so the only bicyclic parallelogram is a square.