There are two main types of quadrilaterals: parallelograms and non-parallelograms. Parallelograms can also be divided into many sub-groups, including rectangles, trapezoids, and squares. Each type of parallelogram has its own unique set of properties.
Quadrilateral: A polygon with four sides.
Parallelogram: A quadrilateral with two pairs of parallel sides.
Rectangle: Parallelogram with four right angles
Rhombus: Parallelogram with four congruent sides
Square: Rectangle with four congruent sides
Since parallelograms are quadrilaterals, the sum of the interior angles is 360°. Some examples of parallelograms include:
There are several theorems that describe the properties of parallelograms.Theorems:
If ABCD is a parallelogram, then x° + y° = 180°.
4. If a quadrilateral is a parallelogram, then the diagonals bisect each other.
If ABCD is a parallelogram, then \(\overline{AE} \cong \overline{CE}\) and \(\overline{BE} \cong \overline{DE}\).
We can prove a quadrilaterial is a parallelogram by using the definition of a parallelogram.
The converses of the theorems described above can be used to prove that a quadrilateral is a parallelogram.
There is an additional theorem that shows a quadrilateral is a parallelogram:
There are several ways to use the coordinate plane to show that a quadrilateral is a parallelogram:
Method 1: Use the distance formula to find the lengths of the sides and see if opposite sides are congruent.
Method 2: Find the slopes of the sides of the parallelogram.
Method 3: Use the midpoint formula for each diagonal. If the midpoint is the same for both, the figure is a parallelogram.
Method 4: Determine if one pair of opposite sides has the same slope and the same length. If both statements are true, then the figure is a parallelogram.
There are three special types of parallelograms: rhombus, rectangle, and square. These special parallelograms can be identified with these theorems:
The diagonals of special parallelograms have additional properties that can be used to identify them:
Since a square is a rectangle and a rhombus, it has the properties of a rhombus, rectangle, and parallelogram.
Here is one possible way to identify special parallelograms: