1. Definition of a Parallelogram
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. We usually label a parallelogram as \(ABCD\), where \(AB \parallel CD\) and \(BC \parallel AD\).
Parallel sides ensure that the shape looks like a “slanted rectangle,” but without the requirement of right angles.
2. Properties of a Parallelogram
Parallelograms have several useful geometric properties that help in solving problems easily.
2.1. Opposite Sides are Equal
In parallelogram \(ABCD\):
\(AB = CD \quad \text{and} \quad BC = AD\)
2.2. Opposite Angles are Equal
Angles facing each other are equal:
\(\angle A = \angle C \quad \text{and} \quad \angle B = \angle D\)
2.3. Adjacent Angles are Supplementary
Any two angles next to each other add up to:
\(180^\circ\)
Example: \(\angle A + \angle B = 180^\circ\).
2.4. Diagonals Bisect Each Other
The diagonals of a parallelogram cut each other exactly in half:
\(AO = OC\) and \(BO = OD\) where the diagonals intersect at point \(O\).
3. Understanding Angle Properties
Angle relationships in a parallelogram make many questions easier to solve.
3.1. Example: Finding Missing Angle
If \(\angle A = 70^\circ\), then:
\(\angle C = 70^\circ\)
Because opposite angles are equal.
Also, adjacent angle:
\(\angle B = 180^\circ - 70^\circ = 110^\circ\)
4. Diagonals of a Parallelogram
Diagonals help in proving many geometric results. They create two congruent triangles inside the parallelogram.
4.1. Diagonal Properties
- They bisect each other.
- They are not necessarily equal.
- They divide the parallelogram into two congruent triangles.
4.2. Example with Diagonals
If diagonal \(AC = 10\) and it is bisected at \(O\), then:
\(AO = OC = 5\)
5. Perimeter and Area of a Parallelogram
A parallelogram has well-defined formulas for perimeter and area.
5.1. Perimeter
If the sides are \(a\) and \(b\), then:
\(\text{Perimeter} = 2(a + b)\)
5.2. Area (Using Base and Height)
If \(b\) is the base and \(h\) is the corresponding height, then:
\(\text{Area} = b \times h\)
5.3. Quick Example
A parallelogram has base \(8\text{ cm}\) and height \(5\text{ cm}\). Its area is:
\(8 \times 5 = 40\text{ cm}^2\)
6. Special Cases of a Parallelogram
A parallelogram becomes more specific when additional conditions are satisfied.
6.1. Rectangle
A parallelogram with all angles equal to \(90^\circ\).
6.2. Rhombus
A parallelogram with all four sides equal.
6.3. Square
A parallelogram with all sides equal and all angles \(90^\circ\).