Short Answer 
A parallelogram is a quadrilateral with opposite sides that are parallel and equal in length, and opposite angles that are equal. Additionally, adjacent angles are supplementary, and one can use coordinate geometry to determine a shape’s status as a parallelogram by checking the slopes and lengths of opposite sides.
Step 1: Understand the Definition of a Parallelogram
A parallelogram is a specific type of polygon that has four sides, four angles, and four vertices, classified under quadrilaterals. The defining characteristics include:
- Opposite sides are parallel to each other.
- The lengths of opposite sides are equal.
- Opposite angles are also equal, ensuring that the geometry remains consistent.
Step 2: Identify the Properties and Proves of Angles
In a parallelogram, certain angle properties hold true. For example:
- Adjacent angles are supplementary, meaning they add up to 180 degrees.
- Opposite angles are equal, which can be proved using algebraic equations based on their measures.
- For instance, angles like ABC and ADC can be shown to be equal through congruency proof methods.
Step 3: Utilize Coordinate Geometry for Identification
You can also determine if a shape is a parallelogram using coordinate geometry. This involves checking:
- The slopes of opposite sides; they should be equal, indicating parallelism.
- The lengths of opposite sides should also be equal, confirming that they are the same segment length.
- If both conditions hold, it proves the shape is a parallelogram regardless of its positioning.