Parallelograms are special quadrilaterals with two pairs of parallel sides. They have cool properties like opposite sides being equal and diagonals cutting each other in half. These shapes pop up everywhere in geometry and real life.

Understanding parallelograms is key to grasping more complex shapes. We can use their properties to solve tricky problems and even build them from scratch. It's like having a secret weapon in your geometry toolkit!

Parallelogram Definition and Properties

Properties of parallelograms

• A parallelogram is a quadrilateral with two pairs of parallel sides
  • Opposite sides are parallel to each other (top and bottom sides, left and right sides)
• Opposite sides are congruent meaning they have the same length (top and bottom sides, left and right sides)
• Opposite angles are congruent meaning they have the same measure (top left and bottom right angles, top right and bottom left angles)
• Consecutive angles are supplementary meaning they add up to 180° (any two adjacent angles in the parallelogram)
• Diagonals bisect each other meaning they intersect at their midpoints (the point where the diagonals cross is the midpoint of each diagonal)

Theorems for parallelogram properties

• Theorem: Opposite sides of a parallelogram are congruent
  • Proven using the Alternate Interior Angles Theorem and the Congruent Triangle Theorem (ASA or AAS)
• Theorem: Diagonals of a parallelogram bisect each other
  • Proven using the Alternate Interior Angles Theorem and the Congruent Triangle Theorem (ASA or SSS)

Applications of parallelogram properties

• The properties of parallelograms can be applied to solve for unknown side lengths and angle measures
  • If one side length is known, the opposite side has the same length (if the top side is 5 cm, the bottom side is also 5 cm)
  • If one angle measure is known, the opposite angle has the same measure (if the top left angle is 70°, the bottom right angle is also 70°)
  • If two consecutive angles are known, the remaining angles can be found using the supplementary property (if two adjacent angles are 50° and 130°, the other two angles must also be 50° and 130°)
• The properties of diagonals in parallelograms divide the parallelogram into four congruent triangles
  • The point of intersection of the diagonals is the midpoint of each diagonal (if the diagonals are 8 cm each, they intersect at the 4 cm mark)

Construction of parallelograms

• Parallelograms can be constructed given specific conditions:
  1. Two adjacent sides and the included angle (a side, the angle next to it, and the side adjacent to that angle)
  2. Two opposite sides and one angle (top and bottom side lengths and one of the angles)
  3. One side and two adjacent angles (a side length and the measures of the two angles next to it)
• Steps for constructing a parallelogram:
  1. Draw one side of the parallelogram
  2. Use a compass and straightedge to construct parallel lines and mark off the given lengths or angles
  3. Connect the vertices to complete the parallelogram
  4. Verify that the constructed figure satisfies the properties of a parallelogram (opposite sides parallel and congruent, opposite angles congruent, consecutive angles supplementary, diagonals bisect each other)