Learning Objectives
- Understand properties of different types of quadrilaterals
- Learn properties of parallelograms
- Prove and apply the Mid-Point Theorem
- Establish conditions for a quadrilateral to be a parallelogram
Key Concepts
Angle Sum Property
The sum of the interior angles of a quadrilateral is 360°.
Types of Quadrilaterals
- Parallelogram: Both pairs of opposite sides are parallel.
- Rectangle: A parallelogram with all angles 90°.
- Rhombus: A parallelogram with all sides equal.
- Square: A parallelogram with all sides equal and all angles 90°.
- Trapezium: One pair of opposite sides is parallel.
- Kite: Two pairs of adjacent sides are equal.
Properties of a Parallelogram
- Opposite sides are equal and parallel.
- Opposite angles are equal.
- Consecutive (adjacent) angles are supplementary.
- Diagonals bisect each other.
Conditions for a Parallelogram
A quadrilateral is a parallelogram if any one of these holds:
- Both pairs of opposite sides are equal.
- Both pairs of opposite angles are equal.
- Diagonals bisect each other.
- One pair of opposite sides is both equal and parallel.
Special Properties
Rectangle: Diagonals are equal and bisect each other.
Rhombus: Diagonals bisect each other at right angles.
Square: Diagonals are equal and bisect each other at right angles.
Mid-Point Theorem
Theorem: The line segment joining the mid-points of two sides of a triangle is parallel to the third side and is half of it.
Converse: A line drawn through the mid-point of one side of a triangle, parallel to another side, bisects the third side.
Summary
Quadrilaterals have an angle sum of 360°. Parallelograms have several special properties including equal opposite sides and angles, and diagonals that bisect each other. Rectangles, rhombuses, and squares are special parallelograms with additional properties. The Mid-Point Theorem connects midpoints of triangle sides to properties of parallel lines.
Important Terms
- Parallelogram: Quadrilateral with both pairs of opposite sides parallel
- Diagonal: A line segment connecting two non-adjacent vertices
- Mid-Point Theorem: Line joining midpoints of two sides of a triangle is parallel to and half of the third side
- Trapezium: Quadrilateral with exactly one pair of parallel sides
Quick Revision
- Angle sum of quadrilateral = 360°
- Parallelogram: opposite sides equal and parallel, diagonals bisect each other
- Rectangle diagonals are equal; Rhombus diagonals are perpendicular
- Square has properties of both rectangle and rhombus
- Mid-Point Theorem: midpoint segment is parallel to and half the third side