UNDERSTANDING QUADRILATERALS

Curves are fundamental shapes in geometry. They can be classified based on how they are drawn and their structure.

Types of Curves

• Open Curve: A curve that does not end at its starting point. It has distinct start and end points.
• _Example:_ A letter 'C' or 'U'.
• Closed Curve: A curve that starts and ends at the same point, forming a complete loop.
• _Example:_ A circle, a square, a triangle.

• Simple Curve: A curve that does not cross itself.
• _Example:_ A circle, a straight line, a letter 'S' (if drawn without crossing).
• Non-Simple Curve: A curve that crosses itself.
• _Example:_ A figure-eight shape, a loop-de-loop.

• Simple Closed Curve: A closed curve that does not cross itself. These are the basis for polygons.
• _Example:_ Triangle, square, circle.
• Not Simple Closed Curve: A closed curve that crosses itself.

Polygons

• A polygon is a simple closed curve made up entirely of line segments.
• Vertices: The points where two line segments (sides) meet.
• Sides: The line segments forming the polygon.
• Adjacent Sides: Any two sides with a common endpoint.
• Adjacent Vertices: The endpoints of the same side.

Classification of Polygons (Based on Number of Sides)

| Number of Sides | Name of Polygon | |:---------------:|:---------------:| | 3 | Triangle | | 4 | Quadrilateral | | 5 | Pentagon | | 6 | Hexagon | | 7 | Heptagon | | 8 | Octagon | | 9 | Nonagon | | 10 | Decagon | | n | n-gon |

Regular vs. Irregular Polygons

• Regular Polygon: A polygon that is both equiangular (all angles are equal) and equilateral (all sides are equal).
• _Example:_ Square, equilateral triangle, regular hexagon.
• Irregular Polygon: A polygon that is not regular (either its sides are not equal, or its angles are not equal, or both).
• _Example:_ Rectangle (equiangular but not always equilateral), rhombus (equilateral but not always equiangular), scalene triangle.

Convex vs. Concave Polygons

• Convex Polygon: A polygon where all interior angles are less than 180°. All diagonals lie entirely inside the polygon.
• _Example:_ Square, triangle, regular hexagon.
• Concave Polygon: A polygon where at least one interior angle is greater than 180° (a reflex angle). At least one part of a diagonal lies outside the polygon.
• _Example:_ A star shape, an arrowhead shape.

What are Diagonals?

• A diagonal is a line segment that connects two non-consecutive vertices of a polygon.

Formula for Number of Diagonals

• For a polygon with 'n' sides (or 'n' vertices), the number of diagonals is given by the formula:

\( \text{Number of diagonals} = \frac{n(n-3)}{2} \)

Explanation of the Formula:

1. From each vertex, you can draw a line segment to \(n-1\) other vertices.
2. However, you cannot draw a diagonal to itself or to its two adjacent vertices. So, \(n-3\) diagonals can be drawn from each vertex.
3. Since there are 'n' vertices, the total would seem to be \(n(n-3)\).
4. But this counts each diagonal twice (e.g., diagonal from A to C is the same as C to A). So, we divide by 2.

Examples:

• Triangle (n=3): \( \frac{3(3-3)}{2} = \frac{3 \times 0}{2} = 0 \) diagonals. (Correct, a triangle has no diagonals).
• Quadrilateral (n=4): \( \frac{4(4-3)}{2} = \frac{4 \times 1}{2} = 2 \) diagonals. (Correct, a quadrilateral has two diagonals).
• Pentagon (n=5): \( \frac{5(5-3)}{2} = \frac{5 \times 2}{2} = 5 \) diagonals.
• Hexagon (n=6): \( \frac{6(6-3)}{2} = \frac{6 \times 3}{2} = 9 \) diagonals.

Interior Angle Sum Property

• The sum of the interior angles of a polygon depends on the number of its sides.
• We can divide any polygon into triangles by drawing diagonals from one vertex.
• A polygon with 'n' sides can be divided into \((n-2)\) triangles.
• Since the sum of angles in a triangle is \(180^\circ\), the sum of interior angles of an n-sided polygon is \((n-2) \times 180^\circ\).

Formula:

• Sum of interior angles of an n-sided polygon = \((n-2) \times 180^\circ\)

Examples:

• Triangle (n=3): \((3-2) \times 180^\circ = 1 \times 180^\circ = 180^\circ\)
• Quadrilateral (n=4): \((4-2) \times 180^\circ = 2 \times 180^\circ = 360^\circ\)
• This is a very important property for quadrilaterals!
• Pentagon (n=5): \((5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ\)
• Hexagon (n=6): \((6-2) \times 180^\circ = 4 \times 180^\circ = 720^\circ\)

Each Interior Angle of a Regular Polygon

• For a regular n-sided polygon, all interior angles are equal.
• Each interior angle = \( \frac{(n-2) \times 180^\circ}{n} \)

Exterior Angle Sum Property

• The sum of the measures of the exterior angles of any convex polygon (one at each vertex) is always \(360^\circ\).
• This is a remarkable property as it holds true for any convex polygon, regardless of the number of sides.

Each Exterior Angle of a Regular Polygon

• For a regular n-sided polygon, all exterior angles are equal.
• Each exterior angle = \( \frac{360^\circ}{n} \)

Relationship between Interior and Exterior Angles

• At each vertex, the interior angle and its corresponding exterior angle form a linear pair (they lie on a straight line).
• Therefore, Interior Angle + Exterior Angle = \(180^\circ\).
• This relationship is useful for finding one if the other is known.

A quadrilateral is a polygon with four sides, four angles, and four vertices. The sum of its interior angles is \(360^\circ\).

Hierarchy of Quadrilaterals

• Quadrilateral is the parent category.
• Specific types are defined by additional properties.

1. Trapezium (or Trapezoid)

• Definition: A quadrilateral with at least one pair of parallel sides.
• Properties:
• One pair of opposite sides is parallel (called bases).
• The non-parallel sides are called legs.
• Consecutive angles between parallel sides are supplementary (sum to \(180^\circ\)).
• Isosceles Trapezium: A trapezium whose non-parallel sides are equal in length.
• Base angles are equal.
• Diagonals are equal.

2. Kite

• Definition: A quadrilateral where two pairs of adjacent sides are equal in length.
• Properties:
• Two distinct pairs of equal-length adjacent sides.
• One pair of opposite angles are equal (the angles between the unequal sides).
• Diagonals are perpendicular to each other.
• One diagonal bisects the other diagonal.
• One diagonal bisects the angles at the vertices it connects.

3. Parallelogram

• Definition: A quadrilateral where both pairs of opposite sides are parallel.
• Properties:
• Opposite sides are parallel (AB || DC, AD || BC).
• Opposite sides are equal in length (AB = DC, AD = BC).
• Opposite angles are equal (∠A = ∠C, ∠B = ∠D).
• Consecutive angles are supplementary (sum to \(180^\circ\)).
• Diagonals bisect each other (they cut each other into two equal parts).

4. Rhombus

• Definition: A parallelogram with all four sides equal in length.
• Properties: (Inherits all properties of a parallelogram, plus these)
• All four sides are equal (AB = BC = CD = DA).
• Opposite angles are equal.
• Diagonals bisect each other at right angles (90°).
• Diagonals bisect the angles at the vertices.

5. Rectangle

• Definition: A parallelogram with all four angles equal to \(90^\circ\).
• Properties: (Inherits all properties of a parallelogram, plus these)
• All four angles are \(90^\circ\).
• Opposite sides are equal and parallel.
• Diagonals are equal in length.
• Diagonals bisect each other.

6. Square

• Definition: A quadrilateral that is both a rhombus and a rectangle.
• Properties: (Inherits all properties of a parallelogram, rhombus, and rectangle)
• All four sides are equal.
• All four angles are \(90^\circ\).
• Opposite sides are parallel.
• Diagonals are equal in length.
• Diagonals bisect each other at right angles.
• Diagonals bisect the angles at the vertices (each angle is divided into two \(45^\circ\) angles).

Summary of Quadrilateral Properties