Mathematics Notes on Quadrilaterals

QUADRILATERALS

• A quadrilateral is defined as a polygon with four sides.
• General quadrilaterals have no parallel sides.

TYPES OF QUADRILATERALS

• Parallelogram: A quadrilateral with two pairs of parallel sides.
  • Examples include quadrilaterals ABCD and KNML.

KINDS OF PARALLELOGRAMS

• Rectangle:

  • A parallelogram with four right angles.
  • Opposite sides are congruent.

• Square:

  • A parallelogram with four right angles and four equal sides.
  • Example: ABCD is a square.

• Rhombus:

  • A parallelogram with four equal sides.
  • Opposite angles are congruent.
  • Example: ABCD is a rhombus.

TRAPEZOID AND KITE

• Trapezoid:

  • A quadrilateral with exactly one pair of parallel sides.
  • If non-parallel sides are congruent, it is an isosceles trapezoid.

• Kite:

  • A quadrilateral with two distinct pairs of congruent sides.

HIERARCHY OF QUADRILATERALS

• Quadrilaterals can be categorized as:
  • Parallelogram
  • Rectangle
  • Square
  • Rhombus
  • Trapezoid
  • Isosceles Trapezoid
  • Kite

INTERIOR ANGLES OF QUADRILATERAL

• The sum of the interior angles of a quadrilateral is 360^ ext{o}.

EXAMPLES

• Example 1: Solve for the value of x in the polygon HOME where:
  • H: 80^ ext{o}
  • E: 70^ ext{o}
  • M: x
  • O: 2x
  • Equation: 80 + 70 + x + 2x = 360 \
    150 + 3x = 360 \
    3x = 210 \
    x = 70
• Further angles are calculated as follows: - ZM = 2(70) = 140^ ext{o}
  • LE = 70^ ext{o}.

PROPERTIES OF PARALLELOGRAM

• Opposite sides are congruent and parallel.
  • AB \cong DC
  • BC \parallel AD
• Opposite angles are congruent. Diagonals bisect each other.
• Consecutive angles are supplementary:
  • LA + LB = 180^ ext{o}

EXAMPLES

• Example 1: Given parallelogram ABCD, find the lengths:
  • Example calculations show congruence in sides (DC = 8 ext{ cm} and EB = 6 ext{ cm}), leading to DB = EB + DE = 12 ext{ cm}.

RECTANGLE, SQUARE, AND RHOMBUS PROPERTIES

• Rectangle:
  • Opposite sides are congruent and parallel with all angles as right angles. Diagonals are congruent.
• Square:
  • All properties of rectangle and rhombus combined.
• Rhombus:
  • All sides are equal and opposite angles congruent. Diagonals are perpendicular and bisect each angle.

EXAMPLES IN RECTANGLES AND RHOMBUSES

• Example 1 for Rhombus:
  • 2y + 3 = 5y - 6 solving yields lengths and angle measures.
• Example 2 for Rectangle:
  • Given dimension relations, substituted values yield the length and measure.

TRAPEZOID PROPERTIES

• Trapezoid: Two parallel sides known as bases.
• Isosceles Trapezoid: Has congruent non-parallel sides (legs).
• Base angles are congruent; leg angles are supplementary.

EXAMPLES IN TRAPEZOIDS

• Example calculations relating angle measures follow properties of congruencies.

KITE PROPERTIES

• A kite features two distinct pairs of consecutive congruent sides.
• Its diagonals are perpendicular, and one diagonal bisects the angles.
• Example: Find the missing angles through equation setups based on congruencies and supplementary angle rules.

MIDSEGMENT OF A TRAPEZOID

• The midsegment runs parallel to both bases and is calculated as an average of the bases’ lengths.
• Example: Calculation yields midsegment lengths with substituted expressions.

SUMMARY OF QUADRILATERALS

• Detailed analysis reveals the properties of various quadrilateral types and their interrelations, with emphasis on congruencies, parallelism, and angle measures.