Quadrilaterals: Lesson 1

Quadrilaterals

• A quadrilateral is a four-sided polygon.
• Parts of a polygon include sides, vertices, angles, and diagonals.

Types of Quadrilaterals

• Parallelogram: a quadrilateral with two pairs of parallel sides.
• Trapezoid: a quadrilateral with one pair of parallel sides.
  • Parts of a trapezoid include parallel sides (bases), non-parallel sides (legs), and base angles.
• Isosceles trapezoid: a trapezoid with congruent legs.
• Kite: a quadrilateral with two distinct pairs of congruent and adjacent sides.
• Rectangle: a parallelogram with four right angles.
• Rhombus: a parallelogram with four congruent sides.

Properties of Parallelograms

• A diagonal forms two congruent triangles.
• Any two opposite angles are congruent.
• Any two consecutive angles are supplementary.
• Any two opposite sides are congruent.
• The diagonals bisect each other.

Triangle Congruence

• Ways to prove a theorem include:
  • A statement of the theorem
  • A drawing under consideration
  • A statement of what is given
  • A statement of what is to be proved
  • An orderly arrangement of statements and reasons
• Triangle congruence theorems include SSS, SAS, ASA, and AAS.

Properties of Special Parallelograms

• Conditions guaranteeing that a quadrilateral is a parallelogram include:
  • Congruent opposite sides
  • One pair of congruent and parallel opposite sides
  • Congruent opposite angles
  • Bisecting diagonals
• Properties of rectangles include:
  • Congruent diagonals
  • At least one right angle
• Properties of rhombuses include:
  • Perpendicular diagonals
  • Bisecting diagonals
  • Each diagonal bisects the opposite angles
• Properties of squares include:
  • Bisecting diagonals that bisect the vertex angle

Properties of Trapezoids

• Properties of isosceles trapezoids include:
  • Congruent base angles
  • Congruent diagonals
• The median of a trapezoid is parallel to its bases and is half of the sum of the lengths of the bases.

Ratio and Proportion

• A ratio is a comparison between two positive quantities, expressed as a:b or a/b.
• A proportion is a statement of equality between two ratios.
• The first and fourth terms of a proportion are called the extremes, while the second and third terms are called the means.

Triangle Proportionality

• The Basic Proportionality Theorem states that if a line parallel to one side of a triangle intersects the other two sides in distinct points, then it divides the sides proportionally.

Similar Figures

• Correspondence refers to the relationship between similar parts of two polygons.
• Similar polygons have congruent corresponding angles and proportional corresponding sides.
• Similar triangles have congruent corresponding angles and proportional corresponding sides.
• Similarity theorems include the AAA Similarity Theorem.