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Advanced Geometry
5.4 / 5 Four Sided Polygons /
Properties of Quadrilaterals
Learner Objective: I will solve problems using properties of special
quadrilaterals.
Polygon
Plane Figure
Consists only of segments
Consecutive sides intersect only at endpoints
Non-Consecutive sides do not intersect
Each vertex belongs to exactly two sides
Convex Polygon
A polygon in which each interior angle has a measure less than 180.
Concave Polygon
A polygon in which at least one interior angle has a measure greater than 180.
Diagonal of a Polygon
Any segment that connects two nonconsecutive (nonadjacent) vertices of the polygon is a Diagonal of the
Polygon.
Quadrilateral: A four sided Polygon
Special Quadrilaterals
Parallelogram:
Definition: A Parallelogram is a Quadrilateral in which both pairs of
Properties:
Opposite sides are congruent
Opposite angles are congruent
Diagonals bisect each other
Any pair of consecutive angles are supplementary
opposite sides are parallel.
Rectangle:
Definition: A Rectangle is a Parallelogram in which at least one angle is a
Properties:
All properties of a Parallelogram apply by definition
All angles are right angles
The diagonals are congruent
right angle.
Rhombus:
Definition: A Rhombus is a Parallelogram in which at least two consecutive
Properties:
All properties of a Parallelogram apply by definition
All sides are congruent
Diagonals are perpendicular bisectors of each other
Diagonals bisect the angles
Diagonals divide the rhombus into four congruent right triangles
sides are congruent.
Square:
Definition: A Square is a Parallelogram that is both a Rectangle and a
Properties:
All properties of a rectangle apply by definition
All properties of a rhombus apply by definition
The diagonals form four isosceles right triangles (45⁰-45⁰-90⁰)
Rhombus.
Kite:
Definition: A Kite is a quadrilateral in which two disjoint pairs of consecutive
Properties:
Diagonals are perpendicular
One diagonal is the perpendicular bisector of the other
One diagonal bisects a pair of opposite angles
One pair of opposite angles are congruent
sides are congruent.
Trapezoid:
Definition: A Trapezoid is a quadrilateral with exactly one pair of parallel
the trapezoid.
sides. The parallel sides are called bases of
Isosceles Trapezoid:
Definition: An Isosceles Trapezoid is a trapezoid in which the nonparallel
Properties:
Legs congruent by definition
Bases parallel by definition
Lower base angles congruent
Upper base angles congruent
Diagonals are congruent
Any lower base angle is supplementary to any upper base angle
sides (legs) are congruent.
quadrilateral
parallelogram
trapezoid
isosceles trapezoid
rectangle
square
rhombus
kite
kite
kite
Some
All
No
_______ trapezoids are parallelograms
_______ rectangles are rhombuses
_______ squares are rectangles
_______ parallelograms are quadrilaterals
_______ isosceles trapezoids are parallelograms
_______ rhombuses are squares
_______ squares are rhombuses
_______ rhombuses are kites
_______ trapezoids are rectangles
_______ rectangles are parallelograms
_______ kites are rhombuses
STATEMENTS
REASONS