1. Advanced Geometry
5.4 / 5 Four Sided Polygons /  
 Properties of Quadrilaterals
Learner Objective: I will solve problems using properties 
 of special quadrilaterals.

2. 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.

3. Quadrilateral: A four sided Polygon
Special Quadrilaterals
Parallelogram:
 Definition: A Parallelogram is a Quadrilateral in which both pairs of 
 opposite sides are parallel.
Properties:
 Opposite sides are congruent
 Opposite angles are congruent
 Diagonals bisect each other
Any pair of consecutive angles are supplementary

4. Rectangle:
 Definition: A Rectangle is a Parallelogram in which at least one angle is a 
 right angle.
 Properties:
 All properties of a Parallelogram apply by definition
 All angles are right angles
 The diagonals are congruent

5. Rhombus:
 Definition: A Rhombus is a Parallelogram in which at least two consecutive 
 sides are congruent.
 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

6. Square:
 Definition: A Square is a Parallelogram that is both a Rectangle and a 
 Rhombus.
 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)

7. Kite:
 Definition: A Kite is a quadrilateral in which two disjoint pairs of consecutive 
 sides are congruent.
 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

9. Trapezoid:
 Definition: A Trapezoid is a quadrilateral with exactly one pair of parallel 
 sides. The parallel sides are called bases of the trapezoid.

10. Isosceles Trapezoid:
 Definition: An Isosceles Trapezoid is a trapezoid in which the nonparallel 
 sides (legs) are congruent.
 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

12. _______ rhombuses are squares _______ squares are rhombuses
quadrilateral
parallelogram
trapezoid
isosceles trapezoid
rectangle
square
rhombus
kite
kite
kite
Some
All No
_______ rhombuses are squares
_______ squares are rhombuses
_______ rhombuses are kites
_______ trapezoids are rectangles
_______ rectangles are parallelograms
_______ kites are rhombuses
_______ trapezoids are parallelograms
_______ rectangles are rhombuses
_______ squares are rectangles
_______ parallelograms are quadrilaterals
_______ isosceles trapezoids are parallelograms

22. STATEMENTS
REASONS