## Tuesday, July 24, 2007

### Geometry Chapter 6.2 - 6.6 Properties of Quadrilaterals

I) 6.2 Properties of Parallelograms
A) Parallelogram: A quadrilateral with both pairs of opposite sides parallel.
B) theorems for Parallelograms:
If a quadrilateral is a parallelogram, then
1. both pairs of opposite sides are congruent.
2. both pairs of opposite angles are congruent.
3. both pairs of consecutive angles are supplementary.
4. the diagonals bisect each other.
5. a diagonal bisects the parallelogram into 2 congruent triangles
6. if one pair of opposite sides are parallel and congruent.

II) 6.3 Proving Quadrilaterals are Parallelograms:
1. Show that both pairs of opposite sides are parallel.
2. Show that both pairs of opposite sides are congruent.
3. Show that both pairs of opposite angles are congruent.
4. Show that one angle is supplementary to both consecutive angles.
5. Show that the diagonals bisect each other.
6. Show that one pair of opposite sides are congruent and parallel.

III) 6.4 Rhombuses, Rectangle, and Squares
A) Rhombus: A parallelogram with 4 congruent sides
1) the diagonals are perpendicular
2) the diagonals bisect opposite angles
B) Rectangle: A parallelogram with 4 congruent right angles.
1) the diagonals are congruent
C) Square: a parallelogram with 4 congruent sides and 4 congruent right angles.

IV) 6.5 Trapezoids and Kites
A) Trapezoid: A quadrilateral with exactly one pair of parallel sides called the bases. It has 2 pairs of base angles. The nonparallel sides are called the legs.
B) Isosceles Trapezoid: A trapezoid that has:
1) nonparalel sides are congruent
2) base angles are congruent
3) diagonals are congruent.
C) Kite: a quadrilateral with 2 pairs of consecutive congruent sides but the opposite sides are not congruent.
1) the diagonals are perpendicular
2) it has exactly one pair of opposite angles congruent. (the pair of angles that is between the noncongruent consecutive sides)
D) Midsegment of a trapezoid: the midsegment of a trapezoid connects the midpoints of the legs and is parallel to both bases. Its length is the average of the 2 bases.
Example: given trapezoid ABCD with BC ll AD and midsegment MN where M is the midpoint of AB and N is the midpoint of CD, then we know:

A) Quadrilateral - 4 sided polygon

1) Kite
2) Parallelogram
a) Rhombus
b) Rectangle
i) Square
3) Trapezoid
a) Isosceles Trapezoid

VI) Areas of Triangles and Quadrilaterals:
A) Rectangles, Square,Parallelogram
A = (base)(height) where the base and height must be perpendicular
B) Triangle
A = (1/2)(base)(height) where the base and height must be perpendicular
C) Trapezoid
A = (1/2)h (B1 + B2)
D) Rhombus and Kite
A =(1/2)( d1 )(d2 )