Download Unit 8

Geometry
Polygons
Lesson 8-4: Parallelograms, Trapezoids and Kites proofs
Objectives:
 Students will be able apply definitions and theorems all
parallelograms and kites and trapezoid proofs.
Procedure:
 HW Discussion
 Quiz: polygon formulas
 Explore the kite (Math open reference)
 Note organizer: problems/example proofs
 Practice proofs
Homework: All quad proofs WS 8-4
Mathopenref.com Quadrilaterals
Kite
Definition: A
QUADRILATERAL WITH 2 DISTINCT PAIRS
OF CONSECUTIVE CONGRUENT SIDES
Draw AND Label diagram of kite: (include diagonals)
A kite is a member of the quadrilateral family, and while easy to understand visually, is a little tricky to
define in precise mathematical terms. It has two pairs of equal sides. Each pair must be adjacent sides
(sharing a common vertex) and each pair must be distinct. That is, the pairs cannot have a side in
common.
Properties of a kite:
1. Diagonals – are perpendicular
-One diagonal bisects the other
2. Angles – angles between the unequal sides are equal
3. Area -
d1 d 2
2
where d1 and d2 are the diagonals
4.Perimeter - the sum of the sides
Geometry Lesson 8-4: parallelograms, trapezoids, kites
Ways to prove each quadrilateral!
Rectangle
Rhombus
Square
If a parallelogram contains
a right angle, then it is a
rectangle.
If a quadrilateral is
equiangular, then it is a
rectangle.
If a parallelogram is a
rhombus, then it has two
consecutive congruent
sides.
If a quadrilateral is
equilateral, then it is a
rhombus.
If a rectangle is a square,
then it has
two consecutive
congruent sides
If a rhombus is a square,
then it has
a right angle
If the diagonals of a
parallelogram are
congruent,
then the parallelogram is a
rectangle.
If the diagonals of a
parallelogram are
perpendicular to each other,
then the parallelogram is a
rhombus.
Area= s2
Area = length x width
Area=
d1 d 2
2
Trapezoid
Kite
If a quadrilateral is a trapezoid then it
has exactly two parallel sides.
If a quadrilateral is a kite then it has two pairs
of distinct adjacent congruent sides
Properties:
b1  b2 
h
 2 
Area= 
Right trapezoid
If a trapezoid is a right trapezoid then
it has a right angle.
Isosceles trapezoid
A trapezoid is isosceles if and only if
the diagonals are congruent
If a trapezoid is an isosceles trapezoid
then the nonparallel sides are
congruent.
If a trapezoid is an isosceles trapezoid
then its opposite angles which are
supplementary.
A trapezoid is isosceles if and only if
the base angles are congruent.


Diagonals are perpendicular
angles between the unequal sides are
equal
d1 d 2
Area:
2
Median of a trapezoid
The median (also called the mid-segment)
of a trapezoid is a segment that connects the
midpoint of one leg to the midpoint of the
other leg.
;
Theorem: The median (or mid-segment) of
a trapezoid is parallel to each base and its
length is one half the sum of the lengths of
the bases.
(True for ALL trapezoids.)
Examples:
1) EF is the median (mid-segment) of trapezoid ABCD.
EF = 25 and AD = 40. Find BC.
40  x
2
50  40  x
25 
x  10
BC  10
2)
2 x  3  2 x  11
2
28  4 x  8
4 x  20
x5
PQ  21
14 
Example proofs:
Statements
Reasons
1.
1. Given
2. AM  BM
2. Def. of midpoint
3. A  B
3. Rectangles are equiangular.
4. DA  CB
5. DAM  CBM
4. Rectangles have opposite
congruent sides.
5. SAS  SAS
6. DM  CM
6.CPCTC
Statements
Reasons
1.
1. Given
2. EB  CB
2. In a triangle, if 2 angles are equal,
their opposites sides are equal
3. AB  CB
4. ABCD is a rhombus
3. Substitution
4. A parallelogram with two
consecutive congruent sides is a
rhombus.
Given: LORI is a rectangle
BIRD is a trapezoid with bases BD and IR
LD  OB
Prove: BIRD is an isosceles trapezoid
Statements
Reasons
1. LORI is a rectangle
LD  OB ;
BIRD is a trapezoid
1. Given
2. L  O
2. A rectangle is equiangular.
3. LI  OR
3. Opposite sides of a rectangle are congruent.
4. LB + BD = LD; BD + DO = BO
4. Partition
5. LB + BD = BD + DO
5. Substitution
6. BD = BD
6. Reflexive
7. LB = OD
7. Subtraction
8. BLI  DOR
9. BI  DR
10. BIRD is an isosceles trapezoid
8. SAS  SAS
9.CPCTC
10. A trapezoid with nonparallel congruent sides is
isosceles.
Given: PQRS is a kite.
Prove: QTR  STR
Statements
Reasons
1. PQRS is a kite.
1. Given
2. PR  QS
2. Kites have perpendicular diagonals
3. QTR  STR
3. Right angles are congruent
4. RT  RT
4. Reflexive
5. ST  QT
5. The diagonal that joins the non-congruent
sides of a kite is bisected by the other
diagonal
6. SAS  SAS
6. QTR  STR