Chapter 8.notebook
name
# of
sides
December 10, 2014
diagonals
sum of
total
from 1
diagonals interior
vertex
angles
sum of each ind each ind
exterior int angle ext angle
angles for reg for reg
3
4
5
6
7
8
9
10
n­gonn
Diagonal of a polygon: segment joining 2 non­consecutive vertices of the polygon.
(A segment joining two vertices that is not a side)
Chapter 8.notebook
December 10, 2014
Outline
Quadrilateral=4 sided polygon
A.) Parallelogram: Quad w/ 2 pairs of // sides
1.) Rectangle: Par. w/ 4 rt angles
a.) Square: Rect. with 4 = sides
2.) Rhombus: Par w/ 4 = sides
a.) Square: Rhombus w/ 4 rt angles
B.) Trapezoid: Quad w/ 1 pair of // sides
1.) Isosceles Trapezoid: Trap. with one pair of
= sides
C.) Kite: Quad w/ 2 pairs of congruent consecutive sides.
Q
P
T
R S H
I
K
Q=Quadrilaterals
P=Parallelograms
R=Rectangles
H =rHombus
S=Square
T=Trapezoid
I = Isosceles Trapezoid
K = Kite
Chapter 8.notebook
December 10, 2014
A
B
D
Given: ABCD is a
C
What we Know
Reason
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
And 2 more......
A
B
Given: ABCD is a D
What we Know
11:
12:
C
Reason
Chapter 8.notebook
December 10, 2014
18
16
770
Given: This is a x + 80
4x + 1
2x + 9
5x + 20
22 ­ x
3x + 12
7x + 38
3x + 2
Given: Each figure shown is a
Chapter 8.notebook
December 10, 2014
19
2
0
1 54
1200
3
4
28
e
c
f
18
10
11
12
13
a
15
d
9
7
8
6
330
5
b
Given: this is a Specifics About Special
RECTANGLE: A rectangle is a parallelogram
All Facts about Parallelograms are true about rectangles!!!
+
Definition of Rectangle: All 4 main angles are 900
Theorem: Diagonals of a rectangle are congruent
Extra Fact: The diagonals of a rectangle form 4 isosceles triangles
Chapter 8.notebook
December 10, 2014
Given: This is a rectangle
a
1
67o
2
5
e
12
15
6
14
10
4
5
f
11
3
g
d
13
b
c
8
7
9
6
8
Specifics About Special
RHOMBUS: A rectangle is a parallelogram
All Facts about Parallelograms are true about rhombuses!!!
+
Definition of Rhombus: All 4 sides are congruent
Theorem: Diagonals of a rhombus are perpendicular
Extra Fact: The diagonals of a rectangle form 4 congruent triangles
Extra Extra Fact: Each diagonal of a rhombus bisects the angles of the rhombus
Chapter 8.notebook
December 10, 2014
1 2
3
25
a
e
g
730 11
10
7
48
12 13
15 14
6
d
4
5
f
b
c
9
8 7
Specifics About Special
SQUARE
All Facts about Parallelograms are true about squares!!!
+
All Facts about Rectangles are true about squares
+
All Facts about Rhombuses are true about squares
Chapter 8.notebook
December 10, 2014
a
1
4
2
5
3
g
6
9 2
d
h
14
13
18
16
b
15
e
f
9
12
11
8
10
c
OTHER Quadrilaterals
Trapezoid
base
leg
leg
base
A segment joining the midpoints of the legs of a trapezoid is known as the midsegment of the trapezoid
7
Chapter 8.notebook
December 10, 2014
Midsegment Theorem for Trapezoids: The midsegment of a trapezoid is parallel to each base and has length equal to one half the sum of the lengths of the two bases
Given: MN is a midsegment of trapezoid ABCD
A
10
B
1130
4
M
14
N
720
D
22
C
Chapter 8.notebook
December 10, 2014
An isosceles trapezoid has congruent legs.
Isosceles Trapezoid Theorems
Theorem 8.14: If a trapezoid is isosceles then each pair of base angles is congruent
Theorem 8.15: If a trapezoid has a pair of congruent base angles then it is isosceles
Theorem 8.16: A trapezoid is isosceles, if and only if, its diagonals are congruent.
Chapter 8.notebook
December 10, 2014
Given: This is an isosceles trapezoid
8
0
1 37
2
3
b
5
12
11 13
14
d
4
5
e
a
13
c
11
840
10
8
9
6
7
21
Another special Quadrilateral: A Kite, is a quadrilateral with 2 pairs of congruent consecutive sides.
Chapter 8.notebook
December 10, 2014
Theorems about Kites:
*Diagonals of a Kite are perpendicular
*On pair of opposite angles of a Kite are congruent (the angles formed by non­congruent consecutive sides)
* One diagonal of a Kite bisects the angles of the kite (the angles bisected are the ones formed by congruent sides)
* The diagonals of a Kite form 2 pairs of congruent triangles
13
210
9 10
12
1 2
3
5
11 12
13 14
d
b
a
4
6
c
e
16
20
480
8 7
5
Chapter 8.notebook
December 10, 2014
If these two triangles are congruent, then the shape is a parallelogram
What information would make the two triangles congruent?
Th 8.7
Th 8.8
TH 8.9
Plus, one more Th 8.10
Ways to prove a quadrilateral is a parallelogram
1.) Show both pairs of opposite sides are parallel (Definition of parallelogram)
2.) Show both pairs of opposite sides are congruent (Theorem 8.7)
3.) Show both pairs of opposite angles are congruent (Theorem 8.8)
4.) Show one pair of opposite sides are congruent and parallel (Theorem 8.9)
5.) Show the diagonals bisect each other (Theorem 8.10)
Chapter 8.notebook
December 10, 2014
A
D
Given:
~
~
1.) AB = CD and AD = BC
~
2.) AB = CB and AB // DC
~
3.) AB // DC and AD = BC
~
4.) AD // BC and AD = BC
5.) AB // DC and AD // BC
B
C
Is ABCD a ?