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Chapter 8: Exploring
Quadrilaterals
4 sided polygons
8.1 Polygons
Def of Polygon: Closed figure formed by a finite
number of coplanar segments such that:
-The sides that have a common
endpoint are noncollinear.
-Each side intersects exactly 2 other
sides at their endpoints
NOT CONVEX
No line containing a side
of the polygon contains a
point in the interior of the
polygon.
# of sides
3
4
5
6
7
8
9
10
11
12
13
14
•
•
•
•
•
•
•
•
•
•
•
•
•
Name
Regular Polygon
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Undecagon
Dodecagon
13-gon
14-gon
The pattern continues
A Polygon that
is
Both
equilateral
And equiangular
(Remember: If a triangle
Is equilateral, then it is
Equiangular. This only
Works for triangles.)
-The sum of the measures
of the angles in a triangle is?
-Look at the quadrilateral, how
many triangles are formed?
-So, what is the sum of the
measures of the angles in a
quadrilateral?
-Look at the pentagon, how many
triangles are formed?
-So, what is the sum of the
measures of the angles in a
pentagon?
Do you see a
pattern? How
does the number
of triangles
relate to the
number of sides?
Can you write a
formula?
-Look at the hexagon, how many
triangles are formed?
-So what is the sum of the measures of the angles in a hexagon?
1. Using a template to draw each of these
figures on a piece of paper.
2. Extend out the sides for each figure (like the
triangle.)
3. Measure each of the exterior angles that are
formed.
4. What can you conclude about the sum of the
exterior angles of a convex polygon, one
at each vertex?
Finding angle measures
# of sides
n
Sum of
interior <‘s
One interior <
(if regular)
(n – 2)180
(n – 2)180
sum of
One exterior <
exterior <‘s (if regular)
360
360
n
n
SAMPLES OF APPLYING THE FORMULAS
3
12
(3-2)180 = 180 (3-2)180= 60
3
(12-2)180 =1800 (12-2)180 =150
12
360
360
360=120
3
360=30
12
Examples
1. Find the sum of the measures of the
interior angles of a convex 26-gon.
2. Name the regular polygon with an
exterior angle measuring 45.
3. Find the measure of an interior angle and an
exterior angle for a regular 16-gon.
4. If the measure of an interior angle of a
regular polygon is 144, classify the polygon
according to the number of sides.
Try p. 518: 7 - 18
7. AB, BC, CD, DE, EF, FA
13. 135 and 45
8. Convex
14. 180(a-2) , 360
a
9. 1440
10. 3420
15. 6
11. 12
16. 12
12. 45
17. 34, 102,67,199,138
18. 72
Section 8.2
Parallelograms
Definition of Parallelogram:
A quadrilateral with both pairs of
opposite sides parallel.
II. PROPERTIES OF
PARALLELOGRAMS
Both pair of
opposite sides
are 
Parallelogram
Consecutive
angles are
supplementary
Def: Both pair
of opposite
sides are //
Both pair of
opposite <‘s
are 
Diagonals
bisect each
other
Examples
1. WXYZ is a parallelogram. Find the
M<ZWX = 100
indicated information.
3a + 5
W
X
c
16b -3
8b + 13
V
17
Z
7a -7
A. Find the value of a. 3
C. Find the value of c. 17
E. Find m<XYZ
100
Y
2 B. Find the value of b.
80 D. Find m<WZY
80 F. Find m<WXZ
2. Use the definition of a parallelogram to
determine if RSTV is a parallelogram.
S (3,6)
(1,1) R
To use the definition, we
must see if opposite sides
are parallel.
Slope RS = (6-1)/(3-1) = 5/2
Slope VT = (8-3)/(8-6) = 5/2
V (6, 3)
T
(8, 8)
Slope RV = (3-1)/(6-1) = 2/5
Slope ST = (8-6)/(8-3) = 2/5
Since RS // VT (have the same slope) and RV // ST (have the same
slope), RSTV is a parallelogram.
The Probability of an event
is the ratio of the number of favorable
outcomes to the total number of possible
outcomes.
Favorable Outcomes
Total Outcomes
3. Two sides of
ABCD are chosen
at random. What is the
probability that the two sides are
not congruent?
A
D
B
C
AB and BC not congruent
AB and DC are congruent
AB and AD not congruent
BC and DC not congruent
BC and AD are congruent
DC and AD not congruent
Not congruent/Total 4 /6
So the probability is 2/3.
4. Find the values of w, x, y, and z for the
parallelogram.
y
w
110
z
x
Answers:
w = 110 (opposite angles are congruent)
x = 70 (linear pair with w)
y = 70 (consecutive angles are supplementary)
z = 70 (opposite angles are congruent and consecutive
angles are supplementary)
5. Find the indicated values for the parallelogram
g
h
i
f
100
b
c
a
70
e
20
d
Answers
a = 100 b = 80
e = 30
f = 70
i = 30
c = 80
g = 20
d = 60
h = 60
6. Find all possible values for the 4th vertex of
the parallelogram 3 of the vertices are
(0,0), (4,4), and (8,0)
(12,4)
(-4,4)
(4,4)
(0,0)
(4, -4)
(8,0)
Answers to 6 – 14
__
6. HF
__
7. DC
8. <DFG
__
9. GF
10. <CDF and
<CGF
11. HDF
12. AB = CD so 2x + 5 = 21 and
2x = 16, so x = 8
Since <B = 120,
m<BAC + m<CAD = 60
2y + 21 = 60, then 2y = 39
and y = 19.5
13. m<Y = 47
m<X = 133
m<Z = 133
14. SLOPES
PT and QR = 5
QP and TR = -1 therefore opposite
sides are // and it is a parallelogram
8-3 TESTS FOR
PARALLELOGRAMS
I.
HOW DO YOU KNOW IF A
QUADRILATERAL IS A
PARALLELOGRAM?
PERFORM A
TEST
Opposite
sides are
parallel
(both pair)
Opposite
Sides are
Opposite
angles are
congruent
congruent
(both pair)
(both pair)
Diagonals The same
Bisect
Each
other
pair of
opposite
sides
 and //
II. EXAMPLES
 1.
IS CUTE A PARALLELOGRAM?
WHY?
C
U
62
T
118
118
62
E
YES, BECAUSE OPPOSITE ANGLES ARE
CONGRUENT AND CONSECUTIVE
ANGLES ARE SUPPLEMENTARY
 2.
Find x and y so that the
quadrilateral is a parallelogram.
3x + 17
4x-8
4
4x – 8 = 4
2y
So 3( 3) + 17 = 26
4 x = 12
26 = 2 y
X=3
13 = y
A
3. IS ABCD A
PARALLELOGRAM?
 A(5,6)
 AB=
 CD=
B
D
B(9,0) C(8, - 5) D (3, -2)
(9  5) 2  (0  6) 2  52
(3  8) 2  (2  5) 2  34
NO- OPPOSITE SIDES ARE NOT
CONGRUENT.
C
4. Determine if PZRD is a parallelogram.
P(-1,9)
D(2,3)
Z(3,8)
R(6,2)
a. SLOPES
slope PZ and DR = -1/4
slope PD and ZR = -2
It is a parallelogram because both
pair of opposite sides are //
b. DISTANCES
distance PZ and DR = 17
distance PD and ZR = 35
It is a parallelogram because both
pair of opposite sides are
congruent.
c. MIDPOINTS
DZ (2.5,5.5)
PR (2.5,5.5)
It is a parallelogram because the
diagonals bisect each other.
d. Slope PZ and DR = -1/4 and
Distance PZ and DR = 17.
Since the same pair of opposite
sides are parallel and
congruent it is a parallelogram.
e. Slope PD and ZR = -2 and
Distance PD and ZR = 35.
Since the same pair of opposite
sides are parallel and
congruent it is a parallelogram.
7. Since the triangles will be congruent by SAS, the other
pair of opposite sides will be congruent and it is a
parallelogram because both pair of opposite sides are
congruent.
8. No, the same pair are not congruent and parallel.
9. 6x = 4x + 8 so 2x = 8 and x = 4
y² = y so y² - y = 0 and y(y – 1) = 0 so either
y = 0 or y – 1 = 0 which means y = 0 or y = 1.
Distances are positive, so y = 1.
10. 2x + 8 = 120 so 2x = 112 and x = 56. 5y = 60 and y = 12.
11. False: It could have congruent diagonals and be another
type of quadrilateral (trapezoid).
12. No, not a parallelogram. One method of showing this is to
show the diagonals do not bisect each other.
midpoint of GJ = (2, 2.5) midpoint of HK = (1.5, -1.5)
The diagonals do not bisect each other.
8-4 Rectangles
Properties of Rectangles
– How do you
know if a
quadrilateral
is a
rectangle?
RECTANGLE
PROPERTIES
Definition:
Quadrilateral
with 4 right
angles
A
Parallelogram
with
congruent
diagonals
Examples
1. Find x and y.
SOLUTION
X– Y=9
2X + Y = 36
Add the two equations
3X + 0Y = 45
3X = 45
SO X = 15
36
9
2x+ y
SINCE X = 15 AND X – Y = 9, THEN
15 – Y = 9 OR –Y = -6 OR Y = 6
OR
SINCE X = 15 AND 2X + Y = 36, THEN
2(15) + Y = 36 OR 30 + Y = 36 Y = 6
x -y
• 2. Find x.
• AC = x 2
• DB = 6x - 8
C
D
E
A
B
SOLUTION
Since the diagonals of a rectangle are congruent, x² = 6x – 8
x² = 6x – 8 set equal to 0 and factor
x² - 6x + 8 = 0 SO (x – 2)(x – 4) = 0 THEN
x – 2 = 0 or x – 4 = 0 SO x = 2 or x = 4
C(-4,8)
B(10, 8)
D(-4, 4)
A(10,4)
3.Is ABCD a rectangle? Prove
it. A (10,4) B (10,8) C (-4,8) D
(-4,4)
Since a slope of 0
SOLUTION:
Slope of AB = (8-4)/(10-10) = 4/0 = undefined
Slope of CD = (8-4)/(-4- -4) = 4/0 = undefined
Slope of BC = (8-8)/(10 - -4) = 0/14 = 0
Slope of AD = (4-4)/(10- -4) = 0/14 = 0
and an undefined
slope make the
consecutive sides
, ABCD is a
rectangle because it
has 4 right angles (
form 4
Right angles)
4. Find all of the numbered angles
60º
1
4
2
7
9
8
6
3
10
SOLUTION: angles 1, 4, 5, and 10 = 30º
angles 3, 9, and 11 = 60º
angles 2 and 8 =
60º
angles 6 and 7 =
120º
5
11
5.
6.
7.
8.
X = 15.5
X = 5 or –2
X = 13.5
False: This is a property of a parallelogram. The
parallelogram
might not be a rectangle.
9. Slope of AB and CD = 1
Slope of AD and BC = -1
Since consecutive sides are , the quadrilateral has 4 right
angles and it is a rectangle.
10. m<2 = 20
m<5 = 70
m<6 = 20
11. m<6 = 26
m<7 = 26
m<8 = 64
12. m<2 = 54
m<3 = 54
8-5 Squares and Rhombi
I. Rhombus
Def:
quadrilateral
with 4 =
sides
Parallelogram
with 
diagonals
Parallelogram
with diagonals.
that bisect a
pair of opposite
angles
II. Square
Rectangle
+
rhombus
=
Square
D
L
O
P
M
III. Examples
rhombus DLPM
1. DM = 26
13
a. OM= _______
b. MD is congruent to PL.
True or false? False, this is not a rectangle
c. <DLO is congruent to
<MLO. True or false?
True, diagonals
bisect the angles.
2. Use rhombus BCDE and the given information
to find the missing value.
B
a. If m<1 = 2x + 20 and m<2 = 5x – 4,
find the value of x, m<1 and m<2.
C
3
F
1
2
b. If BD = 15, find BF
E
D
c. If m<3 = y² + 26, find y.
ANSWERS:
a. 2x + 20 = 5x – 4 so 20 = 3x – 4 then 24 = 3x and 8 = x
m<1 = 2(8) + 20 = 36 and m<2 = 5(8) – 4 = 36
b. Since it is also a parallelogram, BF = 7.5 (diag. bisect each
other.)
c. y² + 26 = 90 so y² = 64 then y = 8 or -8
3. What type of quadrilateral
is ABCD?
A (-4, 3) B (-2,3)
C(-2, 1) D (-4,1)
Justify.
A(-4,3)
D(-4,1)
B(-2,3)
AB = 2
BC = 2
CD = 2
AD = 2
Slope AB = 0
Slope BC = undefined
Slope CD = 0
Slope AD = undefined
C(-2,1)
Therefore ABCD is a parallelogram, rhombus, rectangle, and
square because all sides are  and it has 4 right <‘s.
Try p. 316 – 317:4, 10-14, 17, 18
Answers:
4. Yes yes
no yes
no no
no no
yes
no
yes
yes
yes
yes
yes
yes
17. rectangle
square
18. rhombus
square
10. m<RSW = 33.5
11. m<SVT = 22.5
12. X = 41
13. X = 12
14. PA = 5 AR = 5 RK = 5 PK = 5
Slopes PA = -1/2 AR = 2
RK = -1/2 PK = 2
It is a parallelogram, rectangle, rhombus, and square
because all sides are congruent and it forms 4 right
angles.
8-6 Trapezoids
I. Properties
base
and
are one pair of
base angles
Trapezoid
leg
Definition: A
quadrilateral with
exactly one pair
of parallel sides.
base
leg
and
are another pair
of base angles
Isosceles trapezoid
Congruent legs
Both pair of base <‘s are
congruent
Diagonals are
congruent
Medians of trapezoids
BASE
MEDIAN
The length of a
median of a trapezoid
is the average of
the base lengths.
BASE
BASE + BASE
2
= MEDIAN
Examples
1. Verify that MATH is an isosceles
trapezoid. M(0,0) A(0,3) T(4,4) H (4,-1)
A(0,3)
M(0,0)
Slope of AM = undefined Slope of AT = 1/4
T(4,4) Slope of TH = undefined Slope of MH = 1/4
H(4, -1) So MATH is a trapezoid
(one pair of parallel sides)
AT = 17 MH = 17
So MATH is isosceles
(legs are congruent)
2. Find x.
29.5
3x - 1
7x + 10
(3x – 1) + (7x + 10)
2
= 29.5
10x + 9
= 59
10x =
50
x=5
3.Find the value of x.
5
5 + 3x – 5 = 2(12)
12
3x = 24
X=8
3x - 5
4. Decide whether each statement is
sometimes, always, or never true.
a. A trapezoid is a parallelogram.
never
b. The length of the median of a trapezoid
is one-half the sum of the lengths of the
bases. always
c. The bases of any trapezoid are parallel.
always
d. The legs of a trapezoid are congruent.
sometimes
Try p. 324: 5 - 13
5. True: Diagonals of an isosceles trapezoid are congruent.
6. True: The legs of an isosceles trapezoid are congruent.
7. False: If the diagonals bisected each other it would be a
8. x² = 16 so x = 4 or –4
9. 17 the legs are congruent and the median cuts the legs in half
10. 180 – 62 = 118
11. R(5, 3)
S(2.5, 8)
12. RS = approx 5.59 (use distance formula for the points in #11)
13. NO, MNPQ is not a trapezoid.