1. Chapter 8: Exploring Quadrilaterals
4 sided polygons

3. 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

4. Polygons
Not Polygons

5. Concave polygons Convex polygons
No line containing a side of the polygon contains a point in the interior of the polygon.
NOT CONVEX

6. Naming Polygons Regular Polygon # of sides Name A Polygon that is3 4 5 6 7 8 9 10 11 12 13 14
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.)

8. -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?
pentagon?
-Look at the hexagon, how many
-So what is the sum of the measures of the angles in a hexagon?
Do you see a
pattern? How
does the number
of triangles
relate to the
number of sides?
Can you write a
formula?

9. 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.)
Measure each of the exterior angles that are
formed.
What can you conclude about the sum of the
exterior angles of a convex polygon, one
at each vertex?

10. Finding angle measures
# of sides Sum of One interior < sum of One exterior <
interior <‘s (if regular) exterior <‘s (if regular)
n (n – 2) (n – 2)
n n
SAMPLES OF APPLYING THE FORMULAS
(3-2)180 = (3-2)180= =120
(12-2)180 = (12-2)180 = =30

11. Examples Name the regular polygon with an exterior angle measuring 45.
Find the sum of the measures of the interior angles of a convex 26-gon.
Name the regular polygon with an exterior angle measuring 45.

12. 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.

13. Try p. 518: 7 - 18 AB, BC, CD, DE, EF, FA 13. 135 and 45
Convex (a-2) , 360 a
1440
, 102,67,199,

14. Section 8.2 Parallelograms
Definition of Parallelogram:
A quadrilateral with both pairs of
opposite sides parallel.

15. II. PROPERTIES OF PARALLELOGRAMS
Both pair of
opposite sides
are 
Parallelogram
Both pair of
opposite <‘s
are 
Consecutive
angles are
supplementary
Def: Both pair
of opposite
sides are //
Diagonals
bisect each
other

16. Examples 1. WXYZ is a parallelogram. Find the indicated information.
M<ZWX = 100
3a + 5 W X c
16b -3
8b + 13 17 V Z
7a -7 Y
Find the value of a. B. Find the value of b.
Find the value of c. D. Find m<WZY
E. Find m<XYZ F. Find m<WXZ 2 80

17. Use the definition of a parallelogram to
determine if RSTV is a parallelogram.
(3,6)
(1,1) S 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
Slope RV = (3-1)/(6-1) = 2/5
Slope ST = (8-6)/(8-3) = 2/5 T
(6, 3) V
(8, 8)
Since RS // VT (have the same slope) and RV // ST (have the same
slope), RSTV is a parallelogram.

18. 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

19. 3. Two sides of ABCD are chosen at random
3. Two sides of ABCD are chosen at random. What is the probability that the two sides are not congruent?
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 A B
Not congruent/Total 4 /6
So the probability is 2/3. C D

20. Find the values of w, x, y, and z for the parallelogram.
110 w 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)

21. 5. Find the indicated values for the parallelogram g i f b c a e d
100 b c a 70 e d 20
Answers
a = 100 b = 80 c = d = 60
e = 30 f = 70 g = h = 60
i = 30

22. 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)
(8,0)
(4, -4)

23. Try p. 295: 6 - 14 AB = CD so 2x + 5 = 21 and Answers to 6 – 14
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
SLOPES
PT and QR = 5
QP and TR = -1 therefore opposite
sides are // and it is a parallelogram
Answers to 6 – 14 __ HF DC
<DFG GF
<CDF and <CGF
HDF

24. 8-3 TESTS FOR PARALLELOGRAMS

25. HOW DO YOU KNOW IF A QUADRILATERAL IS A PARALLELOGRAM?
PERFORM A
TEST
Diagonals
Bisect
Each
other
The same
pair of
opposite
sides
Opposite sides are parallel
Opposite
Sides are
congruent
Opposite angles are
congruent
(both pair)
(both pair)
(both pair)
 and //

26. II. EXAMPLES 1. IS CUTE A PARALLELOGRAM? WHY? C T 118 62 118 62 U E
YES, BECAUSE OPPOSITE ANGLES ARE CONGRUENT AND CONSECUTIVE ANGLES ARE SUPPLEMENTARY

27. 2. Find x and y so that the quadrilateral is a parallelogram.4 2y
4x – 8 = 4
4 x = 12
X = 3
So 3( 3) + 17 = 26
26 = 2 y
13 = y

28. 3. IS ABCD A PARALLELOGRAM?
NO- OPPOSITE SIDES ARE NOT CONGRUENT.

29. 4. Determine if PZRD is a parallelogram.

30. solutions MIDPOINTS DZ (2.5,5.5) PR (2.5,5.5)
It is a parallelogram because the
diagonals bisect each other.
solutions
SLOPES
slope PZ and DR = -1/4
slope PD and ZR = -2
It is a parallelogram because both pair of opposite sides are //
DISTANCES
distance PZ and DR = 17
distance PD and ZR = 35
It is a parallelogram because both pair of opposite sides are congruent.
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.

31. 7. Since the triangles will be congruent by SAS, the other
Try p. 301:
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.
No, the same pair are not congruent and parallel.
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 = y = 60 and y = 12.
False: It could have congruent diagonals and be another type of quadrilateral (trapezoid).
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.

32. 8-4 Rectangles Properties of Rectangles
How do you know if a quadrilateral is a rectangle?

33. RECTANGLE
PROPERTIES
Definition:
Quadrilateral
with 4 right angles
A Parallelogram with
congruent
diagonals

34. Examples 1. Find x and y. 36 9 x -y 2x+ y X – Y = 9 2X + Y = 36
SOLUTION
X – Y = 9
2X + Y = 36 9
x -y
2x+ y
Add the two equations
3X + 0Y = 45
3X = SO X = 15
SINCE X = 15 AND X – Y = 9, THEN
15 – Y = 9 OR –Y = OR Y = 6 OR
SINCE X = 15 AND 2X + Y = 36, THEN
2(15) + Y = OR Y = Y = 6

35. 2.    Find x.
AC = x
DB = 6x - 8 C D E B A
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

36. Is ABCD a rectangle? Prove it. A (10,4) B (10,8) C (-4,8) D (-4,4)
Since a slope of 0
and an undefined
slope make the
consecutive sides
, ABCD is a
rectangle because it has 4 right angles ( form 4
Right angles)
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)/( ) = 0/14 = 0
Slope of AD = (4-4)/(10- -4) = 0/14 = 0

37. 4. Find all of the numbered angles1 4
60º 9 7 2 8 6 11 3 10 5
SOLUTION: angles 1, 4, 5, and 10 = 30º
angles 3, 9, and 11 = º
angles 2 and 8 = º
angles 6 and 7 = º

38. Try p. 309:
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.
Slope of AB and CD = Slope of AD and BC = -1
Since consecutive sides are , the quadrilateral has 4 right angles and it is a rectangle.
10. m<2 = m<5 = m<6 = 20
11. m<6 = m<7 = 26 m<8 = 64
12. m<2 = 54 m<3 = 54

39. 8-5 Squares and Rhombi I. Rhombus
Def:
Parallelogram
with diagonals.
that bisect a
pair of opposite
angles
Parallelogram
with 
diagonals
quadrilateral
with 4 =
sides

40. II. Square
Square
Rectangle = +
rhombus

41. D L O P M 13
False, this is not a rectangle
III. Examples rhombus DLPM 1. DM = a. OM= _______ b. MD is congruent to PL True or false? c. <DLO is congruent to <MLO. True or false?
True, diagonals
bisect the angles.

42. Use rhombus BCDE and the given information to find the missing value.
a. If m<1 = 2x + 20 and m<2 = 5x – 4,
find the value of x, m<1 and m<2.
b. If BD = 15, find BF
c. If m<3 = y² + 26, find y. B C 1 2 3 F D E
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
Since it is also a parallelogram, BF = 7.5 (diag. bisect each other.)
y² + 26 = 90 so y² = 64 then y = 8 or -8

43. 3. What type of quadrilateral is ABCD
3. What type of quadrilateral is ABCD? A (-4, 3) B (-2,3) C(-2, 1) D (-4,1) Justify.
AB = 2
BC = 2
CD = 2
AD = 2
A(-4,3)
B(-2,3)
Slope AB = 0
Slope BC = undefined
Slope CD = 0
Slope AD = undefined
D(-4,1)
C(-2,1)
Therefore ABCD is a parallelogram, rhombus, rectangle, and square because all sides are  and it has 4 right <‘s.

44. Try p. 316 – 317:4, 10-14, 17, 18
Answers:
4. Yes yes yes yes rectangle
no yes no yes square
no no yes yes
no no yes yes rhombus
square
m<RSW = 33.5
m<SVT = 22.5
X = 41
X = 12
PA = 5 AR = 5 RK = 5 PK = 5
Slopes PA = -1/2 AR = RK = -1/ PK = 2
It is a parallelogram, rectangle, rhombus, and square because all sides are congruent and it forms 4 right angles.

45. 8-6 Trapezoids

46. I. Properties base leg leg base Trapezoid Definition: A
and
are one pair of
base angles
base
Trapezoid
Definition: A
quadrilateral with
exactly one pair
of parallel sides.
leg
leg
and
are another pair
of base angles
base

47. Isosceles trapezoid Both pair of base <‘s are congruent
Congruent legs
Both pair of base <‘s are
congruent
Diagonals are

48. Medians of trapezoids = MEDIAN BASE + BASE 2 BASE The length of a
median of a trapezoid
is the average of
the base lengths.
MEDIAN
BASE
= MEDIAN
BASE + BASE 2

49. Examples
1. Verify that MATH is an isosceles trapezoid. M(0,0) A(0,3) T(4,4) H (4,-1)
Slope of AM = undefined Slope of AT = 1/4
Slope of TH = undefined Slope of MH = -1/4
So MATH is a trapezoid
(one pair of parallel sides)
AT =  MH = 17
So MATH is isosceles
(legs are congruent)
A(0,3)
T(4,4)
M(0,0)
H(4, -1)

50. 2. Find x.
(3x – 1) + (7x + 10) = 29.5 2
10x = 59
10x =
x = 5
29.5
3x - 1
7x + 10

51. Find the value of x. 5
5 + 3x – 5 = 2(12)
3x = 24
X = 8 12
3x - 5

52. 4. Decide whether each statement is sometimes, always, or never true.
a. A trapezoid is a parallelogram.
b. The length of the median of a trapezoid is one-half the sum of the lengths of the bases.
c. The bases of any trapezoid are parallel.
d. The legs of a trapezoid are congruent.
never
always
always
sometimes

53. Try p. 324:
True: Diagonals of an isosceles trapezoid are congruent.
True: The legs of an isosceles trapezoid are congruent.
False: If the diagonals bisected each other it would be a
x² = 16 so x = 4 or –4
17 the legs are congruent and the median cuts the legs in half
180 – 62 = 118
R(5, 3) S(2.5, 8)
RS = approx 5.59 (use distance formula for the points in #11)
NO, MNPQ is not a trapezoid.