Download Section Li Patterns and Inductive Reasoning a. 3,6,12,24

Section Li
Patterns and Inductive Reasoning
Inductive Reasoning:
Conjecture:
Example 1: Finding and Using a Pattern
Find a pattern for each sequence. Use the pattern to show the next two terms of the
sequence.
a. 3,6,12,24,...
b.
•
N
7
\j_i
/
/\ i
\__v
Example 2: Using Inductive Reasoning
Make a conjecture about the sum of the first 30 odd numbers.
Counterexample:
Example 3: Finding a Counterexample
Find a counterexample for each conjecture.
a. The square of any number is greater than the original number.
b. You can connect any three points to form a triangle.
c. Any number and its absolute value are opposites.
Example 4: Real-World Connection
A skateboard shop finds that over a period of five consecutive months, sales of smallwheeled skateboards decreased. Use inductive reasoning to make a conjecture about the
number of small-wheeled skateboards the shop will sell in June.
Skateboards Sold
-
601
55j
50j
45 1
J
r
M A M
Month
2
_____
______
______
Nets for Three-Dimensional Figures
Section 1.2
Net:
Example 1: Identifying Solids From Nets
Given the net below, which figure will be formed by folding the net?
a.
c
b.
d
Example 2: Drawing a Net
Draw a net for the graham cracker box. Label the net with its dimensions.
:4
Section 1.3
Points, Lines, and Planes
Point:
Line:.
B
-—4--
Collinear points:
Example 1: Identifying Collinear Points
a. Are points E, F, and C collinear? If
so, name the line on which they lie.
b. Are points E, F, and D collinear? If
so, name the line on which they lie.
Plane:
p
Coplanar:
B
A
Example 2: Naming a Plane
Each surface of the ice cube represents part of a plane. Name the plane represented by
the front of the ice cube.
H
F
D
C-
A
4
Postulate:
Postulate 1-2:
If two lines intersect, then they intersect in exactly one point.
p
Example 3: Finding the Intersection of Two Planes
What is the intersection of plane HGFE and plane BCGF?
A
B
Postulate 1-4:
Through any thiee noncollinear points there is exactly one plane.
Example 4: Using Postulate 1-4
Name the plane that is represented by the shaded area.
a.
b.
Ni
\
Section 1.4
Segments, Rays, Parallel Lines and Planes
Segment:
-
-a--——-
—-—. -
-
Ray:
Opposite Rays:
p
_.—__-__:_
_-
R
Example 1: Naming Segments and Rays
Name the segments and rays in the figure.
Segments:
Rays:
-
Parallel lines:
Skew lines
Parallel planes:
Example 2: Identifying Parallel and Skew Segments and Parallel Planes
a. Name all segments that are parallel to DC:
b. Name all segments that are skew to DC:
H
B
c. Name two pairs of parallel planes:
d. Name a line that is parallel to plane GHIJ:
6
_____________
Section 1.5
-
Measuring Segments
Ruler Postulate
The points of a line can be put into one-to-one correspondence with the real numbers
so that the distance between any two points is the absolute value of the difference of the
corresponding numbers.
B
L
-
Congruent()
(
Congruent segments:
i’ Xb
i1;
IQ
•k--:
C
Example 1: Comparing Segment Lengths
—
Find AR and BC. Are AR and BC congruent?
—A
I}
•1-
I
91
fl
—3----(——4——2—I
0
1
2
‘
-
I
Segment Addition Postulate:
If three points A, B, and C are collinear and B is between A and C, then
B
..
Example 2: Using the Segment Addition Postulate
If DT = 60, find the value ofx. Then find DS and ST.
N
c
T
7
_______--
Midpoint:
Example 3: Using the Midpoint
C is the midpoint of AB. Find AC, CB, and AR
2 + I
p
B
8
Section 1.6
Measuring Angles
Angle (L):
I
r
.
Example 1: Naming Angles
Name the following angle four different ways.
C
_—
A
Classifying angles:
Example 2: Measuring and Classifying Angles
Using a protractor, find the measure of each angie. Classify as either acute, obtuse, right,
or straight.
a.
I
//
b.
C.
-
9
Congruent angles:
Angle Addition Postulate:
If point B is in the interior ofL4OC then
.
•4\
“B
,
f-i
(
If L4OC is a straight angle, then
—
A
Example 3: Using the Angle Addition Postulate
a. What is LTSW if mZRST = 50 and mLPSW
=
125?
b. Suppose that m1 =42 and mL4BC =88. Find mL2.
1
A
//
B
•
c
10
...
0
(
Angle pairs:
Z4
-
4
ii;:
4
/
/
/ 4
4
Example 4: Identifying Angle Pairs
In the diagram identify pairs of numbered angles that are related as follows:
a. Complimentary
I
b. Supplementary
c. Vertical
Example 5: Making Conclusions From a Diagram
What can you conclude from the information in the diagram?
/
/
3/
——
11
Section 1.7
-
Basic Constructions
Example 1: Constructing Congruent Segments
Construct a segment congruent to the given segment:
Il
Step 1: Using a straightedge, draw a ray with endpoint C
Step 2: Open the compass to the length of AB
A
B
Step 3: With the same compass setting, put the compass point on point C. Draw an arc
that intersects the ray. Label the point of intersection D.
12
Example 2: Constructing Congruent Angles
Construct an angle congruent to the given angle:
Step 1: Draw a ray with endpoint S.
Step 2: With the compass on point A, draw an arc that intersects
the sides ofLA. Label the points of intersection B and C.
A ‘-
Step 3: With the same compass setting, put the compass point on
point S. Draw an arc and label its point of intersection
with the ray as R.
S
Step 4: Open the compass to the length of BC. Keeping the same
compass setting, put the compass point on 1?. Draw an arc
to locate point T.
c
Step 5: Draw ST.
I
R
13
Perpendicular lines:
c\
/\\\
Perpendicular bisector:
B
Example 3: Cdnstructing the Per,endicular liseêtor
Construct the perpendicular bisector of segment AB:
-
U.
Step 1: Put the compass on point A and draw a long arc as
shown. Be sure the opening is greater than half of
—
A
—
B
Step 2: With the same compass setting, put the compass
point on point B and draw another long arc. Label
the points where the two arcs intersect as X and Y.
A
B
•1
Step 3: Draw XY. The point of intersection of AB and
XY is M, the midpoint of AR.
4
.
II
V
Li
Angle bisector:
Example 4: Finding Angle Measures
KíV bisects LJKL so that mLJKJ’/ = 5x —25 and mLNKL
mLJKN.
/
‘,
=
3x +5. Solve for x and find
r
,
Example 5: Constructing the Angle Bisector
Construct the bisector of L4:
Step 1: Put the compass point on vertex A. Draw an arc that
intersects the sides of LA. Label the points of intersection
BandC
Step 2: Put the compass on point C and draw an arc. With the same
compass setting, draw an arc using point B. Be sure the
arcs intersect. Label the point where the two arcs intersect
asX
Step 3: Draw AX.
V.
Section 1.8
The Coordinate Plane
f_
I
I
—)
(÷
The Distance Formula:
The distance d between two points A(x
1 y) and 2
B(x 2
y is
)
,
,
I
Example 1: Finding Distance
Find the distance between T(5, 2) and R(-4, -1) to the nearest tenth.
Example 2:
Each morning Juanita takes the “Blue Line” subway from Oak Station to Jackson Station
.
As the maps below shows, Oak Station is 1 mile west and 2 miles south of City Plaza.
Jackson Station is 2 miles east and 4 miles north of City Plaza. Find the distance Juanita
travels between Oak Station and Jackson Station.
I-’
4
—4
—.-
r
llt.t
(U
The Midpoint Formula:
The coordinates of the midpoint M of AB with endpoints A(x
1
following:
—
,
)
1
y
and 2
B(x y,) are the
,
Example 3: Finding the Midpoint
QS has endpoints Q(3, 5) and S(7, -9). Find the coordinates of its midpoint M
Example 4: Finding an Endpoint
The midpoint of AB is M(3, 4). One endpoint is A(-3, -2). Find the coordinates of the
other endpoint B.
Section 1.9
Perimeter Circumference and Area
Perimeter and Area:
/
. ,/.‘
I)
I
.1
N•
Square with side
length s
Rectangle with base b
and height h
Circle with radius r and
diameter d
Perimeter P =
Perimeter P
Circumference C
AreaA
AreaA
=
=
AreaA
=
=‘-
fi
Example 1:
Your pool is 15 ft wide and 20 ft long with a 3-ft wide deck surrounding
it. You want to
build a fence around the deck. How much fencing will you need?
-
\
___4
-‘=
——
-
—
-
b
/
.
:
--
i
Example 2: Finding Circumference
Find the circumference o(
A in terms ofg. Then find the circumference to the nearest
5
tenth.
/
LA
>4
\.
\/*.
A1:
12
1
-
Example 3: Finding Perimeter in the Coordinate Plane
Find the perimeter of AABC.
ii
tiii
r2 c
11 V
-.
1—
I!
/
/
—
A
C
‘LA
2
.A:
x
—,
no.)
jy
c_ti.
4/
I
IF
I
—
—
4,
1
A-
iaYiE
—
::•i/’4
F
A
I
[t
V:c1:
—
A
I —t
—I
tThC
t(.
1
‘5:
I,
<;
A
.
‘I •4,
‘4’
.
Cr
.:4,E1
I
—;
C
-.
—
—
A
‘1
1.
IA!
o
:I.Il.
...
A
4
A
Example 4: Finding Area of a Rectangle
You are designing a rectangular banner for the front of a museum. The banner
will be 4
ft wide and 7 yd high. How much material do you need?
Example 5: Finding Area of a Circle
The diameter of a circle is 10 in. Find the area in terms of
Example 6: Finding Area of an Irregular Shape
What is the area of the figure below?
ir.