Transcript Geometry

Chapter 1
1
What are the 3 BASIC UNDEFINED TERMS IN
GEOMETRY?
Answer:
Point, line, & plane
They do not have any shape or size.
They are generally defined using
examples.
2
A point
 is a location.
 has no shape or size.
 is named using one capital
letter.
EX:
A
3
A line:
 is made up of infinitely many
points.
 has no thickness.
4
A line is named 2 ways.
1. Using 2 capital printed letters
representing 2 points on the line
E
D
DE
5
2. Using a lowercase, cursive letter.
line m
6
A plane:


Is a flat surface made up of
points that extends infinitely
in all directions.
has no thickness.
7
A plane is named 2 ways.
1.) using 3 capital printed letters
representing 3 points that are not all on
the same line
EX. plane BCD or plane DBC or
plane CBD, etc
8

2.
using 1 capital, cursive letter
plane N
9
COLLINEAR POINTS:
Points that lie on the same line
COPLANAR POINTS:
Points that lie on the same plane
10
INTERSECTION:
The set of points that 2 or more geometric
figures have in common
point .
1.) 2 lines intersect in a ________
In the diagram, line a and line b
intersect at point R.
11
2.) A line and a plane intersect in
a point OR the line if the line
_________________________________
lies in the plane.
___________________
EX
In the first diagram, In the second diagram,
line k and plane B
line p lies completely in
intersect at point P. plane Q , so their
intersection is line p.
12
3.) 2 planes intersect in a
line
.
EX
13
POSTULATE OR AXIOM:
An accepted statement of fact
POSTULATE
Through any 2 points there is exactly one line.
EX
14
POSTULATE
If 2 lines intersect, then they intersect in
exactly one point
.
EX
15
POSTULATE
If 2 planes intersect, then they
intersect in
exactly one line
.
16
POSTULATE
Through any 3 noncollinear points there is
exactly one plane .
17
18
Assignment:
Worksheet
Front: #1, 7, 10 – 13, 15, 19
Back: #1, 4, 5, 8, 24, 33, 35, 40, 43
19
Segment:
The part of a line consisting of 2 endpoints and
all points between them
P
PQ
Q
consists of points P and Q and all
of the points between them
To name a line segment use 2 capital
letters and a segment above them.
20
What is the difference between
AB
AB
SEGMENT AB is at
the right in blue
LINE AB is at
the right in red
AB and AB
A
?
B
A
B
21
AB
is thesame segment as
BA.
22
AB
is the same line as
BA.
23
IN CLASS:
Front of worksheet
#1,7,10-13,15,19
Assignment:
Back of worksheet
#1,4,5,8,24,33,35
24
Ray:
The part of a line consisting of
one endpoint and all the points of
the line on one side of the
endpoint
25
Naming a RAY:
FIRST:
Name the endpoint.
SECOND:
Name another point on the line
closer to the arrow.
THIRD:
Write  over the letters.
26
Name the rays.
EX
N
D
M
F
MN
FD
FD is not the same as DF .
27
Opposite Rays:
2 collinear rays with the same
endpoint
Opposite rays ALWAYS form a line!
A
B
C
BA and BC are oppositerays.
28
QP and QR are OPPOSITE RAYS!!!
R
Q
P
Together, they make PR .
29
Let’s do #2 – 6, 14, 17, & 18
front of the worksheet
together for practice!
30
Parallel lines:
Coplanar lines that do not intersect
Give some real life examples
of parallel lines.
31
Skew lines:
Non-coplanar lines that do not
intersect
Give some real life examples
of skew lines.
32
Parallel Planes:
Planes that do not intersect
Give some real life examples
of parallel planes.
33
Let’s do #8,9, & 21 on the front
of the worksheet.
34
Assignment:
Do the rest of the back of the
worksheet.
#2,3,6,7,9-23,25-32,34,
37-39,41,42,44-47
When you have finished this, you
should have the front and back of the
worksheet completed.
35
A
B C D E
F
‒4 ‒3 ‒2 ‒1 0 1 2 3 4
To find the length of a segment,
think RIGHT minus LEFT.
36
A
B C D E
F
‒4 ‒3 ‒2 ‒1 0 1 2 3 4
The length of
EF
is
F - E or 4 – 2 = 2.
37
A
B C D E
F
‒4 ‒3 ‒2 ‒1 0 1 2 3 4
1.
2.
3.
4.
DF
CE
BC
AE
38
A
B C D E
F
‒4 ‒3 ‒2 ‒1 0 1 2 3 4
Remember, RIGHT MINUS LEFT.
1.
2.
3.
4.
DF = 4 – 1 = 3
CE = 2 – 0 = 2
BC = 0 – (‒1) = 1
AE = 2 – (-3) = 5
39
When we found the lengths of the segments on
the previous slide, notice how we wrote the
lengths.
1.
2.
3.
4.
DF = 4 – 1 = 3
CE = 2 – 0 = 2
BC = 0 – (‒1) = 1
AE = 2 – (-3) = 5
We wrote DF = 3 and not DF = 3.
40
When we write the length of a segment, we do
NOT write “
―
“ over the letters.
Find AD & AF.
AD = 4 & AF = 7
41
Congruent segments
Segments with the same length
42
A
B C D E
F
‒4 ‒3 ‒2 ‒1 0 1 2 3 4
EF is congruent to what other
segments on the number line?
CE DB AB
43
Since EF CE DB AB
all have length 2, they are
congruent.
The symbol for congruent is ≅ .
EF  CE  DB  AB
44
SEGMENT ADDITION POSTULATE
If 3 points A, B, and C are collinear,
and B is between A and C, then
AB + BC = AC
A
B
C
45
If 3 points A, B, and C are collinear
and B is between A and C, then
AB + BC = AC
A
B
C
If AB = 5, and BC = 8, then we write….
5 + 8 = 13… AC = 13
46
X, Y, AND Z are collinear and Y lies between X
and Z. Sketch a diagram representing this
information, and use it to write an equation
using the letters X, Y, and Z. Then do the
problems on the following slides.
X
Y
Z
XY + YZ = XZ
47
1. XY = 5, YZ = 3; Find XZ.
X
Y
Z
XY + YZ = XZ
5 + 3 = XZ
8 = XZ
48
2. XY = 4, XZ = 11; Find YZ.
XY + YZ = XZ
4 + YZ = 11
YZ = 7
49
3. If XZ = 70, XY = 3a - 2, and YZ = 5a,
find the value of a, XY, and YZ.
XY + YZ = XZ
(3a – 2) + (5a) = 70
8a – 2 = 70
8a = 72
a=9
XY = 3a - 2 = 3(9) – 2 = 25
YZ = 5a = 5(9) = 45
50
4. If XY = 9a + 1, YZ = 3a + 6, and XZ = 4a + 23,
find the value of a, XY, YZ, and XZ.
XY + YZ = XZ
(9a + 1) + (3a + 6) = (4a + 23)
12a + 7 = 4a + 23
8a + 7 = 23
8a = 16
a=2
51
XY = 9a + 1 = 9(2) + 1 = 19
YZ = 3a + 6 = 3(2) + 6 = 12
XZ = 4a + 23 = 4(2) + 23 = 31
Notice that 19 + 12 = 31.
52
ASSIGNMENT:
Worksheet (Measuring Segments & Angles)
FRONT: #1 – 11, 36
BACK: evens
53
Midpoint
The point halfway between the endpoints of a
segment
Notation
Use “ | ” on the diagram to show which
segments are congruent.
54
D
E
F
The red marks mean DE  EF .
E is the midpoint of DF.
55
Use the diagram to set up an equation with the
letters on the diagram and another equation
with the variable x. Find the values of x, JK,
KL, and JL.
10X
J
JK = KL
10X = 16X – 30
16X - 30
K
L
continued on next slide
56
10X = 16X – 30
-6x = -30
x=5
JK = 10x = 10(5) = 50
KL = 16x – 30 = 16(5) – 30 = 50
JL = 50 + 50 = 100
57
Find JK if KL = 3y + 5 and JL = 9y + 1.
3y + 5
J
K
L
9y + 1
58
JK + KL = JL
Because the diagram has congruent marks,
we know that JK ≅ KL.
Thus, instead of writing JK + KL = JL, we can
write
2(KL) = JL
JK = KL = 3(3) + 5 = 14
2(3y + 5) = 9y + 1
JL = 9(3) + 1 = 28
6y + 10 = 9y + 1
-3y + 10 = 1
-3y = -9
y=3
59
Find the coordinate of the midpoint
of the two points.
-6
To find the midpoint,
find the average of
the two numbers.
65
2
5
1

2
60
Find the midpoint of points A and B.
A(-3, 5)
B(8, 13)
38 5
x

2
2
5  13
y
9
2
 5 
The midpoint is  , 9 .
 2 
61
62
Assignment:
Worksheet (Measuring Segments & Angles)
Do
Front: #12 – 15, 29 – 32, 42 – 46.
Back: odds
63
An angle is formed by
2 rays with the same endpoint.
64
Name the
angle at the
right 4 ways.
1.
2.
3.
4.
∠
∠
∠
∠
B
CBA
ABC
1
C
B
1
A
Notice that the vertex
(point of the angle) is
in the middle when
using 3 letters to name
an angle.
65
∠2 cannot be named ∠T.
Why not?
∠T is not specific
enough.
Does ∠ T mean ∠STR?
Does ∠ T mean ∠ STP?
Does ∠ T mean ∠PTQ?
Does ∠ T mean ∠QTR?
P
4
Q
3
T
S
2
5
R
66
∠2 cannot be named ∠T.
To be sure we know
which angle we want,
we must name the
shaded angle by one of
these three names.
∠STR
∠RTS
∠2
P
4
Q
3
T
S
2
5
R
67
CLASSIFYING ANGLES
ACUTE ANGLE
an angle measuring between 0º and 90º
OBTUSE ANGLE
an angle measuring between 90º and 180º
RIGHT ANGLE
an angle measuring 90º
STRAIGHT ANGLE
an angle measuring 180º
68
Match the angle with its classification.
OBTUSE
RIGHT
STRAIGHT
3
ACUTE
1
2
4
69
90º .
A right angle measures _______
180º
A straight angle measures _______ .
70
What does the little black square
mean?
2
It means that ∠2 is
a right angle
a 90º angle
71
A
ANGLE ADDITION POSTULATE
B
O
C
interior of ∠AOC,
If point B is in the __________
m∠AOB + m∠BOC = m∠AOC
then ___________________________________.
72
Use the given information to find m∠AOC.
m ∠AOB = 30
m ∠BOC = 45
?
30
A
O
B
45
C
m ∠AOB + m ∠BOC = m ∠AOC
30 + 45
= m ∠AOC
75 = m ∠AOC
73
Find the angle measures using the
given information.
H
m∠DPH = 35
m∠HPW = 100
1.)
2.)
3.)
4.)
m∠DPM = 180
m∠WPM = 45
m∠DPW = 135
m∠MPH = 145
D
35
100
W
P
M
45
74
Assignment:
Worksheet (Measuring Angles)
Front: 17,19,21,23-28, 47,48,65,66,
12-17 on the right
Back: all
75
Recall, on a number line, the
length of a segment is found by
doing right minus left .
76
On a number line, finding the
distance between 2 points
___________________________uses
the same process as finding
the length of a segment .
___________________________
77
A
B C D E
F
‒4 ‒3 ‒2 ‒1 0 1 2 3 4
The length of
EA
is
E - A or 2 – (-3) = 5.
78
length of a
When we write the ________
segment, we do NOT write “ ― “
over the letters.
Find BD & BE.
BD = 2 & BE = 3
79
A
B C D E
F
‒4 ‒3 ‒2 ‒1 0 1 2 3 4
Find the length of these segments.
1.
2.
3.
4.
ED
CF
AF
BA
80
A
B C D E
F
‒4 ‒3 ‒2 ‒1 0 1 2 3 4
Remember, RIGHT MINUS LEFT.
1.
2.
3.
4.
ED
CF
AF
BA
 2 – 11
4–04
 4 – (-3) 7
 - 1 – (-3) 2
81
When we found the lengths of the segments on
the previous slide, notice how we wrote the
lengths.
1.
2.
3.
4.
ED = 2 – 1 = 1
CF = 4 – 0 = 4
AF = 4 – (‒3) = 7
BA= -1 – (-3) = 2
We wrote ED = 3 and not ED = 3.
82
If the points are ordered pairs, you must use
distance formula
the ________________________to
find the
distance between 2 points.
83
distance formula for ordered pairs
x2 , y2 
x1 , y1 
d
x1  x2    y1  y2 
2
2
84
distance between 2 points is the
The ______________________________
_______________
same as the length of the segment
connecting those 2 points.
85
d
x1  x2    y1  y2 
2
2
distance formula to
Use the __________________
find the distance between the
ordered pairs.
x1 y1
x2 y2
5,3 and  8, 1
d
5   8   3  1
2
2
86
d
5   8   3  1
2
d
2
13   4
2
2
d  169 16
d  185  13.6
The distance between
5,3 and  8, 1 is 13.6 .
87
In your book…
p.31-32 (#14-32 evens)
88
Midpoint
The point halfway between the
endpoints of a segment
The midpoint is the middle point of a
segment.
Notation
Use “ | ” on the diagram to show which
segments are congruent.
89
A
B
C
The green marks mean AB  BC , so
B is the midpoint of AC.
90
Use the diagram to set up an equation
with the letters on the diagram and
another equation with the variable x.
Find the values of x, PQ, QR, and PR.
6x + 2
9x - 10
P
Q
R
Because of the l marks, we know PQ  QR
so PQ = QR
91
PQ = QR
6x + 2 = 9x – 10
2 = 3x - 10
12 = 3x
4=x
PQ = 6x + 2 = 6(4) + 2 = 26
QR = 9x – 10 = 9(4) – 10 = 26
PR= 26 + 26 = 52
92
Use the diagram to find the
indicated information.
K
L
M
5
If KM = 10, then LM = ________
.
9
If KM = 18, then KL = ________
.
4
If KL = 4, then LM = ________
.
93
continued… Use the diagram to
find the indicated information.
K
L
7
If LM = 7, then KL = ________
.
10
If KL = 5, then KM = ________
.
16
If LM = 8, then KM = ________
.
M
94
Find DF if DE = 2a + 5 and DF = 5a + 3.
2a + 5
D
E
F
5a + 3
95
Because the diagram has congruent marks,
we know that DE = EF.
Thus, instead of writing DE + EF = DF, we
can write 2(DE) = DF.
2(DE) = DF
2(2a + 5) = 5a + 3
4a + 10 = 5a + 3
-1a + 10 = 3
-1a = -7
a=7
DE = EF
= 2a + 5
= 2(7) + 5
= 19
DF = 5a + 3
= 5(7) + 3
= 38
96
Find the coordinate of the midpoint
of the two points.
-1
To find the midpoint,
find the average of
the two numbers.
 1 8
2
8
7

2
97
Find the midpoint of points P and Q.
P(12, -6)
Q(-5, 6)
12  (5) 7

6

6
x
 y
0
2
2
2
 7 
The midpoint is  , 0 .
 2 
98
(-4, 3)
L
(-2, 1)
Find the
coordinates of N.
ENDPOINT L ( -4, 3 )
+2 -2
MIDPOINT M ( -2, 1 )
+2 -2
N ( 0 , -1 )
M
N
99
Find the coordinates of P.
ENDPOINT
R ( -2, 1)
-8 -4
(-10, -3)
MIDPOINT Q (-10, -3 )
( -2, 1 )
-8 -4
P( -18 , -7 )
10
0
Assignment:
p.32(#34-56 even)
p.45(1-8 all)
10
1
If AB = 3x – 4, BC = 5x, and AC = 28, set
up and solve an equation to find the value
of x, AB, and BC.
3x – 4
A
5x
B
C
28
AB + BC = AC
3x – 4 + 5x = 28
8x – 4 = 28
8x = 32
x=4
AB = 3x – 4 = 3(4) – 4 = 8
BC = 5x = 5(4) = 20
10
2
Given info
Sketch and label a diagram and set up and solve an
equation to find the value of x and AC.
6x – 1
B is the midpoint of AC . 3x + 5
AB = 3x + 5
BC = 6x – 1
A
B
C
3x + 5 = 6x – 1
5 = 3x – 1
6 = 3x
x=2
10
3
Continued…
x=2
AB = 3x + 5 = 3(2) + 5 = 11
BC = 6x – 1 = 6(2) – 1 = 11
Since AC = AB + BC,
AC = 11 + 11 = 22
10
4
On PQ , A is the midpoint. Use the given
information to find the coordinates of Q.
P(5, -6)
A(1, 7)
Endpoint P( 5, -6)
Midpoint
-4+13
A( 1, 7)
-4 +13
Q(-3 , 20 )
10
5
ANGLE ADDITION POSTULATE
P
Q
N
R
interior of ∠PQR,
If point N is in the __________
m∠PQN +m∠NQR =m∠PQR
then ________________________________.
10
6
Find the value of x, m∠PQN,
m∠NQR, and m∠PQR.
m∠PQN = 9x - 10
m∠NQR = 10x + 2
m∠PQR = 16x + 10
P
Q
N
R
∠PQN + m ∠NQR = m ∠PQR
9x – 10 + 10x + 2 = 16x + 10
10
7
9x – 10 +10x +2 = 16x + 10
19x – 8 = 16x + 10
3x – 8 = 10
3x = 18
x=6
m∠PQN = 9x – 10 = 9(6) – 10 = 44
m∠NQR = 10x + 2 = 10(6) + 2 = 62
m∠PQR = 16x + 10 = 16(6) + 10 =106
10
8
Assignment:
Worksheet (Midpoint, Angle & Segment
Addition)
10
9
D
E
F
E
vertex of ∠DEF is ___.
The ________
EF .
ED and ____
The sides of ∠DEF are ____
11
0
Assignment:
p.41-42
(#12-26 even, 30-34 even,
38-42 even)
For #3- - 34, you do not need to copy the
diagram of use a protractor.
#38 – 42 Set up and solve an equation to
find the indicated information.
11
1
Adjacent Angles
same plane
2 angles that lie in the ____________
common vertex and a
and have a ________________
common side but no common
______________
interior points
∠1 and∠2 are
adjacent angles .
________________
1
2
11
2
4 3
are NOT adjacent angles.
∠3 and ∠4 _________
Why not???
common side
They do not share a ______________.
11
3
5
6
are NOT adjacent angles.
∠5 and ∠6 _________
Why not???
common vertex
They do not share a _________________.
11
4
Complimentary Angles
2 angles whose sum is 90
90
Complimentary
Supplementary Angles
8
2 angles whose sum in 180
Supplementary
11
5
L
M
N
3
4
right angle , we
Since ∠LMN is a ____________
90 .
know m∠LMN = ___
Therefore, ∠3 and ∠4 are
complimentary
________________.
11
6
A
B
C
opposite rays .
BA and BC are ______________
straight angle .
∠ABC is a _______________
180°
What is m∠ABC? _____
11
7
Linear Pair
a pair of adjacent angles whose
noncommon sides are opposite rays (a
D
line)
∠1 and ∠2
1
2
form a
B
A
C
linear pair .
___________
Angles that form a linear pair
supplementary .
are ________________
11
8
P
1
Q
2
R
straight angle , we
Since ∠PQR is a _______________
180 .
know m ∠PQR = _____
Therefore, ∠1 and ∠2 are
supplementary and they form a
________________,
linear pair .
___________
11
9
Which pairs of angles
form a linear pair?
Several answers:
∠1 & ∠2, ∠2 & ∠3,
∠3 & ∠4, ∠4 & ∠1
1
2
4
3
12
0
Use the given information & the diagram
on the previous slide to set up and solve
an equation to find the value of x and
the measures of those two angles.
m∠1 = 4x + 12
m∠2 = 12x + 8
Since ∠1 & ∠2 form a linear pair,
their sum is 180
____, and we can write
m∠1 + m∠2 = 180 .
_____________________
12
1
Since ∠1 & ∠2 form a linear pair,
their sum is 180, and we can write
m∠1 = 4x + 12
m∠2 = 12x + 8
Angle Relationships (1-5)
m∠1 + m∠2
= 180
4x + 12 + 12x + 8 = 180
16x + 20 = 180
16x = 160
x = 10
12
2
x = 10
m∠1 = 4x + 12 = 4(10) + 12 = 52
m∠2 = 12x + 8 = 12(10) + 8 = 128
12
3
4
1
3
2
vertical angles .
∠1 and ∠3 are _______________
vertical angles .
∠2 and ∠4 are _______________
12
4
Vertical Angles
2 nonadjacent angles formed by 2
intersecting lines
Which angles
in the diagram
1
6
2
are vertical
5 4 3
angles?
∠1 and ∠4
12
5
Which pairs of angles are
vertical angles?
Answer: ∠1 & ∠3,
∠2 & ∠4
1
2
4
3
Therefore, ∠1≅ ∠3 and ∠2≅ ∠4.
12
6
Use the given
information below to
2
1
3
4
set up and solve an
equation to find the
value of x and the
measures of those two
angles.
m∠1 = 4x + 12
m∠3 = 6x - 8
∠1≅ ∠3
m∠1=m ∠3
12
7
∠1≅ ∠3
m∠1=m ∠3
4x + 12 = 6x – 8
12 = 2x – 8
20 = 2x
10 = x
m∠1 = 4x + 12 = 4(10) + 12 = 52
m∠3 = 6x – 8 = 6(10) – 8 = 52
12
8
right angles are
Lines that form ____________
perpendicular
______________.
m
perpendicular to m .
l is ______________
This can be written l  m.
l
12
9
Assignment:
13
0