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1.2 – Points, Lines, and Planes
• Warm – Up, Do if you finished checking
homework
Find the slope of the line going through these two points.
Because I know some of you forgot:
• (4,5) (-1,3)
(-3,1) (2,-5)
• Does order matter? When? Why?
𝑦2 −𝑦1
𝑥2 −𝑥1
=𝑚
A definition uses known words to describe a new word.
In geometry,
are undefined
terms.
Point. Technically, it has no size, but
we use a dot that has size to
represent it. You use a capital letter
to label it. Such as Point A
All figures are made of points. This
is a LINE. It goes both ways, forever
without ending. Once again, it has
no thickness, but we use a picture
with thickness to describe it. Arrows
on both ends say it goes on forever.
K
I
M
PLANE, goes on forever,
A once again has no
thickness. Even though
it goes on forever, we
usually use a
parallelogram shape to
draw it.
To label it, a capital
cursive letter can be
used, or you can use
three points that don’t
line up (also known as
non-collinear points)
Collinear points, points all in one line.
Noncollinear points, points NOT all in one line.
Coplanar points, points all in one plane.
Noncoplanar points, points NOT all in one plane.
S
C
D
U
K
R
T
F
This is a line
segment, it is a
segment, or part of
a line
It is labeled
TR
B
I
This is a ray
It is labeled
FI or BI
For ray FI
F is called the
initial point
T, R are
ENDPOINTS
For ray BI
B is called the
initial point
l
M
N
O
OPPOSITE RAYS –
NO and NM
are called opposite
rays cuz N is between
M and O.
Name four coplanar points
l
P
B
Name three collinear points
A
D
C
What is the intersection of line
l and plane P?
E
Q
G
I
Which plane has points F,H,I?
H
F
J
• Warm – Up, Do if you finished checking
homework
• Solve for y. Remember, that means make it ‘y=‘
2x + 3y = 6
y – 3 = -2(x – 1)
What does collinear mean?
Why do two points HAVE to be collinear?
Draw three noncolline ar points A, B, and C.
Draw AB, BC, AC
Two or more geometric figures intersect if they have
one or more points in common. The intersection
of the figures is the set of points the figures have
in common.
Draw three noncollinear points A, B, C on plane P. Draw
line l not on plane P going through point C.
Draw three planes M,
N, P meeting at point P.
Draw three planes M,
N, P meeting on line l.
In 3-D, sometimes it helps to imagine a box, or look
around the room (but not during a test)
• Three lines l, m, n meet at
point P, where lines m, n
are on plane P.
Four points, A, B, C, D, where A,
B, and C are coplanar but not
collinear on plane P, and D is
noncoplanar
1.3 Segments and Their Measures
Warm – Up: Find the distance between the points
B
A
D
C
E
F
Where your ruler is doesn’t matter. Two points are the same distance
apart no matter how you line up the ruler.
Two find the distance between two points, you can just subtract the
distance and take the absolute value.
Like if on a number line, I have a points at Q and R, and they are different,
to find the distance, all I have to do ___________ and then ____________
Postulate \ Axiom – A rule that is accepted
WITHOUT PROOF.
Postulate 1 – Ruler Postulate
The points on a line can be matched one to one with
the real numbers. The real number that corresponds
to a point is the coordinate of the point.
The distance between points A and B, written as AB,
is the absolute value of the difference between the
coordinates of A and B.
AB is also called the length of AB.
A
B
-1
0
1
SEGMENT ADDITION POSTULATE
If B is BETWEEN A and C, then AB + BC = AC. Also
If AB + BC = AC, then B is between A and C
AC
A
AB
B
BC
C
Why does this only work if it’s ___________?
_____ 50 ______
E
4x
A
2x  8
T
_____ 4 x ______
B
x
O
2x  8
X
DS = 30
DU = 5
KS = 7
UC = .5CK
UK =
UC =
DC =
US =
D
U
C
K
S
boardwork
T
5
H
R
5
S
TR  HS
This is CONGRUENCE
Congruence is shown with marks. The marks say that they
are the same size and shape
TR  HS
This is EQUAL
Equals means they have equal length, number value.
They are equivalent. Definition of congruent segments:
Congruent segments have equal lengths
Superposition:
What is Pythagorean Theorem?
(x2,y2)
What does a, b, and c represent?
Looking at the graph, one way to
represent the length of the horizontal
leg is:
(x1,y1)
Using the same logic, another way to represent the vertical leg is:
Replace a and b with what we just found and solve for c
DISTANCE FORMULA
(-5, -2) (4, 1)
x1 y1
Why do they use d instead of c?
How come order doesn’t really
matter for this formula?
Why do you think they set it up
this way?
x2 y2
Find the distance between Mr. Kim and each food location.
(0, 12)
(8, 6)
(0, 0)
(8, 0)
(16, 0)
From where Mr. Kim starts, if he goes to In-N-Out, Der Veener, and Carl’s, and
back to where he started, how far does he walk?
What’s nice about finding distance when lines are horizontal or vertical?
1.4 – Angles and Their Measures
• Warm – Up
• Graph these lines. We will use them later.
• x=0
• y=0
• y=x
• x = 0 is also known as:
• y = 0 is also known as:
L
A
E
N
1
G
S
Angles are formed by two rays with the same initial
point.
Two rays are called the sides.
The initial point is called the vertex
Definition of congruent angles
m1  m 2
1   2
If two angles are congruent, their measures are equal.
If the measure of two angles are equal, they are
congruent
D
U
1
R
E
C
2
X
How to use a protractor to measure angles.
Protractor Postulate
A
O
B
Consider a point A on one side of OB. The rays of the form OA
can be matched one to one with the real numbers from 0 to 180.
The measure of AOB is equal to the absolute value of the
difference between the real numbers for OA and OB.
Acute – Angle is between __ and __ degrees
Right – Angle is exactly __ degrees
Obtuse – Angle is between __ and
___ degrees
90
180
20
0
180
Straight – Angle is ___ degrees
90
120
A point is in the _______ of an angle if it is
between points that lie on each side of the
angle.
A points is in the _______ of an angle if it is not
on the angle or its interior
D
U
C
Let’s look at the warm-up and identify angels, interior, and exterior points.
BUC and CUD
are adjacent.
BUC and BUD
are not adjacent.
Adjacent angles, share
common side and
vertex, but share NO
interior points.
Angle Addition Postulate
If B is in the interior of
AOC, then
mAOB  mBOC  mAOC
C
O
B
A
In the future with proofs, angle
and segment addition
postulates will be important in
putting together and breaking
apart angles.
Find the measure of the unknown angles, state if they are
acute, right, or obtuse.
B
C
D
A 1
4
m2  20
mDAF  70
2
mCAF 
3
E
F
mBAE  90
mCAE  65
m1 
1
76o
Draw angle ABC that is 90o. Draw right angle DBF so
that angle ABF and DBA is 45o and A is in the interior of
angle DBF and F is in the interior of angle ABC.
Find
mDBA 
mDBC 
mFBC 
• Draw a right angle KIM. Draw angle JIQ such
that M is in the interior of angle JIQ and Q is in
the interior of KIM and JIM is 30 degrees and
MIQ is 60 degrees
1.5 – Segment and Angle Bisectors
Warm – Up: What coordinate is in the MIDDLE of these two points?
(-3, -2) (5, -1)
x1 y1
How did you find it?
What’s another way to think of the
‘middle’ of two numbers?
x2 y2
Find the midpoint.
(-2, -1) (2, 5)
(5, -2) (3, 6)
D
A
E
B
C
DE , DE , BE ,etc
are segment
bise ctors
SEGMENT BISECTOR – A line, segment, or ray
that INTERSECTS THE
_____________________________________!
The ___________ of a segment divides the
segment into __________________parts.
Definition of midpoin t : AB  BC
AB  BC
B is midpoin t of AC
Given an endpoint and the midpoint, find the other
endpoint. A is an endpoint, M is a midpoint
A (5, -2) M (3, 6) B (x, y)
A (2, 6) M (-1, 4) B (x, y)
B
T
20
20
A
R
ANGLE BISECTOR – is a ray that divides an
angle into two adjacent angles that are
congruent.
Definition
of angle bisector
mBTA  mATR
BTA  ATR
TA bisects BTR
BD bisects ABC, find x
A
A
x5
D
D
B
C
1
x  10
2
x 4
2
7x  6
B
C
Constructing a perpendicular bisector.
1) Point on one end, arc up and down.
2) Switch ends and do the same
3) Draw line through intersection
This is DIFFERENT from book (slightly).
Bisect an angle
1) Draw an arc going across both sides of the angle.
2) Put point on one intersection, pencil on other, draw an arc
so that it goes past at least the middle.
3) Flip it around and to the same.
4) Line from vertex to intersection.
1.7 – Introduction to Perimeter, Circumference, and Area
Warm – Up: Things you should know from your past, fill in the blanks
Square
A=
P=
s
Rectangle
l
A=
P=
w
Perimeter of a
Area of a triangle triangle, add up
A
the sides
Circumference is the
distance around the
circle. (Like perimeter)
C = πd = 2πr
Area of a circle:
A = πr2
Find Perimeter\Circumference, and Area for each shape
13 cm
15 cm
5 ft
12 cm
14 cm
3 ft
3 in
6 ft
Find the area and perimeter
12 cm
8 cm
17 cm
Find the area of the figure described
Find the area of a circle with
diameter 10 m
Find the area of a rectangle
with base 4 ft and height 2 ft
Find the area of a triangle
with base 2 in and height 6 in
Find the area of a square
with perimeter 8 miles
Write on board
Finding Area
Mr. Kim needs to make a moat
How many square yards of flooring
around his castle. The radius of
are needed to cover a room that is
the outer circle is 50 feet, the
18 ft by 21 ft?
radius of the inner circle is 40 feet.
What is the area of his moat?