Download Geometry Notes Math 2 through March 9

Geometry
Math II, Unit 3
Points
Lines
Planes
There are three undefined terms in geometry:
Point
Line
Plane
*They are undefined because they have to be
explained using examples and descriptions.
Point
• Simply a location
A
• Drawn as a dot and named
with a
capital letter
• A point has neither shape nor size
• Say "point A"
Line
is made up of points and has neither thickness or width
Drawn with arrows on both ends
Labeled with capital letters representing points or a lowercase script letter
say "line AB" or "line l "
write AB or l
B
A
There is exactly one line through any two points
l
Plane
a flat surface made up of points
Has no edges or sides, but is drawn as a 4-sided figure
There is exactly one plane through any 3 noncollinear points.
Name a plane with 3 noncollinear letters
or one capital script letter.
T
X
Y
Plane XYZ, Plane ZXY, Plane T
Say "Plane XYZ" or "Plane T "
Z
Other important terms:
Collinear points: points all in the same line
B
Which points are collinear?
A
Non-collinear points: Points not all in the same line.
C
Other important terms:
Coplanar Points: Points all in the same plane
Non-Coplanar Points: points not all in the same plane
B
C
D
A
E
H
F
G
Other important terms:
Space: a bound-less, 3-dimensional set of all points.
Can contain lines and planes.
Line Segment
• Line Segment: The part of a line that connects two points.
B
• It has definite end points.
• Adding the word "segment" is important,Abecause a line normally
extends in both directions without end.
• Write AB
Ray
• Ray: A line with a start point but no end point (it goes to infinity)
• Write: AB
B
A
C
Angle Pair
Relationships
Angle Pair Relationship Essential
Questions
How are special angle pairs
identified?
Straight Angles
Opposite
rays are two rays that are part of a the same line and have
___________
only their endpoints in common.
Y
X
Z
opposite rays
XY and XZ are ____________.
The figure formed by opposite rays is also referred to as a
straight angle A straight angle measures 180 degrees.
____________.
Angles – sides and vertex
There is another case where two rays can have a common endpoint.
angle
This figure is called an _____.
S
Some parts of angles have special names.
vertex
The common endpoint is called the ______,
and the two rays that make up the sides of
the angle are called the sides of the angle.
R
vertex
side
T
Naming Angles
There are several ways to name this angle.
1) Use the vertex and a point from each side.
SRT or
TRS
The vertex letter is always in the middle.
R
R
If there is only one angle at a vertex, then the
angle can be named with that vertex.
1
1
side
2) Use the vertex only.
3) Use a number.
S
vertex
T
Angles
An angle is a figure formed by two noncollinear rays that
have a common endpoint.
Symbols:
D
Definition
of Angle
DEF
FED
E
2
E
F
2
Angles
1) Name the angle in four ways.
ABC
CBA
B
C
A
1
1
2) Identify the vertex and sides of this angle.
vertex: Point B
sides:
BA
and
BC
B
Angles
X
1
2
1) Name all angles having W as their vertex.
W
Y
2) What are other names for
1 ?
3) Is there an angle that can be named
Z
W?
Angle Measure
Once the measure of an angle is known, the angle can be
classified as one of three types of angles. These types are defined
in relation to a right angle.
Types of Angles
A
obtuse angle
90 < m
A < 180
A
A
right angle
m
A = 90
acute angle
0 < m A < 90
Angle Measure
Classify each angle as acute, obtuse, or right.
110°
40°
90°
50°
130°
75°
Adjacent Angles
When you “split” an angle, you create two angles.
The two angles are called
_____________
adjacent angles
adjacent = next to, joining.
A
B
2
1
1 and 2 are examples of adjacent angles.
They share a common ray.
Name the ray that 1 and 2 have in common.
C
BD
____
Adjacent Angles
Adjacent angles are angles that:
A) share a common side
B) have the same vertex, and
C) have no interior points in common
Definition of
Adjacent
Angles
J
R
1 and 2 are adjacent
with the same vertex R and
2
1
common side RM
N
Adjacent Angles
Determine whether 1 and 2 are adjacent angles.
They have a common vertex B, but
_____________
2
1
B
1
They have the same vertex G and a
common side with no interior points in
common.
2
G
N
L
J
2
1
They do not have a common vertex or
____________
Adjacent Angles and Linear Pairs of Angles
Determine whether 1 and 2 are adjacent angles.
1
2
1
X
2
D
Z
In this example, the noncommon sides of the adjacent angles form a
___________.
These angles are called a _________
Linear Pairs of Angles
Two angles form a linear pair if and only if (iff):
A) they are adjacent and
B) their noncommon sides are opposite rays
A
1
Definition of
Linear Pairs
D
B
2
1 and 2 are a linear pair.
BA and BD form AD
1  2  180
Linear Pairs of Angles
In the figure, CM and CE are opposite rays.
H
1) Name the angle that forms a
linear pair with 1.
T
A
2
1
3 4
C
M
2) Do 3 and TCM form a linear pair? Justify your answer.
E
Complementary and Supplementary Angles
Two angles are complementary if and only if (iff)
The sum of their degree measure is 90.
E
Definition of
Complementary
Angles
A
B
D
30°
C
60°
F
mABC + mDEF = 30 + 60 = 90
Complementary and Supplementary Angles
If two angles are complementary, each angle is a
complement of the other.
ABC is the complement of DEF and DEF is the
complement of ABC.
E
A
B
D
30°
C
60°
F
Complementary angles DO NOT need to have a common side
or even the same vertex.
Complementary and Supplementary
Angles
Some examples of complementary angles are shown below.
15°
H
75°
P
I
mH + mI = 90
Q
40°
mPHQ + mQHS = 90
50°
H
S
U
T
60°
V
mTZU + mVZW = 90
30°
Z
W
Complementary and Supplementary Angles
If the sum of the measure of two angles is 180, they form a
special pair of angles called supplementary angles.
Two angles are supplementary if and only if (iff) the
sum of their degree measure is 180.
D
C
Definition of
Supplementary
Angles
50°
A
130°
B
E
mABC + mDEF = 50 + 130 = 180
F
Complementary and Supplementary Angles
Some examples of supplementary angles are shown below.
H
75°
105°
I
mH + mI = 180
Q
130°
50°
H
P
S
U
V
60°
120°
60°
Z
T
mPHQ + mQHS = 180
mTZU + mUZV = 180
and
W
mTZU + mVZW = 180
Congruent Angles
measure
Recall that congruent segments have the same ________.
Congruent
angles
_______________
also have the same measure.
Congruent Angles
Two angles are congruent if, they have the same
degree measure
______________.
Definition of
Congruent
Angles
B  V iff
50°
50°
B
V
mB = mV
Congruent Angles
To show that 1 is congruent to 2, we usearcs
____.
1
2
To show that there is a second set of congruent angles, X and Z,
we use double arcs.
This “arc” notation states that:
X  Z
X
mX = mZ
Z
Vertical Angles
four angles are formed.
When two lines intersect, ____
There are two pair of nonadjacent angles.
vertical angles
These pairs are called _____________.
4
1
3
2
Vertical Angles
Two angles are vertical iff they are two
nonadjacent angles formed by a pair of
intersecting lines.
Vertical angles:
Definition of
Vertical
Angles
4
1
3
1 and 3
2
2 and 4
Vertical Angles
Vertical angles are congruent.
Theorem 3-1
Vertical
Angle
Theorem
n
2
m
1  3
3
1
4
2  4
Vertical Angles
Find the value of x in the figure:
130°
x°
The angles are vertical angles.
So, the value of x is 130°.
Vertical Angles
Find the value of x in the figure:
(x – 10)°
125°
The angles are vertical angles.
(x – 10) = 125.
x – 10 = 125.
x = 135.
Congruent Angles
Suppose A  B and mA = 52.
Find the measure of an angle that is supplementary to B.
1
52°
A
B + 1 = 180
1 = 180 – B
1 = 180 – 52
1 = 128°
52°
B
Knowledge Check!!
G
D
1) If m1 = 2x + 3 and the
m3 = 3x + 2, then find the
m3
1
A
4
3
2
B
C
E
2) If mABD = 4x + 5 and the mDBC = 2x +
1, then find the mEBC
H
3) If m1 = 4x - 13 and the m3 = 2x + 19, then find the m4
4) If mEBG = 7x + 11 and the mEBH = 2x + 7, then find the m1