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Chapter 1.notebook
September 02, 2014
Undefined Terms
Line
l
A
Point
A B
A point has no length, no width, no height. Represented by a dot. Named with a capital letter
A line has infinite length, no width, no height. Represented by a line with two arrowheads to show that it extends infinitely. A line is named after any two points which lie on the line and the line symbol is written above it or with a script lower case letter
AB
line l
Lines may have more than one proper name, but no two lines may have the same name.
B
A
C
E
D
Plane
A
C
B
A plane has infinite length and infinite width, but no height. Represented by a shape that looks like a floor or a wall (slanted rectangle to obtain, 3D effect. Planes are named by any 3 noncollinear points which lie in the plane.
Plane ABC
Chapter 1.notebook
September 02, 2014
Collinear Points: Points that lie on the same line
2 points are always collinear
A
F
3 points are sometimes collinear
Coplanar points are points that lie in the same plane
2 or 3 points are always coplanar
4 or more points are sometimes coplanar
E
D
C B
H
G
Defined Terms: Terms that can be described using the undefined terms and other defined terms
A
B
Line Segment (segment): Part of aline
consisting of two end and all points
points on AB between A and B.
Line segments are named after the two endpoints and have a line segment AB
symbol over them
Chapter 1.notebook
September 02, 2014
point
Ray: part of a consisting of one end line
and all points on AB that lie on the same side of A as B.
A
B
A ray is named with its endpoint first and then any other point on the ray and the ray symbol is put above the two points
2 Final Definitions
Opposite Rays: two collinear rays with the same endpoint but going in opposite directions.
A
A ray can have more than one name
A B C
D
AB AC AD Since rays must be named with the endpoint first AB and BA are not the same ray.
B
A
AB
B
Special notes about naming rays
Intersection: Two or more geometric figures if they have two or more intersect
points in common. The intersection of the figures is the set of all points that they share.
C
AB and AC are opposite rays
the intersection of two different lines is a ____________
the intersection of two different planes is a _______
Chapter 1.notebook
September 02, 2014
5 questions...... 1.) What has infinite length, but no measurable height or width?
m
Y
Z
X
2.) Give 2 other names for line m
3.) True or False: 3 points are always collinear.
4.) True or False: AB and AD are opposite rays.
D
A
B
P
5.) Give an acceptable name for the plane.
M
A
B
AB = BC =
BD = AD =
DC = CA =
C
D
The distance between point A and B, written as AB, is the absolute value of the difference of the coordinates of A and B.
The ruler postulate tells us how to use a ruler. W
K
­4 ­3 ­2 ­1 0 1 2 3 4 5 6 7 8 9
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 called the coordinates of the point.
The distance between two points is represented by writing the 2 points with no symbol over it. AB is read "The distance between A and B" or "The length of line segment AB"
Adding Segment Lengths: When 3 points are collinear, you can say that one point is between the other two.
Segment Addition Postulate: If B is between A and C, then AB + BC = AC (also known as the definition of betweeness of points
Chapter 1.notebook
September 02, 2014
Line segments that have the same length are called CONGRUENT SEGMENTS
Congruent and equal are two totally different things!!!!!!!!!!!!!!!!!!!!
Symbol:
Congruent must be used with Geometric ~
Figures (such as segments) AB = CD
How indicated on a drawing:
Equal must be used with numbers (such as lengths) AB = CD
A couple more definitions!!
A
9
B
9
4
E
5
F G
C
H
D
Midpoint: The midpoint of a segment is the point that divides the segment into two M
B
A
congruent segments
* every segment has exactly one midpoint
True or False
AB = CD AB = CD
~
~
AB = CD
AB = CD
EF = GH EF + GH = CD
~
EF + GH = CD
Segment Bisector: A segment bisector is a point, ray, line, line segment or plane that intersects a segment at its midpoint. Bisects is a verb.
A
B
M
TM bisects AB
T
Chapter 1.notebook
September 02, 2014
A
M
B
Given: M is the midpoint of AB and that AM = 12 then MB = ____
Given: M is the midpoint of AB and that AM = 16 and MB = x + 4, then x = ____
Given: M is the midpoint of AB and that AM = 3x + 2 and MB = 5x ­ 6, then x = ___
and AM = ____ and AB = _____
What is the midpoint of a segment with endpoints (3,4) and (­5,12)?
What is the midpoint of a segment with endpoints (0,5) and (3,­12)?
Find AM and AB
3x + 2
A
M
Given: M is the midpoint of AB
4x ­ 5
Midpoint Formula: To find the midpoint of any two points on a coordinate system, you use the following formula
Midpoint(x , y ) = (x + x , y + y )
2
1
m
1
m
2
2 2
What is the midpoint of a segment with endpoints (4,5) and (10,3)?
What is the midpoint of a segment with endpoints (2,­4) and (­3,8)?
The midpoint of a segment is (3,9) if the one endpoint is (2,4) what is the other endpoint?
Two ways to do it!!!!
Algebra
B
Logic
Chapter 1.notebook
September 02, 2014
Distance formula for coordinate system
B (5,7)
Distance Formula
d = (x1 ­ x2 )2 + (y1 ­ y2 )2
A
(1,4)
Ex 1: Find the distance between the points (3,4) and (7,9)
(x2,y2 )
A2 + B2 = C2
Find the midpoint of a segment with endpoints (2,8) and (­6,4)
(x1,y1 )
d = ( ­ )2 + ( ­ )2
Ex 2: How about between
(­3,2) and (0,­6)
5 Big Things
1.) If M is the midpoint of AB and AM = 8 then AB = _____
If the midpoint of a segment is the point (2,7) and one endpoint is (­1,8), what is the other endpoint?
Find the distance between the points (3,6) and (2,10)
Find the length of a segment with endpoints (8,1) and (3,­4)
The endpoints of two segments are ~
given below: Are AB and CD =?
A: (2,7) & B: (­1,3) C:(8,3) & D: (7,8)
2.) Which of the following is correctly written;
a.) AB = XYb.) AB = XYc.) AB = XY
~
3.) If the endpoints of a segment have coordinates (3,9) and (­1,3) the midpoint of that segment is __________
4.) What is the length of a segment with endpoints (4,9) and (7,5)?
5.) AB =
A
­2 B
0
2
4
6
8
Chapter 1.notebook
September 02, 2014
Angle: An angle consists of two different rays with the same endpoint.
A
B
C
The 2 rays are called the sides of the angle.
Other naming techniques:
An angle can be named after only its vertex if it is the only angle in the diagram with that vertex.
Angles are sometimes named with numbers located inside the angle.
The common endpoint is called the vertex of the angle.
W
Y
1
An angle is named using 3 letters(points) the middle letter is the vertex and one other point from each side is put in the first and third places. The angle symbol is put in front
2
X
Z
F
D
Protractor Postulate: Consider OB and 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
The protractor postulate tells us how to use a protractor
C
E
A
m CAD=
m DAF=
m FAE=
m CAE=
m DAE=
m CAF=
m 1=
1
2
m 2=
Chapter 1.notebook
There are 4 kinds of angles:
Acute:
Right:
September 02, 2014
Angle Addition Postulate (AAP): If P is on the interior of RST, then the measure of RST is equal to the sum of the measures of RSP and PST
Obtuse:
P
R Straight:
S
T
m RSP + m PST = m RST
Congruent Angles: Angles with the same measure. (aka­­­same shape and size)
Symbols:
B
A
D
1
2
C
o
m in front of the angle name means "the measure of angle" This is a number
~
= means congruent and refers to the shapes not the measures Not a number
E
o
Given: m ACE = 139 , m BCD = 99 , ACB = DCE
1.) Give another name for ACB?
2.) (True or False) ACD and DCA are the same angle?
3.) (True or False) ACB = DCE
4.) m ACB =
5.) m DCE =
6.) m ACD =
7.) m 2 =
Chapter 1.notebook
September 02, 2014
A
5 Big Things
Angle Bisector: An angle bisector is a ray that divides an angle into two angles that are congruent.
A
If m ABC = 72 then m ABD=
D
1.) Give another name for /1:
C
1
B
X
2.) Classify /1 (acute, right, obtuse or straight)
3.) m/ 2 = 25o, m/ 3 = 40o, m/ XYZ = _____
2
3
Y
Z
F
4.) m/ DEF = ____
W
5.) m/ FEG = ____
If m ABC = 2x+4
and m DBC=32 then x =
G
D
E
B
C
BD bisects ABC
By Location
Angles Pairs!!!!!!
BY SIZE
Adjacent Angles: 2 angles that share a common vertex and side, but no common interior points.
Complementary Angles: 2 angles with measures that add up to 90o (complement)
The 3 Questions
Are BAC and BAD
adjacent?
Do they have the same vertex?
B
C
Supplementary Angles: 2 angles with measures that add up to 180o
(supplement)
A
D
Are BAC and CAD adjacent?
Do they completely share a side?
Do they not overlap at all?
Chapter 1.notebook
September 02, 2014
Linear Pair: two adjacent angles with non­common sides that are opposite rays.
ABC and CBD are a linear pair
C
Vertical Angles: 2 angles with sides that form 2 pair of opposite rays.
2
4
A
B
D
1
3
1 and 4 are vertical angles
Name 4 different linear pairs
2 and 3 are vertical angles
2
1
3
4
Given: BD bisects /ABC
A
Given: m/XYZ =79o
5 Big Things
D
E
5x + 3
7x ­ 6
F
B
C
4x ­ 3
3x + 15
1.) /ABC and /ABD are adjacent angles. (True or False)
x=
x=
2.) /ABD and /DBC are adjacent angles. (True or False)
3.) /ABC and /EBF are vertical angles. (True or False)
4x + 4
8x + 20
x=
83 ­ x
3x ­ 3
4.) /EBA and /ABD are a linear pair. (True or False)
5.) /FBC and /ABC are a linear pair. (True or False)
x=
Chapter 1.notebook
September 02, 2014
A figure that lies in a plane is called a plane figure
The SIDES of a polygon are the segments that make it
A POLYGON is a closed plane figure formed by 3 or more segments with each segment intersecting exactly 2 segments so that no two segments with a common endpoint are collinear.
The VERTICES (plural of vertex) of a polygon are the endpoints of the sides.
L
A
Name the sides and vertices of this polygon
G
K
There are two classifications of polygons
CONVEX
CONCAVE
POLYGONS are named based on the number of sides they have
Sides
3
4
5
6
7
8
9
10
12
n
Name
Chapter 1.notebook
September 02, 2014
Special Polygons
Perimeter, Circumference and Area
Equilateral:
Equiangular:
Perimeter:
Regular:
Circumference:
Area:
RECTANGLE
Some Basic Shapes and Formulas
SQUARE
1 sq unit
1 sq un
4 units
6 units
6 units
Perimeter
P = 4s
Area
A = s2
10 units
Perimeter
P = 2L + 2W
Area
A=LW
Chapter 1.notebook
September 02, 2014
TRIANGLE
A triangle is really two half rectangles?
10 units
CIRCLE
π = Approx 3.14
6 units
7 units
Circumference
8 units
9 units
Perimeter
P = a + b + c
12 units
C = πd = 2πr
Area
Area
A=bh/2
Converting Regular Units
Converting Square Units
1 ft = ____ in
1 yd = ____ ft
1 yd = ____ in
1 mi = ____ ft
1 m = ____ cm
1 km = ____ m
1 sq ft = ____ sq in
1 sq yd = ____ sq ft
1 sq yd = ____ sq in
1 sq mi = ____ sq ft
1 sq m = ____ sq cm
1 sq km = ____ sq m
A = πr2
Five Big Things:
1.) Name the polygon and tell if it is
convex or concave.
5 m
2.) What is the area of the following square?
3.) What is the perimeter of the following rectangle?
2 in
6 in
4.) What is the area of the following triangle?
14 mm
5.) 54 square feet = ______ square yards
12 mm
4 mm
5 mm
Chapter 1.notebook
September 02, 2014
CHAPTER 1 TEST
1­3: Diagram 4­6: SAP and Def. of Midpoint
7: Find the midpoint
8: Find the other endpoint
9­10: Distance Formula
11­13: Distance and Midpoint w/ Graph
14­15: Using a Protractor
16­17: Complement and Supplement
18­21: Def of Bisector and AAP
22­24: Vertical, Linear or Neither
25­26: Polygons (name and convex or concave)
27­28: Perimeter or Circumference
29­32: Area (work problems)
33: HLP 8 POINTS
34­43: Multiple Choice