Download Geometry Ch 1 Notes

Unit 1 Basics of Geometry
Objective- the students will be able to use undefined terms and definitions to work with
points, lines and planes.
Undefined Terms
1. Point – has no dimension, geometrically looks like a dot, represented by a capital
letter. Read point A
.A
2. Line – extends indefinitely, represented by two points or a scripted letter. Read
line X, Y.
X
Y
3. Plane – is a flat surface, has no thickness, represented by a capital letter. Read
Plane M.
M
Space – set of all points.
Collinear points – a set of points that lie on one line.
Coplanar points - a set of points that lie on one plane.
Here are different ways of expressing relationships between points, lines, and planes.
A lies in k
A lies on k
k contains A
A
m and n intersect at P
m and n intersect in P
P
k is drawn through A
k
the intersection of m and n is P
m
n
S and h lie in Z
Z contains S and h
•S
h
Z
Terminology
Postulate (axiom) – is a statement (basic assumption) assumed to be true without proof.
Theorem – is a statement that has to be proved.
Corollary – is a special case of a theorem.
Basic Assumptions, the first
Postulate –
A line contains at least two points, a plane contains at least three points
not all on one line, space contains at least 4 points not all n one plane.
Postulate –
Through any two points there is exactly one line.
Postulate –
Through any three points not on one line there is exactly one plane.
Postulate –
If two points lie in a plane, then the line joining them is in that plane.
Postulate –
If two planes intersect, then their intersection is a line.
Subsets of a Line
Segment -
Given any two points X and Y, segment XY is the set of all points
consisting of X and Y and all the points that lie between X and Y. X and
Y are the endpoints.
X
Y
Ray-
Ray RS, denoted by RS is the union of RS and all the points X for which
it is true that S lies between R and X.
S
R
Opposite Rays -
!##"
SR and ST are called opposite rays if S lies on RT between R
and T
S
Congruent segments-
T
R
segments with equal lengths. The symbol for length of a
AB is AB.
If AB = XY, then AB ! XY
Midpoint of a segment-
Point M is the midpoint of AB if M lies on AB and
AM = MB
A
Bisector of a segment-
M
B
a line, segment, ray, or plane that intersects AB at its
midpoint is a bisector of AB
Segment Addition Postulate-
If B lies on AC , then AB + BC = AC
A
B
C
Symbols
!##"
AB
line containing A and B
!!!"
AB
ray with endpoint A, through B
AB
segment joining A and B
AB
length of AB
Postulate -
For any two points there is a unique positive number called the distance
between the points
Ruler Postulate
The points on a line can be paired with the Real Numbers in such a
way that:
a.
any desired point can be paired with zero;
b.
the distance between any two points is equal to the
absolute value of the difference of the numbers
paired with those points.
Midpoint
Ex.
Find the coordinate of the midpoint of a segment that connects (10,0) and
(12, 0).
Draw a number line and find the middle of the two coordinates.
10 + 12 22
Using arithmetic
=
2
2
the midpoint is (11,0)
Ex.
Find the coordinates of the midpoint of a segment with A(2,3) and B (4,5)
as the endpoints.
Draw a number line, find the middle of the line segment connecting 2 and
4.
! 2 + 4 3+ 5$
! 6 8$
,
Using arithmetic, we have #
&% ' #" , &% or ( 3, 4 )
" 2
2
2 2
Midpoint Formula
Those examples suggest the midpoint formula-
! x + x2 y1 + y2 $
midpo int = # 1
,
&
" 2
2 %
Distance Formula
d = x2 ! x1 )2 + (y2 ! y1 )2
Ex.
Find the distance between (4,7) and (8,3)
d = x2 ! x1 )2 + (y2 ! y1 )2
d = (8 ! 4)2 + (3 ! 7)2
d = 4 2 + (!4)2
d = 16 + 16
d = 32 = 16 • 2
d=4 2
Reasoning – Logic
Deductive reasoning –
(deduce) reasoning by which a conclusion is reached based
on accepted statements.
Inductive reasoning-
conclusions are based on observing individual cases and
then stating a general principle.
Objective-
students will learn how to write angles, learn definitions and classification
of angles and angle pairs
Angle -
is the union of two rays with a common endpoint. The common endpoint
is called the vertex.
X
vertex
A
Y
3 Ways to Name an Angle
1.
Name an angle by the vertex - !B
A
1
B
C
2.
Name an angle by three letters, one point on each ray and the vertex being the
middle
!ABC or !CBA
3.
Name the angle by a number written in the interior of the angle
!1
Angle Classification
Acute angles are angles less than 90º.
Right angles are angles whose measure is 90º.
Obtuse angles are greater than 90º, but less than 180º.
Straight angles measure 180º.
Postulate –
For every angle there is a unique number between 0 and 180 called the
measure of the angle.
Protractor Postulate-
The set of rays which have a common endpoint O can be
paired with the numbers between 0 and 180 inclusive in
such a way that;
a. one of the rays is paired with zero and the other is
paired with 180;
!!!"
!!!"
b. if OA is paired with x and OB is paired with y, the
m ! AOB = |x–y|
Angle Addition Postulate- If B lies on the interior of ! AOC,
then m ! AOB + m ! BOC = m ! AOC
If point B lies in the interior of ! AOC, then m ! AOB + m ! BOC = m ! AOC.
A
B
O
C
The Angle Addition Postulate just indicates the sum of the parts is equal the whole. That just
seems to make sense.
Ex.
If m!AOB = 30˚ and m!BOC = 15˚ , find the m!AOC .
m!AOB + m!BOC = m!AOC
30˚+15˚= m!AOC
45˚= m!AOC
N.B. The parallel construction of these angle postulates and the segment postulates.
Angle Pairs
Adjacent angles are two angles that have a common vertex, a common side, and no common
interior points.
B
A
X
C
! AXB and ! BXC are adjacent angles. They have a common vertex – X, they have a common
side XB and no common interior points.
!!!"
Angle bisector; AX is said to be the bisector of ! BAC if X lies on the interior of ! BAC and
m ! BAX = m ! XAC.
B
X
A
C
Ex.
!!!"
AX bisects !BAC . If m!BAC = 110˚ , find the m!XAC
m!BAX + m!XAC = m!BAC ;
m!BAX = m!XAC
m!XAC + m!XAC = 110˚
2m!XAC = 110˚
m!XAC = 55˚
Ex.
!!!"
AX bisects !BAC .
value of x
!BAX = 6x + 5 and
m!BAX = m!XAC
6x + 5 = 2x + 13
4x = 8
x=2
!XAC = 2x + 13 , find the
Complementary angles – are two angles whose sum is 90˚.
Ex.
If ! A = 30˚, then the complement of ! A measures 60˚.
Supplementary angles – are two angles whose sum is 180˚
Ex.
If m ! M = 100˚ and m ! S = 80˚, then ! M and ! S are supplementary
angles.
Ex.
Find the value of x, if ! A and ! B are complementary ! ’s and ! A = 3x and
! B = 2x + 10.
! A + ! B = 90˚
3x + (2x + 10) = 90˚
5x + 10 = 90˚
5x = 80
x =16
Theorem
If the exterior sides of two adjacent angles lie in a line, they are supplementary.
These angles, ! 1 and ! 2, are called linear pairs.
1
2
Vertical Angles
The mathematical definition of vertical angles is: two angles whose sides form pairs of opposite
rays.
1
2
! 1 and ! 2 are a pair of vertical angles.