Download Math B Notes: Definitions and Drawing Conclusions

Geometry Notes IGP - 1: Lines and Segments
Undefined Terms
Point: A location in space; no size. (zero dimensions).
Line: A continuous “straight” set of points that extends
indefinitely (forever) in two opposite directions
(one dimension).
Plane: A continuous set of points forming a “flat surface”
extending forever in two dimensions.
Space: All points. Three dimensions. (Not real important until the end of the course. For now, we will do
everything “in a plane.”)
S
Between: In the diagram, P is between A and B; Q and S are not.
P
A
B
Q
Definitions
A ray is a “half line;” it has one endpoint and
extends indefinitely in one direction.
B
P
A
A line segment consists of two endpoints and all the points between them.
B
A
The measure (length) of a line segment is the
Line segment AB (at right) has measure (length) 8:
B
8
A
Note: AB (with the bar over it) represents the actual segment (an object);
AB (without the bar) represents the length of the segment (a number)
Two line segments are congruent () if
E
5
C
D
5
F
Ex: In ABC, AB  BC . If AB = 4x – 10, BC = 2x + 5
and AC = 3x – 2, find the perimeter of the triangle.
The midpoint of a line segment
If M is the midpoint of AB then
Conversely, if
A postulate is a statement (not a definition) that is accepted without proof.
Postulate: Every segment has exactly one midpoint.
Ex: M is the midpoint of line segment PQ . If PM = x2 – 8
and MQ = 2x + 27, find the numerical length of PQ .
Three or more points are collinear if they are all
E
A
B
C
D
F
Ex: Given FLAT , A is the midpoint of LT , FT = 28 and LA is 6 less than twice FL. Find the length of AT .
A bisector of a segment is a line, ray or segment that intersects a segment at its midpoint. Therefore, a bisector
of a segment
Ex:
C
BD bisects CT at A. If BA = 3(x + 1), AD = x2 – 7,
CA = x2 – 2x and AT = 4x – 8, find the length of BD.
x2 – 2x
x2 – 7
3(x + 1)
B
A
4x – 8
T
D
Geometry Notes IGP - 2: Angles
Definitions (continued)
An angle is the union of two rays with a
common endpoint (the vertex).
NOTE: Angles may be named in three ways.
1. By three letters, with the middle letter at the vertex,
2. (Sometimes) by a single letter at the vertex (only if there is no chance of confusion),
3. By a number or lower case letter placed inside the angle near the vertex.
C
Ex: In the diagram at right,
6
A
The measure of an angle is the number of degrees in the angle.
A
4
2 3
D
1
Note: The measure of an angle is a measure of rotation (turning).
It has nothing to do with the “lengths” of the sides.
5
B
50
Acute angle:
Right angle:
Obtuse angle:
Straight angle:
Congruent angles: Two angles that have the same measure.
40
F
40
J
Perpendicular segments (or lines or rays)
B
Ex: Given APC and BP , mAPB = 3x + 20 and
mCPB = 5(x – 2). Determine if PB  APC .
C
P
A
Two angles are complementary if their measures
Two angles are supplementary if their measures
Ex: The measures of two complementary angles are in the ratio 3:5.
Find the measure of the larger angle.
Two adjacent angles share a common ray but have
no interior points in common.
B
A
C
O
A
Angle bisector: A ray that divides an angle into two
congruent angles.
P
Postulate: Every angle has exactly one bisector.
B
Ex: HO  OP , mHOT = 5x + 3 and mPOT = 2x + 28.
Does OT bisect HOP?
O
T
P
H
5x + 3
O
2x + 28
Geometry Notes IGP - 3: Definitions and Drawing Conclusions
Definitions (Review)
In math, a precise definition should work “both ways.” (It is a biconditional.)
Ex: A triangle is a polygon with exactly three sides.
1. If a polygon is a triangle, then it has exactly three sides.
2. If a polygon has exactly three sides, then it’s a triangle.
Ex: A square is a polygon with exactly four sides.
1. If a polygon is a square, then it has exactly four sides.
2. If a polygon has exactly four sides, then it’s a square.
Drawing simple conclusions
We can use definitions to draw simple conclusions.
Ex: Given: M is the midpoint of AB .
.
A
.
M
.B
Conclusion:
Reason:
Note: Remember, in proofs, a “given” is assumed to be true.
Ex: Given: PQR and PQ  QR .
Conclusion:
Reason:
Ex: Given: mABC + mXYZ = 180
Conclusion:
Reason:
.
P
.
Q
.R
Ex: Given: JKL is a right angle.
J
Conclusion:
Reason:
K
L
Conclusion:
Reason:
Conclusion:
Reason:
Ex: Given: BD bisects ABC
A
B
Conclusion:
Reason:
D
C
Not:
Not:
Q
Ex: Given: PQ  QR
Conclusion:
Reason:
P
R
Geometry Notes IGP - 4: Basic Postulates
Postulates (aka Axioms)
A postulate (also called an axiom) is a statement (not a definition) that is accepted without proof.
A theorem is a statement that has been proven using definitions, postulates and previously proven theorems.
Basic Postulates
1. Reflexive Postulate:
2. Transitive Postulate: If two things both equal the same (third) thing, then they equal each other.
Ex: If a = c and b = c then
B
Ex: If AB  BC and BC  CA then
C
A
Ex: Given: AEB  BEC, CED  BEC
E
Conclusion:
A
D
C
B
3. Substitution Postulate: Equal quantities may be substituted for each other in any expression.
Ex: 2x + y = 6
y = 3x + 1
Ex: Given: mAOB + mBOC = 90
mAOB = mCOD
A
B
C
Conclusion:
O
4. Partition Postulate: The whole equals
Ex:
Ex:
.
.
.
.
A
B
C
D
A
B
O
C
D
Ex: For each of the following, name the postulate illustrated.
a. Amy is the same height at Bob. Bob is the same height as Chris. So Amy is the same height as Chris.
b. Amy, Bob, Chris, Don, Emma and Fred are a hockey team. Fred is the goalie. George is another
goalie. So Amy, Bob, Chris, Don, Emma and George are a hockey team.
c. Amy, Bob, Chris, Don, Emma and Fred are a hockey team. Fred is the goalie. Herb is baseball pitcher.
So Amy, Bob, Chris, Don, Emma and Herb are a hockey team.
d. A soccer team is made up three forwards, four midfielders, three fullbacks and a goalkeeper.
e. A basketball team is made up a center, two forwards, two guards and a goalkeeper.
Ex: Which of the following is an example of the reflexive postulate?
(1) Amy looks in the mirror.
(2) Amy is the same height as Amy.
(3) Amy is the same height as Bob.
(4) Amy is taller than Bob. Bob is taller than Chris. So Amy is taller Chris.
(5) None of these.
Ex: Equality is transitive: If a = b and b = c then a = c. Which of the following are also transitive?
a. not equal to ()
b. greater than (>)
c. parallel (||)
d. perpendicular ()
e. “lives in the same town as”
f. “lives next door to”
g. “goes to the same school as”
h. “is related to (by blood)”
Geometry Notes IGP - 5: Addition and Subtraction Postulates
5. Addition Postulate: Equal quantities may be added to both sides of an equation.
Ex: If
and
a=b
x=y
Note: In the Addition Postulate, we always
add two equations to get a new equation.
then
Ex: 2x + 3y = 9
x – 3y = 3
Note: Always line up the equal signs and
add vertically on each side.
A
Ex: Given: ADB , AEC
AD  AE , DB  EC
D
E
F
B
C
Note: For addition of line segments to make sense,
a) They must share an endpoint.
.
A
.
B
.
C
.
D
.C
b) They must be collinear.
.
A
c) They must not overlap.
.
A
AB  BC 
.
B
.
B
.
D
.
C
A
Ex: Given: ABC , FED
AB  ED , BC  FE
F
AB  CD 
AC  BD 
B
E
C
D
Ex: Given: AFB  DCE, BFE  ECB
B
A
F
C
D
E
Note: For addition of angles to make sense, the angles must be adjacent (and non-overlapping).
P
X
R
Y
XOZ + XOY =
Z
O
XOY + YOZ =
YOZ+ PQR =
Q
M
Ex: Given: YDM  NDO
O
Y
N
D
6. Subtraction Postulate: Equal quantities may be subtracted from both sides of an equation.
Ex: If
and
a=b
x=y
Note: In the Subtraction Postulate, we always
subtract two equations to get a new equation.
then
P
Ex: ABCD , AC  BD
A
B
C
D
Note: For subtraction of line segments to make sense,
a) They must share an endpoint.
AC  AB 
.
B
.
A
b) They must be collinear.
..
C
AC  BC 
c) They must overlap.
AB  BC 
Ex: NRT , NGL, NT  NL, RT  GL
N
R
G
T
L
B
A
Ex: ABC  ADC, ABD  CDB
D
C
Note: For subtraction of angles to make sense, the angles must
X
a) share a ray and
XOZ – XOY =
Y
b) overlap
XOZ – YOZ =
YOZ – XOY =
O
Z
P
Ex: Given: QPS  TPR
Q
R
B
D
S
T
C
Geometry Notes IGP - 6: Multiplication and Division Postulates
7. Multiplication Postulate: Both sides of an equation may be multiplied by equal (non-zero) quantities.
Ex: If
and
a=b
x=y
then
x2 2 x 1 1


6
3
2
Ex:
8. Division Postulate: Both sides of an equation may be divided by equal (non-zero) quantities.
Ex: If
and
a=b
x=y
then
Ex:
4y = 3x + 20
Variation: “Halves* of equal quantities are equal.”
(* or thirds or fourths, etc)
If a = b, then
E
A
Ex: Ex: ABC  ADC
DE bisects ADC
BF bisects ABC
2
D
1
F
C
B
Geometry Notes IGP - 7: Statement-Reason Proofs
Proofs
A formal geometry proof is a series of statements in logical order. Each statement is justified by a reason.
Statements
1. Should start with one or more givens
2. Are facts/true that are relevant to the problem
3. Should follow a logical order
Each new statement should either
a. Be a direct conclusion from one or more previous statements or
b. Go together with one or more previous statements to lead to a conclusion
4. The final statement is whatever was to be proved.
Reasons
1. Should explain why the statement is true, often buy referring to previous statements
2. Acceptable reasons are
a. Given (but only if the statement really was given!)
b. Definitions: write them out.
c. Postulates: by name for the few that have a name; otherwise write them out.
d. Previously proven theorems: write them out.
J
Ex: Given: KJM  NJL
Prove: KJL  MJN
1. Mark the givens on the diagram. (See what you know.)
2. Work backwards. (Find out what you need to prove.)
3. Try to have a plan. (Figure out how to get from
what you know to where you need to go.)
4. Write the proof.
K
L
B
D
M
N
C
Ex: Given: AMPL , AM  EX , EX  PL
Prove: AP  ML
E
A
X
M P
L
Geometry Notes IGP - 8: Simple Angle Theorems
A theorem is a statement that has been proven using definitions, postulates and/or previously proven theorems.
Theorem: All right angles are congruent.
Given: A and B are right angles
Prove: A  B
Theorem: All straight angles are congruent.
Theorem: If two adjacent angles form a straight line, they are supplementary.
Given: AOC and BOC, AOB
Prove: AOC and BOC are supplementary
C
A
O
B
Theorem: If two adjacent angles form a right angle, then they are complementary.
Theorem: If two angles are congruent, then their supplements are also congruent.
Given: 1  4, 2 supp. to 1, 3 supp. to 4
Prove: 2  3
2
1
3
4
Theorem: If two angles are supplementary to the same angle, then they are congruent.
Note: The previous two theorems are still true if the words “supplements” and “supplementary” are replaced by
“complements” and “complementary”.
Definition: Vertical angles are non-adjacent angles
formed by two intersecting lines.
1
4
2
3
Theorem: Vertical angles are congruent. (Prove for HW.)
Ex: Given: ABCD , ABP  DCP
Prove: CBP  BCP
P
A
Statement
B
C
Reason
N
Ex: Given: MOR , LOQ , NO  LO , PO  OR
Prove: MON  QOP
Reason
P
Q
M
L
Statement
D
O
R
Geometry Notes IGP – 9/10: Proofs Practice
Ex: Given: AB  AC , AE  AF
Prove: BAE  FAC
D
C
E
A
B
F
T
Ex: Given: PIW , GIN , IT bisects PIG
Prove: NIT  WIT
P
G
I
N
W