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Geometry Section 2.1
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Start Thinking: Aaron looks back in his agenda and notices that two Tuesdays ago he had a pop quiz and
last Tuesday they also had a pop quiz. It is Monday. What might Aaron want to do?
Inductive Reasoning:
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Conjecture:
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REAL WORLD EXAMPLES
Examples: Determine the next two terms in the sequence.
1.) 2,4,6,8,10,……..
2.) 5, 10, 20, 40,…….
3.) 20, 16, 12, 8, ……..
2.) 1,1,2,3,5,8,……..
3.)1, 2, 3, 3, 6, 18, 54, …..
Helpful Hints:
1.)
2.)
3.)
4.)
Harder Ones:
1.) 3,-6, 12, -24, …….
4.)F, S, T, F, F, ……..
5.) M, T, W, T,…….
6.) A, B, D, E, G, H, …
More Practice:
7.) 5, 11, 18, 26, …..
8.) 1,1,2,6,24,120,…..
1 1 1
10.) 1, , , ,......
4 9 16
11.)T,F,S,E,T,T,F, S,……
9.) 30, 20, 11, 3, …..
12.)

NOW IT IS TIME TO THINK OF YOUR OWN…..
1.)
2.)
PROBLEMS WITH INDUCTIVE REASONING:
Counter Example:
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1.) A triangle can be created if you are given 3 points.
2.) When you multiply a number by 2 it becomes larger.
Real World:
1.)
Geometry Section 2.5
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Start thinking: Prove the following with the given information and other facts that you
can think of. You must justify everything you state in your reasoning process.
Background Information: You overhear your aunt talking about how she wants to go to
Vancouver, Canada but she does not know if she is allowed to. You do know that she does
have a passport. Prove that she is legally allowed to go to Vancouver Canada.
Reasoning Put Into Words
Two column Proof
PROPERTIES:
Property
Addition Property:
Subtraction Property:
Multiplication Property:
Reflexive Property:
Example
Property
Example
Symmetric property:
Transitive Property:
Substitution Property:
Distributive Property:
PRACTICE: Name the property that justifies each of the following.
1.) 2x + 1 = 11
2x = 10
2.) 3x = 9
x=3
3.) 3(x + 1)
3x +3
4.) x = -2 and y = 7x +1
5.) AB  AB
6.)
x
 10
2
x = 20
y = 7 (-2) +1

Examples: Write a two-column proof of the following.
Given : 2x +1 =9
Prove: x = 4

Example 1: Fill in the missing parts of the proof.
Example 2:
Example: Fill in the missing parts of the following proof
Geometry Section 2.6
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Start thinking: Solve for x in the following.
1.)
Theorem 2.1: Vertical Angles Theorem:
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Complimentary Angles: Look at the figures to the right. Notice that <4 and <5
are complimentary and <5 and <6 are complimentary.
What can you conclude about <4 and <6?
Theorem 2.2: If two angles are compliments of the same angle then the two angles are
________________
Supplementary Angles: Look at the figures to the right. Notice that <1 and <2
are supplementary and <2 and <3 are supplementary.
What can you conclude about <1 and <3?
Theorem 2.3: If two angles are supplements of the same angle then the two angles are ________________
PROOF:
Given: Figure above and that <1 and <2 are supplementary and <3 and < 2 are supplementary.
Prove:  1  2

OTHER THEOREMS:
Theorem 2.4: All right angles are ____________________________________.
PROOF:
GIVEN: Angle 1 and Angle 2 are right.
Prove : Angle 1 is congruent to Angle 2
Theorem 2.7: Congruent and Supplementary Angles: If two angles are both congruent and
supplementary then the two angles are
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Ex3: Use the Congruent Supplements Theorem
Given 1 and 2 are supplements. 1 and 4 are supplements. m2 = 45°
Prove m4 = 45°
Statements
1. 1 and 2 are supplements. 1 and 4 are supplements.
Reasons
1)
2.
2) Congruent Supplements Theorem
3. m2 = m4
3)
4. m2 = 45°
4)
5.
5) Substitution Property of Equality
Geometry Section 3.1
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Start Thinking: Come up with a correct mathematical definition of parallel lines.
Vocabulary:
Terms/Definition
Notation and Example
Parallel lines:
Skew lines:
Parallel Planes:
Ex1: Identify relationships in space
Think of each segment in the figure as part of a line. Which line(s) or plane(s) in the figure
appear to fit the description?
a.Line(s) parallel to
AF
b. Line(s) skew to AF
c. Line(s) perpendicular to
and containing point E
and containing point E
AF
and containing point E
d. Plane(s) parallel to plane FGH and containing point E
EX2: Identify parallel and perpendicular lines
Use the diagram at the right to answer each question.
a. Name a pair of parallel lines.
b. Name a pair of perpendicular lines.
c. Is AB  BC ? Explain.
Definition:
TRANSVERSALS
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TRANSVERSALS create FOUR special angle pair relationships
l
2
1
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3
4
5
6
8
m
7
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l
2
1
3
4
5
8
6
m
7
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l
2
1
3
4
5
8
6
m
7
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l
2
1
4
5
8
6
7
3
m
Ex 3: Identify the relationship between the angles given:
a. 1 and 9
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b. 8 and 13
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c. 6 and 16
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d.
4 and 10
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e.
8 and 16
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f.
10 and 13 ____________________________________
PRACTICE:
1) Identify all pairs of all of the angles that have the desired relationship to the given angle.
a.) AIA with <20
b.)SSIA with <30
c.)CA with <19
d.) SSIA with <17
e.)AIA with <29
2.) Think of each segment in the diagram as part of a line. Complete the statement with
parallel, skew, or perpendicular.
a.
WZand ZR are ___________________.
WZand ST are ___________________.
c. QT and YS are ____________________.
b.
d.
Plane WZR and plane SYZ are ______________________.
e.
Plane RQT and plane YXW are ______________________.
2) Think of each segment in the diagram as part of a line. Which line(s) or plane(s) appear to
fit the description?
a. Line(s) parallel to EH
b. Line(s) perpendicular to EH
c. Line(s) skew to CD
and containing point F
d. Plane(s) perpendicular to plane AEH
e. Plane(s) parallel to plane FGC
3) Complete the statement with sometimes, always, or never.
a.
If two lines are parallel, then they __________________ intersect.
b.
If one line is skew to another, then they are ___________________ coplanar.
c.
If two lines intersect, then they are ______________________ perpendicular.
d.
If two lines are coplanar, then they are ___________________________ parallel.
Geometry Section 3.2
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Start Thinking:
Give an example of the following types of angles on
the figure to the left.
Alternate Interior Angles: __________________
Same Side Interior Angles: _________________
Alternate Exterior Angles: _________________
Corresponding Angles: _____________________.
In the example above the transversal is intersecting two parallel lines. This is a special case, which
results in special relationships between the different types of angles. Using a piece of tracing paper
trace one of the angles and see what the relationship is.
Postulates / Theorems:
Corresponding Angles Theorem: If two parallel lines are cut by a transversal, then the pairs of
corresponding angles are ____________________________________________.
Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then the pairs of
alternate interior angles are____________________________________.
Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then the pairs of
alternate exterior angles are ________________________________.
Same side Interior Angles Postulate: If two parallel lines are cut by a transversal, then the pairs of
same side interior angles are ___________________________________.
EXAMPLES:
Ex 1:
Find the value of x.
Ex 2: Solve for x
EX 3: Solve for x
EX 4: Solve for X and Y
PROOF of ALTERNATE INTERIOR ANGLES THEOREM using SSIA Postulate.
n
Given: m is parallel to n
Prove: 4  6
m
Statements
Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
4
6
3
Practice:
1)
If m7 = 75°, find m1, m3, and m5. Tell which postulate or theorem you use in each
case. DO NOT USE VERTICAL ANGLES.
m 3 = 75°,
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m 5 = 75°,
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m 1 = 75°,
2)
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Find all the angle measures.
2
1 8
4
6
5
3
105°
7
3.) Application: You are building a wood shed. You have already put up both walls. One is 6 feet tall and the other
is 10 feet tall. You then use a metal brace, which is bent at a 130 degree angle to attach the roof to the smaller
wall. You want to create another brace to attach the roof to the back wall. What angle should the brace be bent
at?
Geometry Section 3.3
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Start Thinking:
IF two parallel lines are cut by a transversal THEN…….
Alternate Interior Angles are ______________________________.
Alternate Exterior Angles are ______________________________.
Corresponding Angles are __________________________________.
Same Side interior Angles are ________________________________.
CONDITIONAL STATEMENT: __________________________________________________________.
CONVERSE: _______________________________________________________________________.
DIAGRAM:
THEOREMS
*Converse of Alternate Interior Angles Theorem:
- IF __________________________________________________________ THEN
________________________________________.
*Converse of Corresponding Angles Theorem:
- IF __________________________________________________________ THEN
________________________________________.
*Converse of Same Side Interior Angles Postulate:
- IF __________________________________________________________ THEN
________________________________________.
*Converse of Alternate Exterior Angles Theorem:
- IF __________________________________________________________ THEN
________________________________________.
PRACTICE: Determine what lines, if any, must be parallel given each piece of information. Justify your
reasoning. Write NEI if there is not enough info.
a.) < 7 and <17 are congruent
b.) <21 and <20 are congruent
c.) <20 and <9 are congruent
d.)< 14 and <3 are supplementary
e.) <22 and <7 are congruent.
f.) <19 and <11 are congruent
g.) m<14 + m<4 = 180
h.) <10 and <7 are congruent
i.) m<17 = 40 and m<20 = 40
j.) m<23 = 120 and m<6 = 120

Hint Use Converse of Corresponding
Angles THM
2.) Complete a 2 column proof using the Converse of the Corresponding Angles Thm.
3.)
4.) A parallelogram is defined as a quadrilateral that has two pairs of opposite parallel sides.
m<A and m<C = 105. M<B = m<D. Is the figure a parallelogram? Justify your answer.
4) 5.) GIVEN: g || h, 1  2
PROVE: p || r
Statements
Reasons
g || h
1. ___________________________________________
1  3
2. ___________________________________________
1  2
3. ___________________________________________
2  3
4. ___________________________________________
p || r
5. ___________________________________________
Geometry Section 3.4
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Start Thinking:
You have a 1 inch wide and 6 foot long piece of lumber. You are trying to create a line with a piece
of chalk that is 8 inches away from a wall so that the line is parallel to the wall. Describe how you
could do this.
Theorem 3.8 Transitive Property of Parallel Lines: _______________________________________
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Example:
Theorem 3.9: If two lines are perpendicular to the same line then they are ____________________.
STATEMENTS
REASONS
PRACTICE:
1.)
2.) PROOF:
GIVEN: <1 is congruent to < 2
< 3 and < 4 are supplementary
PROVE: Line L is parallel to Line N
THEOREM:
Theorem 3-10: In a plane, if a line is perpendicular to one of two parallel lines then..
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Example of why it has to be “IN A PLANE”
PROOF:
1.)
2.) Write a flow proof.
Given:
PQS and QSR are supplementary.
Prove:
Geometry Section 3.5
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Activity: Using a straight edge, and either a pair of scissors or careful tearing create a triangle out of a
piece of paper. Now rip off all of the angles of the triangle. Place then next to one another so that all
of the vertexes are at the same point. What do you notice? What does this tell you about the sum of
the interior angles of a triangle?
Triangle Angle Sum Theorem:
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Postulate 3.2: Parallel Line Postulate:
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Solve the following for x:
1.)
2.)
MORE PRACTICE:
1.)
2.)
DEFINITION:
Exterior Angles of a Triangle:
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Remote Interior Angles:
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PROOF OF THEOREM 3-12:
GIVEN: Diagram to right
Prove: m<2 + m<3 = m<1
PRACTICE PROBLEMS :
1.)
2.)
ONE MORE PROOF:
GIVEN: Diagram to the right
PROVE: m<4+ m<5 +m<6 = 360
Geometry Section 3.7/3.8
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Start Thinking: You are going to go with a bike ride with a buddy. You are a little concerned because he
says that there is a hill. What are some questions you might ask him about the hill?
PRACTICE: Calculate the following slopes.
1.) (3, 10) and (7, 1)
2.) (-1, 6) and (4,-4)
3.)
SPECIAL SLOPES:
1.)
2.)
3.) (3, 1) (3.7)
4.) (3,1) (5,1)
EQUATIONS   GRAPHS
GRAPH
EQUATION
PRACTICE: Graph the following lines.
2.) y  
1.) y = 1/3x – 4
2x
1
3
3.) y = x

PRACTICE: Write the equation of the following lines.
1.)
2.)
3.)
WRITING THE EQUATION OF A LINE GIVEN A POINT AND SLOPE.
EX: Slope = 3 and point (1,5)
METHOD 1 (SOLVING FOR B)
METHOD 2 (POINT SLOPE)
PRACTICE:
1.) Slope= -2 and (3, -6)
2.) Slope= 2/3 and (9 , 2)
3.) Slope= Undefined and (2,3)
WRITING THE EQUATION OF A LINE GIVEN TWO POINTS.
1.) (1, 3) and (3, 13)
2.) (-1,0) and (2, 6)
3.) (2, 7) and (5,7)
Start Thinking: Look at the following lines. How are they
related? How would you relate their steepness? What do
we call the “steepness of a line”?
EX1: Determine if the following lines are parallel?
EX2:
WRITE THE EQUATION FO THE LINE WITH THE FOLOWING CHARACTERISTICS.
1.) Parallel to y = 3x+5 and through (4,2)
2.) Parallel to x = 2 through (-4,7)
EX1: Are the following lines perpendicular?
EX 2: What is the equation of a line perpendicular to y = 2/3x +1 through (10, 5)?
EX3: Determine if the figure with the following vertices is a parallelogram.
A(0,2) B (3,4) C (2,7) D (-1,5)
CONCEPT USING VARIABLES:
EX 1: What is the equation of a line that contains the following points (0,0) and (a,v).
EX 2: Is the line from EX1 parallel, perpendicular, or neither with the following line? Explain.
y

b
xd
a
EX 3: Is a line that goes through the point (0,0) and (a,b) parallel to a line that goes through (1.0) and
(a+1,b)?