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Welcome to LC Math 2 Adv!
• Syllabus
• Supplies
• NB: Learning Logs
Definitions/Postulates
Notes/CW
Homework
Assessments
• Grading Policy
• Remind 101
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• Class Info
• Exploration and Investigation
• Geometry (infused with Algebra) and Probability
• Unit 1 Newsletter
• Unit 1 Learning Log
Points, Lines and Planes
In geometry, several terms are accepted as
intuitive ideas, and are therefore not defined.
They are:
POINT
LINE
PLANE
Point
Line
Plane
Collinear Points
If points are collinear, a single line can be
drawn through them.
Coplanar Points
If points are coplanar, they can be
contained in a single plane.
Sample Coplanar Points:
Sample Noncoplanar Points:
Intersection of Geometric Figures
The intersection of two figures is the set of points
that are in both figures.
Note: Dashes in diagrams indicate parts hidden from view in figures
in space.
Exercises
a.
Give two other names for PQ and for plane R.
b. Name three points that are collinear.
c. Name points that are noncollinear.
d. Name four points that are coplanar.
e. Name four points that are noncoplanar.
Exercises
a.
Sketch line l contained in plane M.
b.
Sketch line l and plane M, which do not intersect..
c.
Sketch plane M that intersects line l at point P.
a.
b.
c.
P
Exercises
Sketch two planes that intersect in a line.
STEP 1
Draw: a vertical plane. Shade the
plane.
STEP 2
Draw: a second plane that is horizontal. Shade this
plane a different color. Use dashed lines to show
where one plane is hidden.
STEP 3
Draw: the line of intersection.
Exercises
Sketch lines n and m that intersect plane P at point Q.
Exercises
Name the intersection of PQ and line k.
Point M
Name the intersection of plane A and plane B.
Line k
Name the intersection of line k and plane A.
Line k
Learning Log Summary
Overarching –
I can define and use appropriate notation to
describe or identify aspects of a geometric diagram
or interpret the notation provided.
I can sketch a geometric diagram using given
information and use a sketch to glean information.
Closure
Homework
Sign and Return Syllabus
Bring Supplies (due Monday 8/22)
pg. 7-9 ~ 2-26 (even), 27-36
Additional Vocabulary
Additional Vocabulary
Additional Vocabulary
Opposite rays - two collinear rays with a
common endpoint.
Additional Vocabulary
Length - The length of a segment is the
distance between the endpoints.
DE = 2cm
Finding the Length of a Segment
Show the work used to find:
AB =
BA=
BC =
CB =
PQ
x y
Use the examples above to write a formula to find the distance
between any points P and Q.
Postulate
Using number lines involves following basic
assumptions. Statements such as these,
that are accepted without proof, are
called postulates or axioms.
Segment Addition Postulate
Implies that B is collinear
with A and C.
If B is between A and C, then AB+BC=AC.
Additional Vocabulary
Congruent - Two objects that have the
same shape and size. Congruent
segments have equal length.
AB  CD
EF  GH
Additional Vocabulary
Midpoint of a Segment - The point that
divides the segment into two congruent
segments.
AM  MB
AM  MB
M is the midpoint of AB
Bisector of a Segment
The bisector of a segment is a line, segment,
ray, or plane that intersects the segment at its
midpoint.
AB is the bisector of PQ
Bisector of a Segment
Using the diagram to the right…
Is M the midpoint of PQ ? Explain.
Make 3 statements about what
geometric figures bisect PQ.
True or False: JM bisects KM
True or False: JM  KM
True or False: PM  MQ
Learning Log Summary
LT 1 - I can state the Segment Addition
Postulate and use it in proof and problem
solving contexts.
The Segment Addition Postulate says…
If two parts of one segment are known…
Closure
Homework
pg. 15 ~ 1-45 (odd), 46-48
Learning Log Summary
LT 2 - I can justify the formulas for
calculating the distance between two
points and the midpoint of a segment on
the coordinate plane and apply them to
solve problems.
The Distance Formula is…
It makes sense because…
The Midpoint Formula is…
It makes sense because…
Learning Log Summary
LT 3 - I can state and explain the
Protractor Postulate and Angle Addition
Postulate. I can apply the Angle Addition
Postulate in proof and problem solving
contexts.
The Protractor Postulate in my own words is…
A diagram and statement that illustrates the Angle
Addition Postulate is…
Deductive Reasoning
Deductive Reasoning
Inductive Reasoning
• Based on definitions,
postulates, and given
information.
• Based on patterns and
observations.
• Requires “proof”.
• Doesn’t require “proof”
• Conclusion must be true.
• Conclusion is probably
true.
Joe observes that spaghetti
is on the school lunch menu
for three Wednesdays in a
row. He concludes that the
school always serves
spaghetti on Wednesdays.
Jane knows that the square
of any real number is
positive. So she concludes
that if x>0, then x2>0.
Learning Log Summary
LT 5 - I can describe the use of inductive
and deductive reasoning in
mathematical sense making.
Deductive reasoning is…
Inductive reasoning is…
Examples of inductive vs. deductive reasoning are…
Deductive Reasoning
“If it is raining after school, then I will give you a ride home.”
Hypothesis
Conclusion
This is an example of an if-then statement.
They are also called conditional statements (or just conditionals)
If p, then q.
Deductive Reasoning
The converse of a conditional is formed by interchanging the
hypothesis and conclusion.
Statement:
Converse:
If p, then q.
If q, then p.
Ex) Find the converse of the statement.
Statement: If Ted lives in California, he lives in the USA.
Deductive Reasoning
Conditionals are not always written in the same way. Here are some
general forms of conditional statements.
Determine which letter represents the hypothesis and conclusion.
If p, then q
p implies q
P if q
p only if q
Learning Log Summary
LT 6 - I can describe the components to a
conditional statement and write its
converse.
The components of a conditional are…
Conditionals can be written as…
The converse to a conditional is…
Deductive Reasoning
A counterexample is an example for which the hypothesis is true but
the conclusion is false.
To disprove a statement, only one counterexample must be found.
Statement: If Ted lives in California, he lives in the USA.
Learning Log Summary
LT 7 - I can determine the truth of a
statement and provide a counterexample
if false.
A counterexample is…
An example of a false statement and possible
counterexample is…
Deductive Reasoning
If a conditional and its converse are both true, they can be
combined into a single statement by using the words “if and only if”.
This single statement is known as a biconditional.
Conditional: If line segments are congruent, they have equal lengths.
Converse:
Biconditional?:
Learning Log Summary
LT 8 - I can determine whether a
statement and its converse can be written
as a biconditional.
A biconditional is…
An example of a true conditional and converse that
can be rewritten as a biconditional is…
Closure
Homework
pg. 35 ~ 1-7 (odd), 17-27 (odd)
and
pg. 35 ~ 9,10, 18-28 (even), 29-30
Deductive Reasoning Practice
Identify the hypothesis and conclusion in each conditional:
• 𝐴𝐵 = 𝐵𝐶 if B is the midpoint of 𝐴𝐶.
• We will go to the beach only if it is sunny.
• 𝑥 = −2 implies 𝑥 2 = 4
Deductive Reasoning Practice
Determine if the statement is true or false. Then write its converse
and determine if that statement is true or false.
• If two angles are right angles, then they are congruent.
• 𝑥 > 7 implies 𝑥 > 2.
• An animal is a penguin only if it is a bird.
• If a number is divisible by 2, then it is divisible by 4.
Deductive Reasoning Practice
Determine if the conditional can be rewritten as a biconditional. If
so, re-write it as such.
• If an angle has a measure of 90°, then it is a right angle.
• 𝐷𝐸 = 𝐹𝐺 only if 𝐷𝐸 = 𝐹𝐺
• Two angles being adjacent implies that they have a common
vertex.
Theorems
Learning Log Summary
LT 10 - I can state and prove the Midpoint
Theorem and Angle Bisector Theorem and
use them in proof and problem solving
contexts.
The Midpoint Theorem is…
The Angle Bisector Theorem is…
To prove the Midpoint or Angle Bisector Theorem…
Special Pairs of Angles
Complementary angles are two angles whose measures have a sum
of 90°. Each angle is called a complement of the other.
A
C
B
STA and ATR
are complementary
C is a
complement of
A
Special Pairs of Angles
Supplementary angles are two angles whose measures have a sum
of 180°. Each angle is called a supplement of the other.
B
and
Q
are supplementary
ABC is a
supplement of
CBD
These angles are a linear pair.
Special Pairs of Angles
Ex) A supplement of an angle is three times as large as a
complement of that angle. Find the measure of the angle.
Learning Log Summary
LT 11 - I can apply the definitions of
complementary and supplementary
angles in problems and proofs.
Complementary angles…
Supplementary angles…
If x is an angle, its supplement/complement is…
Special Pairs of Angles
Vertical Angles are two angles such that the sides of one angle are
opposite rays to the sides of the other angle.
When two lines intersect, they form two pairs of vertical angles.
1
and
3
are vertical angles
2
and
4
are vertical angles
Special Pairs of Angles
What do you notice about the measure of vertical angles?
Special Pairs of Angles
Given:
1
Prove:
1  3
and
3
Statements
are vertical angles
Reasons
Learning Log Summary
LT 12 - I can define vertical angles and
prove that they are congruent.
Vertical angles…
To prove vertical angles are congruent…
Perpendicular Lines
Perpendicular Lines are two lines that intersect to form right angles.
The definition of perpendicular lines can be used in two ways:
If 𝐽𝐾 is perpendicular to 𝑀𝑁, written
JK  MN then each of the
numbered angles is a right angle.
If any one of the numbered angles is
a right angles, then JK  MN .
M
J
2
1
3
4
N
K
Learning Log Summary
LT 13 - I can apply the definition of
perpendicular lines in problems and
proofs.
Perpendicular lines…
To prove an angle is a right angle…
To prove that two lines are perpendicular…
Supplementary Angle Theorem
If two angles are supplementary to the
same angle, then they are congruent.
Complementary Angle Theorem
If two angles are complementary to the
same angle, then they are congruent.
Learning Log Summary
LT 14 - I can prove theorems involving
complementary and supplementary
angles.
If two angles are complementary/supplementary to
the same angle…
Common strategies in proofs involving
complements/supplements are…
Unit 1 Test
• 12 Multiple Choice Questions
• Definitions
• Postulates/Theorems
• Properties
• 6 Free Response Questions
• Proving Known Theorems
• Proving New Statements
• Applying Knowledge to Problems