Title___Section 2-1 Use Inductive Reasoning__________________________________________Page _________
Essential Question – how do you use inductive reasoning in mathematics?
Questions, Ideas, Suggestions
VOCABULARY
1. Inductive Reasoning –
EXAMPLE 1 - Sketch the next figure in each pattern. Describe it.
EXAMPLE 2 - Sketch the next figure in each pattern. Describe it.
Using numbers – find the next number and the pattern.
2.
EX. 3
1, 4, 16, 64, _____
________________________
EX. 4
-5, -2, 4, 13, ______
________________________
EX. 5
17, 15, 12, 8, _____
________________________
Conjecture –
EX 6
Conjecture: The sum of the first n odd positive integers is __________?
Solution: Write out examples
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EX. 7 Conjecture: The product of any two odd integers is _______________?
3.
Counterexample –
EX. 8
For all real numbers x, x2 is greater than or equal to x. Find a counter
example
Ex. 9 Supplementary angles are always adjacent. Find a counter
example
Summary
Title___Section 2-2 Analyze Conditional Statements______________________________________
Essential Question – how do you rewrite a biconditional statement?
Questions, Ideas, Suggestions
VOCABULARY
1.
A conditional Statement
If it is noon in Georgia, then it is 9 a.m. in California.
Hypothesis
conclusion
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Ex. 1 Identify the hypothesis and conclusion “All birds have feathers.”
Write it as an If-then statement
2.
Negation –
The ball is red.
___________________________
The cat is NOT black
___________________________
3.
Converse –
4.
Inverse –
5.
Contrapositive –
Ex. 2 All mammals breathe oxygen. Write in if-then format. Then write the
converse, inverse and contrapositive. Tell whether each statement is true or
false.
If- then
Converse
Inverse
Contrapositive
Ex. 3 Two points are collinear if they lie on the same line. Write in if-then
format. Then write the converse, inverse and contrapositive. Tell whether each
statement is true or false.
If- then
Converse
Inverse
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Contrapositive
6.
Perpendicular Lines
Ex. 4 Decide whether each statement about the diagram is true. Explain your
answer using definitions you have learned.
a.
AC __ BC
b.
<ACD and <DCB are
complementary
c. CD bisects <ACB
7.
Biconditional Statement
Definition – If two lines intersect to form a right angle, then they are
perpendicular.
Converse –
Biconditional
Summary
Title______Section 2-3 Apply Deductive Reasoning ______________________________________ Page _______
Essential Question – How do you construct a logical argument?
Questions, Ideas, Suggestions
VOCABULARY
Deductive reasoning –
We have two laws:
1)
Law of Detachment
If Mrs. Humphreys teaches well, then students will get an A. Mrs. Humphreys
taught well so ___________________________________________
2)
Law of Syllogism
If my girls get an A on their test, then I will take them to Disneyland. If I take
them to Disneyland, they will ride on Space Mountain. My girls go an A on their
test so _______________________________________________
Ex. 1
Use the law of detachment to make a valid conclusion in the true situation.
IF TWO ANGLES ARE VERTICAL, THEN THEY ARE CONGRUENT. You know that <ABC AND
<DBE ARE VERTICAL. _____________________________________________________
Ex. 2 MICHAEL KNOWS THAT IF HE DOES NOT DO HIS CHORES IN THE MORNING, HE
WILL NOT BE ALLOWED TO PLAY VIDEO GAMES THAT DAY. MICHAEL DID NOT DO HIS
CHORES IN THE MORNING _______________________________________________
Ex. 3 If possible, use the law of syllogism to write a new conditional statement that
follows from the pair of true statements.
IF A FISH SWIMS AT 68 MPH, THEN IT SWIMS AT 110 KMPH.
IF A FISH CAN SWIM AT 110KMPH, THEN IT IS A SAILFISH.
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IF A FISH IS THE LARGEST SPECIES OF FISH, THEN IT IS A
GREAT WHITE SHARK.
IF A FISH WEIGHS OVER 2000 LBS, THEN IT IS THE LARGEST
SPECIES OF FISH.
SUMMARY
Title______Section 2.4 Use Postulates and Diagrams_______________________________________
Essential Question – How can you identify postulates illustrated by a diagram?
Questions, Ideas, Suggestions
POSTULATES:
Postulate 5 – Through any two points there exists exactly one line
Postulate 6 - A line contains at least two points.
Postulate 7 – If two lines intersect, then their intersections is exactly one point.
Postulate 8 – Through any three noncollinear points there exists exactly one
plane.
Postulate 9 – A plane contains at least three noncollinear points
Postulate 10 – If two points lie in a plane, then the line containing them lies in
the plane.
Postulate 11 – If two planes intersect, then their intersection is a line.
EXAMPLES:
Ex. 1
If
State the postulate illustrated by the diagram.
then
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Ex. 2
If
A
then
B
A
B
Ex. 3 Sketch a diagram showing TV intersecting PQ at point W, so that TW = WV
Ex. 4
Sketch a diagram showing FH __ EG at its midpoint M.
Ex. 5 Which of the following statements cannot be assumed from the diagram?
A, B and F are collinear
E, B and D are collinear
AB __ plane S
CD __ plane T
AF intersects BC at point B
Summary
Title______Section 2-5 Reason Using Properties from Algebra______________________________Page _______
Essential Question – How do you solve an equation?
Questions, Ideas, Suggestions
Algebraic Properties:
Addition Property
Subtraction Property
Multiplication Property
Division Property
Substitution Property
Distributive Property
Ex. 1 Write reasons for each step
5x – 18 = 3x + 2
Ex. 2 Write reasons for each step
-4(11x + 2) = 80
Reflexive Property
Symmetric Property
Transitive Property
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Ex. 3 NOW GEO - You are designing a logo to sell daffodils. Determine whether
m<EBA = m<DBC if you are given that <1 = <3
Ex. 4 The city is planning to add two stations between the beginning and end of
a commuter train line. Determine whether RS = TU given the information.
Ex. 5
In the diagram, AB=CD. Show that AC = BD
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EX. 6
In the diagram ,<ABD = m<CBE. Show that m<1 = m<3
Summary
Title____________Section 2.6 Prove Statements about Segments and Angles_______________Page _______
Essential Question – How do you write a geometric proof?
Questions, Ideas, Suggestions
**From here on out you are to use Two Column Proof**
Reminder of Reflextive, Symmetric and Transitive with segments and angles.
Reflexive
Symmetric
Transitive
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Ex 1 Guided Practice
Fill in the missing statements or reasons
Given: AC = AB + AB
Ex. 2
Prove: AB = AC
Statements
Reasons
1.
1.
Given
2.
2.
Seg Add
3.
AB + BC = AC
3.
4.
AB = BC
4.
Write a two column proof
Given: Q is the midpoint of PR
Prove: PQ = ½ PR and QR = ½ PR
Statements
Ex. 3
Write a two column proof
Reasons
Given: m<1 + m<2 = 90; m<1 = 59
Prove: m<2 = 31
Summary
Title____Section 2.7 Prove Angle Pair Relationships _____________________________________Page _______
Essential Question – what is the relationship between vertical angles, two angles that are
supplementary to the same angle, and between two angles that are complementary to the same
angle?
Questions, Ideas, Suggestions
THEOREMS:
Theorem 2.3 Right angle congruence theorem
Theorem 2.4 Congruent Supplements theorem
Theorem 2.5 Congruent Complements theorem
Ex. 1