Download Geometry Chapter 2 Blank Notes

2.1: ________________________________________________
Geometry
Date: __________
___________________ _______________ is reasoning based on patterns you observe.
Ex 1). Look for a pattern. What are the next two terms in each sequence?
a)
4, 8, 16, 32, …
b)
A _________________ is used to find the tenth or one-hundredth term in a sequence, it’s a
conclusion you reach using inductive reasoning.
Ex 2).
a)
What conjecture can you make about the 25th term in 1, 2, 3, 1, 2, 3, … ?
b)
What conjecture can you make about the 21st term in R, W, B, R, W, B, … ?
*It is important to gather enough information before making a conjecture.
Ex 3).
a)
What conjecture can you make about the sum of the first 40 even numbers?
b)
What conjecture can you make about the sum of the first 30 even numbers?
Ex 4). Sales of lift tickets at a ski resort increased over a period of four consecutive months. What
conjecture can you make about the number of lift tickets the resort will sell in February?
Not all conjectures are true, you should check it multiple times to be sure. You can prove that a
conjecture is __________ by finding _______ ___________________________. A counterexample is an
example that shows that a conjecture is _________________.
Ex 5). What is a counterexample for each conjecture?
a)
If an animal is green, it is a frog.
b)
Any three segments form a triangle.
c)
When you multiply a number by 2, the product is divisible by 4.
d)
If a flower is red, it is rose.
e)
One and only one plane exists through any three points.
f)
When you multiply a number by 3, the product is divisible by 6.
Homework: pg. 93 #1 – 3, 6 – 14(e), 18 – 27, 29, 30, 32, 33 – 40
2.2 Day 1: __________________________________________
Geometry
Date: __________
A Conditional Statement is an __________
statement.
The __________________ is the part _____
following if.
The __________________ is the part _____
following then.
Ex 1). What are the hypothesis and the conclusion of the conditional statement?
a)
If a number is even, then it is divisible by 2.
Hypothesis: ________________________________________________________________________
Conclusion: ________________________________________________________________________
b)
If an angle measures 130, then the angle is obtuse.
Hypothesis: ________________________________________________________________________
Conclusion: ________________________________________________________________________
Ex 2). How can you write the following statement as a conditional?
a)
Adjacent angles share a side.
b)
Dolphins are mammals.
The __________ ____________ of a conditional statement is either true or false.
* To show the statement is true: show that every time the hypothesis is true, the conclusion is true.
* To show that a statement is false: find ________ ______ counterexample for which the hypothesis is
true, but the conclusion is false.
Ex 3). Is the conditional true or false? If it is false, find a counterexample.
a)
If you live in Copley, then you live in Ohio.
b)
If a number is divisible by 5, then it is odd.
c)
If a month 28 days, then it is February.
d)
If two angles form a linear pair, then they are supplementary.
Homework: pg. 99 #1, 3 – 13, 14 – 28(e)
2.2 Day 2: __________________________________________
Geometry
Date: __________
The ______________ of a statement p is the _____________ of the statement. The symbol ______ is
read “not p.”
Statement
Conditional
Converse
Inverse
Contrapositive
How to Write It
Example
Symbols
How to Read It
Truth Values:
Statement
Example
Truth Value
Conditional
Converse
Inverse
Contrapositive
Ex 1). What are the converse, inverse, and contrapositive of each of the conditional statements
below? What are the truth values of each?
a)
If a figure is a rectangle, then it is a parallelogram.
b)
If a vegetable is a carrot, then it contains beta carotene.
Homework: pg. 103 #1, 3 – 7, 11 – 14, 18 – 25
2.3: ________________________________________________
Geometry
Date: __________
A ________________________ is a single true statement that combines a true conditional and its true
converse. You can write a biconditional by joining the two parts of each conditional with the
phrase ________________________________.
Ex 1). What is the converse of the following true conditional? If the converse is also true, combine
the statements as a biconditional.
Conditional: If the sum of the measures of two angles is 90°, then the angles are complementary.
A biconditional combines 𝑝 → 𝑞 and 𝑞 → 𝑝 as _______________.
ex: A point is a midpoint if and only if it divides a
segment into two congruent segments.
Ex 2). What are the two conditional statements that form this biconditional?
Two numbers are additive inverses if and only if their sum is 0.
What makes a good definition?
1. A good definition uses _________ ____________________ terms.
2. A good definition is _______________.
3. A good definition is _______________. It can be written as a true _______________________.
Ex 3). Is this definition of an equilateral triangle reversible? If yes, write it as a true biconditional.
An equilateral triangle is a triangle with 3 congruent sides.
Ex 4). Which of the following is a good definition?
[A]
Dogs are animals with 4 paws.
[B]
Squares have 4 sides.
[C]
Tuesday is the day before Wednesday.
[D]
An acute angle has a small measure.
Homework: pg. 108 #1 – 3, 8 – 26(e), 45 – 50
2.4: _________________________________________________
Geometry
Date: __________
________________ _______________ is the process of reasoning _____________ from given statements
or facts to make a ___________________.
Law of _________________________:
If the hypothesis of a _______ conditional is _______, then
the conclusion is _______.
Ex 1). What can you conclude from the given true statement?
a) Given: If it is raining outside, then there are clouds in the sky. It is raining today.
b) Given: If Malcolm scores at least 85% on his final exam, then he will earn an A for the
term. Malcolm scores a 90% on his final exam.
c) Given: If two angles are vertical, then they are congruent. Angles 3 and 4 are vertical
angles.
Law of ________________________:
Symbols:
Examples:
Ex 2). What can you conclude from the given information?
a) Given: If a number is divisible by 12, then it is divisible by 6. If a number is divisible by 6,
then it is divisible by 3.
b) Given: If a figure is a square, then the figure is a parallelogram. If a figure is a rectangle,
then the figure is a parallelogram.
Ex 3). What can you conclude from the given information?
a) Given: If you live in Cincinnati, then you live in Ohio. If you live in Ohio, then you live in
the United States. Ken lives in Cincinnati.
b) If a river is more than 4000 mi long, then it is longer than the Amazon. If a river is longer
than the Amazon, then it is the longest river in the world. The Nile is 4132 mi long.
Homework: pg. 117 #1 – 3, 6 – 16(e), 19 – 24, 26, 28
2.5: _______________________________________________
Geometry
Date: __________
Ex 1). What is the value of x?
Ex 2). Use the distributive property to solve each of the following. Justify your steps.
a) 3(2x – 8) = 6
b)
4(x – 7) – 2(3x + 4) = 2
Ex 3). What is the name of the property of equality or congruence that justifies going from
the first statement to the second statement?
a) 7x + 3 = 24
7x = 21
b)
RS = ST and ST = TU
RS = TU
c) 5x = 10
10 = 5x
b)
3x – 4 = x and x = -2
3(-2) – 4 = -2
A _________ is a convincing argument that uses deductive reasoning. A proof
_________________ shows why a conjecture is true. A _______________________ lists each
statement on the left and the reason (justification) on the right. Each statement must
follow logically from the steps ___________ it.
Ex 4). Write a two-column proof.
Given:
AC = BD
Prove:
AB = CD
Ex 5). Write a two-column proof.
Given:
Prove:
𝒎∠𝟏 = 𝒎∠𝟑
𝒎∠𝑨𝑬𝑪 = 𝒎∠𝑫𝑬𝑩
Homework: pg. 124 #1 – 17
2.6: ________________________________________________
Geometry
Date: __________
A ______________ is a conjecture or statement that you prove _________
Ex 1). What is the value of x?
a)
b)
Ex 2). Write a two column proof:
Given:
∠𝟐 ≅ ∠𝟑
Prove:
∠𝟏 ≅ ∠𝟐
Ex 3). Write a two-column proof:
Given:
∠𝟏 𝒂𝒏𝒅 ∠𝟐 𝒂𝒓𝒆 𝒄𝒐𝒎𝒑𝒍𝒆𝒎𝒆𝒏𝒕𝒂𝒓𝒚
∠𝟐 𝒂𝒏𝒅 ∠𝟑 𝒂𝒓𝒆 𝒄𝒐𝒎𝒑𝒍𝒆𝒎𝒆𝒏𝒕𝒂𝒓𝒚
Prove:
∠𝟏 ≅ ∠𝟑
Ex 4). Write a two-column proof:
Given:
∠𝑾 𝒂𝒏𝒅 ∠𝑽 𝒂𝒓𝒆 𝒄𝒐𝒏𝒈𝒓𝒖𝒆𝒏𝒕 𝒂𝒏𝒅 𝒔𝒖𝒑𝒑𝒍𝒆𝒎𝒆𝒏𝒕𝒂𝒓𝒚
Prove:
∠𝑾 𝒂𝒏𝒅 ∠𝑽 𝒂𝒓𝒆 𝒓𝒊𝒈𝒉𝒕 𝒂𝒏𝒈𝒍𝒆𝒔
Homework: pg. 132 #1 – 3, 6 – 13, 16 – 17, 19