Download Packet 3 for Unit 1 M2 Geo

M2 GEOMETRY – PACKET 3 FOR UNIT 1 – SECTIONS 2-3, 2-6, 2-7
Conditional Statements
A conditional statement is a sentence in the form: If _______, then__________.
For example: If an animal is a leopard, then it has spots.
The two parts of a conditional statement are called the hypothesis and the conclusion. In
the conditional statement above “an animal is a leopard” is the hypothesis and “it has spots”
is the conclusion.
1. For each statement below, identify the hypothesis and the conclusion.
a. If Ms. Lee didn’t eat breakfast, then she is hangry.
Hypothesis:
Conclusion:
b. If it is Saturday, then school is closed.
Hypothesis:
Conclusion:
The converse of a conditional statement is formed when the hypothesis and conclusion
switch places. Here’s an example:
Conditional:
If an animal is a leopard, then it has spots.
Converse:
If an animal has spots, then it is a leopard.
2. Write the converse of each conditional statement from #1. Notice that, instead of just
switching the underlined parts, I kept the noun “animal” at the beginning of the sentence
for clarity.
a. Conditional:
If Ms. Lee didn’t eat breakfast, then she is hangry.
Converse:
b. Conditional:
If it is Saturday, then school is closed.
Converse:
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M2 GEOMETRY – PACKET 3 FOR UNIT 1 – SECTIONS 2-3, 2-6, 2-7
If a conditional statement is true, its converse does not have to be true.
3. For each conditional statement (a) determine whether it is true; (b) write the converse,
and (c) determine whether the converse is true.
Conditional Statement
True or
False?
Converse
True or
False?
If two segments are congruent, then
they have the same length.
If you are old enough to get a
driver’s license, then you are old
enough to vote.
If you live in Connecticut, then you
live in the US.
A counterexample is an example that shows why a statement is false. The statement “if an
animal has spots, then it is a leopard” is false. Here’s a counterexample, and an explanation
of how this shows the statement is false:
“A Dalmatian is a counterexample. It is an animal that has spots, but it is not a leopard.”
Notice the parts of my explanation. I showed that my counterexample makes the hypothesis
true but makes the conclusion false.
4. For the false statements in #3 above, give a counterexample, and explain.
Sometimes, we need to restate mathematical properties in “if__, then__” form to make them
easier to use. For example, the sentence “after three strikes, the batter is out” can be
rewritten as “if the batter has three strikes, then he is out.” Notice that the noun is near the
beginning of the sentence for clarity.
5. Convert these statements to “if-then” form.
a. Eighteen-year olds are eligible to vote.
b. Perpendicular lines form 90 degree angles.
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M2 GEOMETRY – PACKET 3 FOR UNIT 1 – SECTIONS 2-3, 2-6, 2-7
Here’s a summary of conditional, converse, and two other forms, called inverse and
contrapositive.
Type of Statement
Format
Example
Conditional
If A, then B.
If an animal has spots, then it is a leopard.
Converse
If B, then A.
If an animal is a leopard, then it has spots.
Inverse
If not A, then not B.
If an animal does not have spots, then it is not a leopard.
Contrapositive
If not B, then not A.
If an animal is not a leopard, then it does not have spots.
6. For the conditional statement “if it is Saturday, then school is closed,” write the converse,
inverse, and contrapositive.
a. Converse:
b. Inverse: (To remember this, think the INverse is IN the same order as the original
statement.)
c. Contrapositive:
7. For the conditional statement “if Ms. Lee didn’t eat breakfast, then she is hangry,” write
the converse, inverse, and contrapositive.
a. Converse:
b. Inverse:
c. Contrapositive:
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M2 GEOMETRY – PACKET 3 FOR UNIT 1 – SECTIONS 2-3, 2-6, 2-7
Algebraic Proof
A proof is an argument that shows how existing rules can be used to explain a new rule.
A two-column proof presents steps in a logical order using statements and reasons.
Here is a list of possible reasons that can be used in a proof:
Addition Property of Equality
If a = b, then a + c = b + c.
Subtraction Property of Equality
If a = b, then a – c = b – c.
Multiplication Property of Equality
If a = b, then ac = bc.
Division Property of Equality
If a = b and c≠0, then
Reflexive Property of Equality
a=a
Symmetric Property of Equality
If a = b, then b = a.
Transitive Property of Equality
If a = b and b = c, then a = c.
Substitution Property of Equality
If a = b, then a may be
replaced by b.
Distributive Property
a (b + c) = ab + ac
a b
= .
c c
8. In the last column of the table above, write a brief summary of what each property
means.
On the next page, you’ll see an example of these statements used in some algebraic proofs.
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M2 GEOMETRY – PACKET 3 FOR UNIT 1 – SECTIONS 2-3, 2-6, 2-7
Example: Prove that if
1
x + 10 = 2 , then x = − 3 2 .
4
Statements
1
x + 10 = 2
4
1
x + 10 − 10 = 2 − 10
4
1
x = −8
4
Reasons
Given
Subtraction property
Substitution
1 
4  x  = 4 ( −8)
4 
Multiplication property
x = −32
Substitution
Sometimes, the Substitution Property may be omitted when the Addition, Subtraction,
Multiplication, or Division Property is applied. Then our proof would look like this:
Statements
1
x + 10 = 2
4
1
x = −8
4
x = −32
Reasons
Given
Subtraction property
Multiplication property
When you write your own proofs, you may choose
either method. However, if you are asked to fill in
the blanks in a pre-written proof, remember that
either of these proofs can correct. You’ll have to
determine which method was used based on the
number of blanks in the proof.
9. Try filling in the missing statements or reasons for this proof now:
Prove that if 3 k + 5 = 17 , then k = 4 .
Statements
Reasons
Given
3 k + 5 − 5 = 17 − 5
Substitution
Division property
k = 4
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M2 GEOMETRY – PACKET 3 FOR UNIT 1 – SECTIONS 2-3, 2-6, 2-7
10. Fill in the missing statements or reasons:
Statements
Prove that if − 3 ( x + 2 ) = 16 − x , then x = − 1 1 .
Reasons
− 3 ( x + 2 ) = 16 − x
− 3 x − 6 = 16 − x
Distributive
− 3 x − 6 + x = 16 − x + x
Substitution
− 2 x − 6 + 6 = 16 + 6
Substitution
Division property
x = −11
In the table on page 4, the letters a, b, and c are understood to represent numbers. Here’s
what some of those properties would look like for segment lengths.
Reflexive Property of Equality
AB = AB
Symmetric Property of Equality
If AB = CD, then CD = AB.
Transitive Property of Equality
If AB = CD and CD = EF, then
AB = EF.
11. Fill in the missing statements and reasons in this geometric proof:
Given the diagram at right, prove that if AB ≅ CD , then x = 2 .
Statements
Reasons
Given
AB = CD
Definition of congruent segments
6 − x = 3x − 2
Substitution
6 = 4x − 2
Addition property
2 = x
Symmetric
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M2 GEOMETRY – PACKET 3 FOR UNIT 1 – SECTIONS 2-3, 2-6, 2-7
Proving Segment Relationships
A postulate is a statement we assume to be true without proving it.
The Segment Addition Postulate says:
If A, B, and C are collinear, then B is between
A and C if and only if AB + BC = AC.
In math, the phrase “if and only if” implies that a statement is true when read in both
directions. Thus, the Segment Addition Postulate tells us that:
(1) If A, B, and C are collinear and B is between A and C, then AB + BC = AC.
(2) If A, B, and C are collinear and AB + BC = AC, then B is between A and C.
We’ll probably use statement (1) most often, but you should be aware that both are true.
Example: Given the diagram at right, prove that if AB ≅ CD , then AC ≅ BD .
Statements
1.
AB ≅ CD
Reasons
1. Given
2. AB = CD
2. Definition of congruent segments
3. BC = BC
3. Reflexive
4.
AB + BC = AC
CD + BC = BD
4. Segment Addition Postulate
5. C D + B C = AC
5. Substitution (in first part of line 4)
6. AC = BD
6. Transitive
7.
AC ≅ BD
7. Definition of congruent segments
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M2 GEOMETRY – PACKET 3 FOR UNIT 1 – SECTIONS 2-3, 2-6, 2-7
Now try filling in the missing statements and reasons in this one:
Prove the following.
Given:
AC = AB
AB = BX
CY = XD
Prove:
AY = BD
Statements
Reasons
1.
1. Given
2.
2. Transitive
3. AC + CY = BX + XD
3.
4.
5.
____ + ____ = ____
____ + ____ = ____
4. Segment Addition Postulate
5. Substitution
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