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GeoGeometr Geometry Geometry Logic and Reasoning

www.njctl.org March 6, 2013

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Table of Contents

Inductive Reasoning Logic (with Truth Tables) Deductive Reasoning Intro to Proofs (t-charts and paragraph) Algebraic Proofs Proofs with Segments and Angles If-Then Statements

click on the topic to go to that section

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Inductive Reasoning

Return to Table of Contents

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When asked a question you don't know the answer to, what 2 courses of action do you have?

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1) You can take a wild guess.

• r

2) Make an educated guess, conjecture, based

• n what is known to be
• true. Using examples to

develop your conjecture is inductive reasoning. Education is based on method #2. When asked a question you don't know the answer to, what 2 courses of action do you have?

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Example: Bob is taller than Carl and Carl is taller than Phil. Use inductive reasoning to make a conjecture about Bob and Phil.

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Make a conjecture of what comes next

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2, 5, 8, 11, 14, 17, ___ Make a conjecture for number comes next: 1, 2, 4, 8, 16, ___ 3, 5, 8, 13, 21, 34, ___ 4, 5, 7, 10, 14, 19, ___ adding 3 multiplying by 2 add 1, then 2, then 3... add 2 terms to get the next

HINT HINT HINT HINT

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are complementary Make a conjecture for each of the following bisects

M is the midpoint of

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1 Which conjecture can be drawn from the given

information:

A B C D

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2 Which conjecture can be drawn from the given

information:

A B C D

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3 Which conjecture can be drawn from the given

information:

A B C D

7, 10, 13, 16, 19, 38 22 23 20

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4 Which conjecture can be drawn from the given

information:

A B C D 18, 13, 9, 6, 4

• 2

1 2 3

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6 Which conjecture can be drawn from the given

information:

A B C D

is a right triangle. U V W M M is the midpoint of hypotenuse 6 8 VM = UV MW = UW VM = 10 MW = 5

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To prove a conjecture true by example, you have to show every example works. This is why we use theorems and postulates and do it in general terms. To prove a conjecture false by example, it takes only one false example. An example that shows a conjecture to be false is a counterexample . Conjecture: When a number is squared it is positive.

Counterexample: 0

Proving and Disproving Conjectures

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7 Using the given information, determine if the

conjecture is true. If conjecture is not necessarily true enter false, and be ready with a counterexample.

True False Given: A and B are on a number line. A is at 6 and B is 3 away. Conjecture: B is at 9

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8 Using the given information, determine if the

conjecture is true. If conjecture is not necessarily true enter false, and be ready with a counterexample.

True False Given: KM = MN and K, M, and N are collinear Conjecture: M is the midpoint

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9 Using the given information, determine if the

conjecture is true. If conjecture is not necessarily true enter false, and be ready with a counterexample.

True False Given: A circle is drawn with center A and a point on the circle C Conjecture: AC is a radius of the circle

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10 Using the given information, determine if the

conjecture is true. If conjecture is not necessarily true enter false, and be ready with a counterexample.

True False Given: xy=6 Conjecture: x=3

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Logic

(with Truth Tables) Return to Table of Contents

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To check the validity of a statement you first must check to see if the statement is always true. Note: It takes only 1 counterexample for the statement to be false. Example What is the validity of the statement: "Trenton is the capital of NJ" ?

The Validity of a Statement

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11 What is the validity of: 2 + 2 = 4.

True False

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12 What is the validity of: Cloudy days are rainy.

True False

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13 What is the validity of: Any number squared is positive.

True False

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Negation is when we take the opposite meaning (like putting a negative sign on a variable). Negations have the opposite validity as the original

• statement. The symbol for negation is ~

The negation of statement p is ~p, said "not p"

Negation of a Statement

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Negate the following statements by stating the validity of the statement and the negation. 1) Cakes are round

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Negate the following statements, state the validity of the statement and the negation. 2) Soda is brown

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Negate the following statements, state the validity of the statement and the negation. 3) Today is not Saturday

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Negate the following statements, state the validity of the statement and the negation. 4) Trees are not tall

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Negate the following statements, state the validity of the statement and the negation. 5) x is a positive number

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Negate the following statements, state the validity of the statement and the negation. 6) The square root of a negative number is real

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And forms a conjunction. The symbol for p and q is p # q or p q For an "AND" statement to be true, both p and q need to be true

Compound Statements using AND

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Or forms a disjunction The symbol for p or q is p∪q or p q For an "OR" statement to be true either p is true, q is true,

• r both are true

Compound Statements using OR

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Venn Diagrams

Venn Diagrams are a visual way

• f looking for

conjunctions and disjunctions.

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Venn Diagrams

Multiples of 2 Multiples of 3 2 4 6 8 10 12 14 16 18 20 3 9 15 21

Multiples of 2 and 3 lie in the intersection Multiples of 2 or 3 lie in the union of the sets

Note: When listing out the union each number appears once.

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Create a Venn Diagram

Types of Cars Planets Types of Cars Types of Cars Planets Planets ∪ #

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14 Given the Venn Diagram. Which letter(s) satisfies p

# q?

A B C D E F

p q

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15 Given the Venn Diagram. Which letter(s) satisfies p

∪ q?

A B C D E F

p q

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16 Given the Venn Diagram. Which letter(s) satisfies p?

A B C D E F

p q

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17 Given the Venn Diagram. Which letter(s) satisfies ~r?

A B C D E F

p r

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18 Given the Venn Diagram. Which letter(s) satisfies ~q?

A B C D E F

p q

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Ands and Ors can also be used with statements. Validity of a compound statement can be found by creating a Truth Table . p q p# q T T T T F F F T F F F F p q p∪ q T T T T F T F T T F F F

Truth Tables

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(1) Start by making the headings, p , q, ~p, and ~p # q. (2) The number of rows needed is 2n where n is the number of variables used. In the example n=2, so 22=4 rows. (3) Fill the first column with T for the first half, F for the second For the second column alternate.

p q ~p ~p# q T T F F T F F F F T T T F F T F

~p# q is true only when p is false and q is true.

(4) Fill in the validity based on the column heading

When is ~p # q true?

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Example: Make a truth table for the following: p ∪ ~q

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Example: Make a truth table for the following: p ∪ (q# r)

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19 When making a truth table for ~p # ~q, how many

columns are needed?

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20 When making a truth table for ~p # ~q, how many rows

are needed?

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21 When is ~p # ~q true?

A

p and q are both true

B

p is true and q is false

C

p is false and q is false

D

p and q are both false

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22 When making a truth table for ~p # (q ∪ r), how many

columns are needed?

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23 When making a truth table for ~p # (q ∪ r), how many

rows are needed?

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24 When is ~p # (q ∪ r) true?

A

p and q true, r false

B

p and q true, r false

C

p, q, and r are all false

D

p true, q and r false

E

p false, q and r true

F

p and r false, q true

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Truth Tables can also be used to show 2 statements are

• equivalent. Example: Show that ~(p # q)= ~p ∪ ~q

p q p# q ~ (p# q) T T T F T F F T F T F T F F F T p q ~p ~q ~p∪ ~ q T T F F F T F F T T F T T F T F F T T T Both statements false only if both p and q are true. They are true for all other cases. The statements are equivalent.

Equivalent statements

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Show that ~p # ~q is equivalent to ~(p ∪ q).

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If-Then Statements

Return to Table of Contents

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If-Then Statements, or Conditional Statements, are in sentences that given a certain condition the following event happens. The if part is called hypothesis or antecedent (If today isTuesday...) The then part is called conclusion or consequent ( ...then tomorrow is Wednesday) Note: The hypothesis does not need to be first in the sentence, its the cause that leads to the effect. (You can go to the movies, if you clean your room.)

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If add 2 and 2, then you'll get 4. If it rains tomorrow, then the picnic will be cancelled. You can use the car, if you'll put gas in it. If the Jets win or the Bengals lose, then the Jets make the playoffs. I'll buy dinner next time, if you buy dinner this time.

Underline the hypothesis with one line and the conclusion with two lines.

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A conditional doesn't have to have if and then in it, just a hypothesis and a conclusion. Underline the hypothesis with one line and the conclusion with two lines. Rewrite the sentence in "if then" form. A square has 4 right angles. Bobby is in big trouble. On a windy day you can fly a kite.

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The symbol notation for conditionals is p⇒ q,which is read "p implies q" We can check the validity of a conditional. The only way to show a conditional false is for T⇒ F We look for this pattern to have a counter example. These cases seem illogical, but we're testing to see if the statement is a lie. p q p⇒ q T T T T F F F T T F F T If you get a 100% on your next test, then I'll give you $1000. The only pay to prove me a liar is to get a 100% and for me not to pay up.

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Find counterexamples for each of the following. (1) If it's summer, then it must be July.

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Find counterexamples for each of the following. (2) If you get a 90% on the test, then you'll get an A for the year.

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Find counterexamples for each of the following. (3) If a figure has 1 right angle, its a right triangle.

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Find counterexamples for each of the following. (4) If today is the 30th, then tomorrow is the 31st.

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Find counterexamples for each of the following. (5) If tomorrow we have no school, then today is Friday.

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25 Is this conditional T or F. If false, be ready with a

counterexample.

True False If points A, B, and C lie in a plane, then they are collinear.

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26 Is this conditional T or F. If false, be ready with a

counterexample.

True False If points A, B, and C lie on a line, then B is between A and C.

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27 Is this conditional T or F. If false, be ready with a

counterexample.

True False If today is May 32, then tomorrow is the 33rd.

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Given the conditional statement of p⇒ q there are 3 associated conditionals Converse: q⇒ p Inverse: ~p⇒ ~q Contrapositive: ~q⇒ ~p The validity of a conditional and its contrapositive are

• equivalent. The validity of the converse and inverse are the
• same. If all four are true, the statement is a definition.

Converse, Inverse, and Contrapositive

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Examples: Conditional: If tomorrow is Tuesday, then today is Monday. Converse: If today is Monday, then tomorrow is Tuesday. Inverse: If tomorrow is not Tuesday, then today is not Monday. Contrapositive: If tomorrow is not Monday, then today is not Tuesday. Which statements are true?

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Examples: Conditional: If angles are vertical angles, then they have the same measure. Converse: If angles have the same measure then they are vertical angles. Inverse: If angles are not vertical angles, then they do not have the same measure. Contrapositive: If angles do not have the same measures then they are not vertical angles. Which statements are true?

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Example: Conditional: If x = 2, then x2 = 4. Converse: Inverse: Contrapositive: Which of these statements are true?

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28 What is the inverse of: If a=0 then ab=0.

A

If a≠0,then ab≠0.

B

If ab≠0, then a≠0.

C

If ab=0, then a=0.

D

If a≠0, then ab=0.

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29 What is the converse of: If a=0 then ab=0.

A

If a≠0,then ab≠0.

B

If ab≠0, then a≠0.

C

If ab=0, then a=0.

D

If a≠0, then ab=0.

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30 What is the validity of the contrapositive of:

If a=0, then ab=0.

True False

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Deductive Reasoning

Return to Table of Contents

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Deductive Reasoning can be used make valid conclusions involving conditionals. If p⇒ q is true and p is true, then q is true. (To be able to conclude that q is true, both p must be true and the conditional must be true.)

Law of Detachment

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Make a conclusion using the Law of Detachment. Statements 1) If it rains, then the game will be cancelled. 2) Its raining. Try underlining parts of the conditional and label them p and q. If the 2nd statement is p, then q can be concluded. Statements 1) If x=4 and y=5, then xy=20. 2) xy=20.

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1) If an angle is a right angle, then its measure is 90 degrees. Angle ABC is a right angle. Conclusion_________________ 2) If you win today, then you'll play tomorrow. You won today. Conclusion_______________________ 3) If a point is a midpoint, then it splits the segment into 2 congruent segments. AM=MB. Conclusion____________________________

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31 Does the following conclusion follow from the first two

statements?

True False 1) If today is Saturday, then I'll wash my car. 2) Today is Saturday. Conclusion: I wash my car.

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32 Does the following conclusion follow from the first two

statements?

True False 1) If angle A is obtuse, then it is greater than 90. 2) Angle A is not 90. Conclusion: Angle A is obtuse.

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33 Does the following conclusion follow from the first two

statements?

True False 1) If an angle is bisected, then there are two angles with the same measure. 2) Angle A and angle B have the same measure. Conclusion: They were made with an angle bisector.

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34 Does the following conclusion follow from the first two

statements?

True False 1) If a triangle has 3 equal sides,then it is equilateral. 2) #ABC has three equal sides. Conclusion: #ABC is equilateral.

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Deductive Reasoning can be used make valid conclusions involving conditionals. If p⇒ q is true and q⇒ r is true, then p⇒ r is true.

Law of Syllogism

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Make a conclusion using the Law of Syllogism. Statements 1) If it rains, then the game will be cancelled. 2) If the game is cancelled, then we can go to the movies. Try underlining parts of the conditional and label them p and q. If in the 2nd statement there is a repeat of the first statement use the same letter to label it. Look for the p⇒ q and q⇒ r so we can conclude p⇒ r. Statements 1) If a tree is an oak, then it grows very tall. 2) If a tree casts a large shadow, then it grows very tall.

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1) If an angle is a right angle, then its measure is 90 degrees. If an angle is 90 degrees, then it is not acute or obtuse. Conclusion_________________

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2) If you win today, then you'll play tomorrow. If you play tomorrow, then you are in the championship game. Conclusion_______________________

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3) If a point is a midpoint, then it splits the segment into 2 congruent segments. If a point is a midpoint then it is collinear with the endpoints

• f a segment.

Conclusion____________________________

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35 Does the following conclusion follow from the first two

statements?

True False 1) If a triangle has 3 equal sides, then it is equilateral. 2) If a triangle is equilateral, then it also has 3 equal angles. Conclusion: If a triangle has 3 equal sides, then it has 3 equal angles.

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36 Does the following conclusion follow from the first two

statements?

True False 1) If Carol goes to Rutgers, then she will major in accounting. 2) If Carol majors in accounting, then she will get a good job. Conclusion: If Carol has a good job,then she went to Rutgers.

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37 Does the following conclusion follow from the first two

statements?

True False 1) If figure is a circle, then it measures 360 degrees. 2) If a figure is a square, then its angles add to 360 degrees Conclusion: If a figure is a square, then it is a circle.

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38 Does the following conclusion follow from the first two

statements?

True False 1) If you get a free soda at Burger Barn, then you won't have to go to the convenience store. 2) If you spend $20 or more at Burger Barn, then you'll get a free soda. Conclusion: If you spend $20 or more at Burger Barn,then you won't have to go to the convenience store.

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Intro to Proofs

(t-charts and paragraph) Return to Table of Contents

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A theorem is a statement proven true. A postulate is a statement that is accepted as true.

Through any 2 points there is exactly one line. Through any 3 points not collinear, there is exactly one plane. A line contains at least 2 points. Planes contain at least 3 non- collinear points. The intersection

• f 2 unique

lines is a point.

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39 Which of the following postulates justifies the statement

is true?

A Through any 2 points there is exactly 1 line. B A line contains at least 2 points. C A plane contains at least 3 noncollinear points. D The intersection of 2 unique lines is a point. E The intersection of 2 unique planes is a line.

P

l R

The intersection of planes P and R is line l.

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40 Which of the following postulates justifies the statement

is true?

A Through any 2 points there is exactly 1 line. B Through any 3 points not on a line, there is exactly one plane. C A line contains at least 2 points. D The intersection of 2 unique lines is a point. E The intersection of 2 unique planes is a line.

Line AB is the only line through both A and B A B

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41 Which of the following postulates justifies the statement

is true?

A Through any 2 points there is exactly 1 line. B Through any 3 points not on a line, there is exactly one plane. C A line contains at least 2 points. D A plane contains at least 3 noncollinear points. E The intersection of 2 unique planes is a line. F If 2 points from the same line lie in a plane, the entire line is in the plane.

A B C S Since AB lies in plane S, point C lies in plane S.

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42 Which of the following postulates justifies the statement

is true?

A Through any 2 points there is exactly 1 line. B Through any 3 points not on a line, there is exactly one plane. C A line contains at least 2 points. D A plane contains at least 3 noncollinear points. E The intersection of 2 unique lines is a point. F The intersection of 2 unique planes is a line.

The bottom of a three-legged stool will sit flat on the floor.

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So how does a postulate become a theorem? Using a series of logical steps, like the Law of Detachment. How do you begin? Begin with what is given. Use definitions, theorems, and postulates to make new statements. So just make statements? Each statements needs a justification, like a lawyer presenting a case needs evidence.

Questions about Proofs

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When do I stop making new statements? When you've made the statement that you were asked to prove. I've heard there is a special form it has to follow? There are a few forms proofs can take. We are going to do T- Charts and Paragraph Proofs.

Questions about Proofs

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T-Chart

Given: ABCD is a square Prove: AB=BC A B C D Statement Reasons 1) ABCD is a square 2) AB≅BC 3) AB=BC 1) Given 2) Definition of a Square 3) Definition of Congruent A T-chart is a way to visually organize your statements and

• justifications. Like writing an outline for a research paper.

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Paragraph Proof

Given: ABCD is a square Prove: AB=BC A B C D Statement 1) ABCD is a square 2) AB≅BC 3) AB=BC 1) Given 2) Definition of a Square 3) Definition of Congruent Reasons Given ABCD is a square, we can use the definition of a square to say that AB≅ BC. By the definition of congruent, AB=BC A paragraph proof is taking the statements and justifications and making them into sentences.

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If M is the midpoint of XY, then XM=MY

Midpoint Theorem

X Y

M

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Justifications Given Definition of Midpoint Definition of Congruent Statement Given: C is the midpoint of AB Prove: AC ≅ BC Now use the T-Chart to write a paragraph proof. A B C Reasons

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Justifications Given Definition of Midpoint Definition of Congruent Statement Given: Prove: Now use the T-Chart to write a paragraph proof. AC ≅ BD, B is the midpoint of AC C is the midpoint of BD AB=CD A B C D Reasons Substitution

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Make a reference sheet for yourself, if you haven't already started. What "justifications" have we learned so far? Segments · Def of Segment Bisector · Def of Midpoint · Def of Congruent Segments

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Make a reference sheet for yourself, if you haven't already started. What "justifications" have we learned so far? Angles · Def of Angle Bisector · Def of Vertical Angles · Def of Complementary · Def of Supplementary · Def of Linear Pair · Def of Perpendicular · Def of Right Angle · Def of Congruent Angles

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Make a reference sheet for yourself, if you haven't already started. What "justifications" have we learned so far? Points,Lines,and Planes · Through any 2 points there is exactly one line. · Through any 3 points not collinear, there is exactly one plane. · A line contains at least 2 points. · Planes contain at least 3 non-collinear points. · The intersection of 2 unique lines is a point. · The intersection of 2 unique planes is a line. · If 2 points from the same line are in a plane, the entire line is in the plane.

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Algebraic Proofs

Return to Table of Contents

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In algebra, you solved equations using real number properties.

Addition Property of Equality If a=b, then a + c = b + c If a=b and c=d, then a + c = b + d Subtraction Property of Equality If a=b, then a - c = b - c If a=b and c=d, then a - c = b - d Multiplication Property of Equality If a=b, then ac = bc Division Property of Equality If a=b and c # 0, then a/c = b/c Substitution Property If a=b, then a can replace b and b can replace a in any equation. Distributive Property a(b + c) = ab + ac Reflexive Property a=a Symmetric Property If a=b, then b=a Transitive Property If a=b and b=c, then a=c.

These properties are the justifications for each step in solving the equation.

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43 Which property justifies the statement.

A Addition Property of Equality B Subtraction Property of Equality

C

Substitution Property D Distributive Property E Reflexive Property F Symmetric Property

If 5x + 7 = 12, then 5x = 5.

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44 Which property justifies the statement.

A Addition Property of Equality B Subtraction Property of Equality C Multiplication Property of Equality D Division Property of Equality E Substitution Property F Distributive Property

If 5x = 5, then x = 1.

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45 Which property justifies the statement.

A Multiplication Property of Equality B Division Property of Equality

C

Substitution Property D Distributive Property E Symmetric Property F Transitive Property

If 5(x + 7) = 12, then 5x +35 = 12.

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46 Which property justifies the statement.

A Addition Property of Equality B Subtraction Property of Equality C Multiplication Property of Equality D Division Property of Equality E Substitution Property F Distributive Property

If x/3 = 20, then x = 60.

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47 Which property justifies the statement.

A Division Property of Equality B Substitution Property C Distributive Property D Reflexive Property E Symmetric Property F Transitive Property

If 8 = x, then x = 8.

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Solve the following equations by making each step in the solution a statement. Justify the step. Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Substitution Property Distributive Property Reflexive Property Symmetric Property Transitive Property given

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Solve the following equations by making each step in the solution a statement. Justify the step. Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Substitution Property Distributive Property Reflexive Property Symmetric Property Transitive Property given

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Solve the following equations by making each step in the solution a statement. Justify the step. Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Substitution Property Distributive Property Reflexive Property Symmetric Property Transitive Property given

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Solve the following equations by making each step in the solution a statement. Justify the step. Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Substitution Property Distributive Property Reflexive Property Symmetric Property Transitive Property given

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Solve the following equations by making each step in the solution a statement. Justify the step. Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality Substitution Property Distributive Property Reflexive Property Symmetric Property Transitive Property given

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Proofs with Segments and Angles

Return to Table of Contents

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Segment Addition Postulate B is between A and C, then AB + BC = AC A B C

Cannot assume AB<BC, only that B is between A and C.

Use Segment Addition Postulate to find x. AB=4, BC=8, and AC=x AB=5, BC=x and AC=10 AB=x, BC=8 and AC= 12 AB=2x, BC=5x, and AC=28

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48 M is between K and N. Use the segment Addition

Postulate to find x, if KN=12, KM= 4, and MN=x.

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49 M is between K and N. Use the Segment Addition

Postulate to find x, if KN=x, KM= 12, and MN=14.

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50 M is between K and N. Use the Segment Addition

Postulate to find x, if KN=4x, KM= 8, and MN=8.

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51 M is between K and N. Use the Segment Addition

Postulate to find x, if KN=6x, KM= 3x, and MN=15.

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There are 2 basic types of proofs with segments: (1) Putting together smaller segments to make bigger segments (2) Breaking bigger segments into smaller ones

proofs with segments

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Putting together smaller segments to make bigger segments Given:AB=CD Prove:AC=BD Statement B 1) AB+BC=AC 2) BC+CD=BD 3) AB=CD 4) AB+BC=BC+CD 5) AC=BD 1) Segment Addition Postulate 2) Segment Addition Postulate 3) Given 4) Addition Prop of Equality 5) Substitution Reasons A C D Smaller ones,add to the bigger, use the given to write an equation, addition, substitution. Proofs

• f this type follow this basic pattern.

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Given:AB=MN, BC=NP Prove:AC=MP Statement Def of Congruent Segments Def of Midpoint Def of Segment Bisector Justifications for Segments Segment Addition Postulate A M B N C P Reasons

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Proofs involving Segments Given:DE=SE, EF=ET Prove:DF=ST Statement Def of Congruent Segments Def of Midpoint Def of Segment Bisector Justifications for Segments Segment Addition Postulate D E F T S Reasons

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Breaking bigger segments into smaller ones Given:AC=BD Prove:AB=CD Statement A B C 1) AB+BC=AC 2) BC+CD=BD 3) AC=BD 4) AB+BC=BC+CD 5) AB=CD 1) Segment Addition Postulate 2) Segment Addition Postulate 3) Given 4) Substitution 5) Subtraction Property of Equality Smaller ones,add to the bigger, use the given to write an equation, substitute, and subtract. Proofs of this type follow this basic pattern. D Reasons

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Proofs involving Segments Given:DF=ST X is the midpoint of DF X is the midpoint of ST Prove:DF=ST Statement Def of Congruent Segments Def of Midpoint Def of Segment Bisector Justifications for Segments Segment Addition Postulate D F T S X Reasons

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Proofs involving Segments Given:AC=XZ, AB=XY Prove:YZ=BC Statement Def of Congruent Segments Def of Midpoint Def of Segment Bisector Justifications for Segments Segment Addition Postulate A X B Y C Z Reasons

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Cannot assume angle size. Can assume a point lies inside.

Find x J H G S

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Find the value of the variable. 5 6 4 is supplemental to is supplemental to 1 2 3 4

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· Perpendicular lines form right angles · Right angles are congruent · Angles that are congruent and supplemental are right angles

Right Angle Theorems

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Find the value of x. 20o x x x y x≅y

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52 Find the value of z.

130o z

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53 Find the value of z.

130o z

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54 Find the value of z.

45o 85o A B C

F

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55 Find the value of z.

35o

z A B C F

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56 Find the value of z.

2z 3z-10

A B C F

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57 Find the value of z.

A B C F 60o bisects

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58 Find the value of z.

A B C F 40o z

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· Putting together smaller angles to make bigger angles · Breaking bigger angles into smaller ones ·Proving 2 angles are congruent because of the their relation to another angle, ie supplement or complement.

There are 3 basic types of proofs with angles:

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Putting together smaller angles to make bigger angles Given: Prove: Statement

1) Angle Addition Postulate 2) Given 3) Reflexive 4) Addition Property of Equality 5) Substitution

Smaller ones,add to the bigger, use the given to write an equation, add the same thing (or congruent things) to both sides, substitution. Proofs

• f this type follow this basic pattern.

A B C X Y Reasons

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Given: Prove: Statement Justifications for Angles Def of Angle Bisector Def of Vertical Angles Def of Complementary Def of Supplementary Def of Linear Pair Def of Perpendicular Def of Right Angle Def of Congruent Angles Segment Addition Postulate Vertical Angle Theorem Right Angle Theorem Supplements of an Angle are ≅ Complements of an Angle are ≅ A B

C X J K L M Reasons

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Given: Prove: Statement Justifications for Angles Def of Angle Bisector Def of Vertical Angles Def of Complementary Def of Supplementary Def of Linear Pair Def of Perpendicular Def of Right Angle Def of Congruent Angles Segment Addition Postulate Vertical Angle Theorem Right Angle Theorem Supplements of an Angle are ≅ Complements of an Angle are ≅ A B C D E F X Reasons

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Breaking bigger angles into smaller ones Given: Prove: Statement

1) Angle Addition Postulate 2) Subtraction Prop 3) Given 4)Subtraction Prop 5) Substitution

A B C D Reasons

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Given: Prove: Statement Justifications for Angles Def of Angle Bisector Def of Vertical Angles Def of Complementary Def of Supplementary Def of Linear Pair Def of Perpendicular Def of Right Angle Def of Congruent Angles Angle Addition Postulate Vertical Angle Theorem Right Angle Theorem Supplements of an Angle are ≅ Complements of an Angle are ≅ A B

C X Y Reasons

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Given: Prove: Statement Justifications for Angles Def of Angle Bisector Def of Vertical Angles Def of Complementary Def of Supplementary Def of Linear Pair Def of Perpendicular Def of Right Angle Def of Congruent Angles Angle Addition Postulate Vertical Angle Theorem Right Angle Theorem Supplements of an Angle are ≅ Complements of an Angle are ≅ A B

C D X Y Reasons

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Proving angles congruent because of their relation to a third Given: Prove: Statement

1) Vertical Angle Theorem 2) Given 3) Substitution

Reasons

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Given: Prove: Statement Justifications for Angles Def of Angle Bisector Def of Vertical Angles Def of Complementary Def of Supplementary Def of Linear Pair Def of Perpendicular Def of Right Angle Def of Congruent Angles Angle Addition Postulate Vertical Angle Theorem Right Angle Theorem Supplements of an Angle are ≅ Complements of an Angle are ≅ are supplemental are supplemental 1 2 3 Reasons

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Statement Justifications for Angles Def of Angle Bisector Def of Vertical Angles Def of Complementary Def of Supplementary Def of Linear Pair Def of Perpendicular Def of Right Angle Def of Congruent Angles Angle Addition Postulate Vertical Angle Theorem Right Angle Theorem Supplements of an Angle are ≅ Complements of an Angle are ≅ Given: Prove: Reasons