Geometry Unit 2

Choose the correct number to continue the pattern. 0, 5, 2, 7, 4, 9, 6, _____

11

True or False: An application of deductive reasoning such that the reasoning is logically correct and undeniably true is a valid argument.

True

True or False: An argument that uses written justification in the form of definitions, properties, postulates and previously proved theorems and corollaries to show that a conclusion is true is a formal proof.

True

True or False: The Associative Property of Addition states that for all expressions a, b, and c , (a+b)+c=a+(b+c) .

True

True or False: The Associative Property of Multiplication states that for all expressions a, b, and c , (ab)c=a(bc) .

True

True or False: The Commutative Property of Multiplication states that for all expressions a and b , ab=ba .

True

True or False: a statement concluded to be true based on logical reasoning is a conjecture.

True

What can be concluded from m∠1=90∘ and m∠2=90∘ by the Transitive Property of Equality?

∠1=∠2

Which property is illustrated? If x=2, then xy=2y

Multiplication Property of Equality

The part of a conditional statement that expresses the action that will result if the conditions of the statement are met is the _____.

conclusion

A statement in which a conclusion is true if the conditions of a particular hypothesis are true is called a _____ statement.

conditional

True or False: An argument that uses logic in the form of definitions, properties, and previously proved principles to show that a conclusion is true is a valid argument.

False

True or False: An argument that uses logic without written justification to show that a conclusion is true is a formal proof.

False

True or False: The Commutative Property of Addition states that for all expressions a and b , ab=ba .

False

True or False: the act of forming conclusions based on available information is a conjecture.

False

Given: 20/5x=2 Prove: x=2 Statements 20/5x=2 E/(20=10x) B/(2=x) D/(x=2)

Given: 20/5x=2 Prove: x=2 Reasons Given C/(Multiplication Property of Equality) A/(Division Property of Equality) F/(Symmetric Property of Equality)

A _____ is a compound logic statement made up of two statements joined together with the word "and."

conjunction

A _____ is a related conditional statement that results from the exchange and negation of the hypothesis and conclusion of a conditional statement.

contrapositive

A related conditional statement in which the hypothesis and the conclusion of a conditional statement have been exchanged is called the ___.

converse of a conditional statement

An example that proves a conjecture false is called a(n) ___.

counterexample

The _____ is the part of a conditional statement that expresses the conditions that must be met.

hypothesis

A _____ is the degree of truth of a conditional statement.

truth value

True or False: An algebraic proof is a proof that uses algebraic properties to reach a conclusion about an algebraic equation.

True

A(n) _____ proof is a proof that uses algebraic properties to reach a conclusion about an algebraic equation.

algebraic

a proof that uses algebraic properties to reach a conclusion about an algebraic equation

algebraic proof

A series of reasons that leads to a conclusion is an _____.

argument

A logical statement formed by the combination of a conditional statement and its converse is a _____.

biconditional statement

The part of the conditional statement that expresses the action that will result if the conditions of the statement are met is called the _____.

conclusion

A(n) _____ is a statement that you conclude to be true based on logical reasoning.

conjecture

The exchange and negation of both the hypothesis and conclusion of a conditional statement results in a related conditional statement called a(n) _____.

contrapositive

An argument that uses written justification in the form of definitions, properties, and previously proved geometric principles to show that a conclusion is true is called a _____ proof.

formal

An argument that uses written justification in the form of definitions, properties, and previously proved geometric principles to show that a conclusion is true: ____

formal proof

A _____ is an argument that uses written justification in the form of definitions, properties, postulates, and previously proved theorems and corollaries to show that a conclusion is true.

geometric proof

An argument that uses written justification in the form of definitions, properties, postulates, and previously proved theorems and corollaries to show that a conclusion is true.

geometric proof

The process of reasoning that a rule, condition, definition, property, or statement is true because specific cases have been observed to be true is called _____ reasoning.

inductive

An argument that uses logic without written justification to show that a conclusion is true: _____.

informal proof

An argument that uses logic in the form of definitions, properties, and previously proved principles to show that a conclusion is true is called a _____.

proof

A _____ is an application of deductive reasoning such that the reasoning is logically correct and undeniably true.

valid argument

Which property is illustrated? For all expressions a , b , and c , if a=b , then a-c=b-c .

Subtraction Property of Equality

Which property is illustrated? If x+3=2x+6 , then x+3-3=2x+6-3 .

Subtraction Property of Equality

Which property of congruence is illustrated? For any geometric figures A and B , if A≅B , then B≅A .

Symmetric

Which property is illustrated? If AB¯≅BC¯, then BC¯≅AB¯

Symmetric Property of Congruence

___ is the act of forming conclusions based on available information.

Reasoning

Choose the correct number to continue the pattern. 1, 3, 5, 7, 9, 11, _____

13

A statement or conjecture that can be proven by undefined terms, definitions, postulates, and previously proven theorems.

theorem

Which property is illustrated? ∠ABC≅∠ABC

Reflexive Property of Congruence

The statement of the reason for each step in a proof: ___.

justification

Which property is illustrated? xy=yx

Commutative Property of Multiplication

Complete the formal algebraic proof: Given: AB=AC-BC; AB=2x+1; BC=8x-1; AC=7x+3 Prove: x=1 Statements AB=AC-BC; AB=2x+1; BC=8x-1; AC=7x+3 2x+1=(7x+3)-(8x-1) 2x+1=7x+3-8x+1 B/(3x=3) F/(x=1)

Complete the formal algebraic proof: Given: AB=AC-BC; AB=2x+1; BC=8x-1; AC=7x+3 Prove: x=1 Reasons Given E/(Substitution Property of Equality) Simplify Addition Property of Equality Division Property of Equality

Complete the proof. Given: BD→ divides ∠ABC ; m∠ABD=60∘ ; m∠DBC=60∘ . Prove: BD→ is an angle bisector. Statements BD→ divides ∠ABC ; m∠ABD=60∘ ; m∠DBC=60∘ m∠ABD=m∠DBC F/(∠ABD≅∠DBC) C/(BD→ is an angle bisector)

Complete the proof. Given: BD→ divides ∠ABC ; m∠ABD=60∘ ; m∠DBC=60∘ . Prove: BD→ is an angle bisector. Reasons Given D/(Substitution) Def. of congruent angles Def. of angle bisector

Given: ∠1 and ∠2 are right angles. What is the reason that m∠1=90∘ and m∠2=90∘ ?

Definition of right angles

Which property is illustrated? 5(a+b)=5a+5b

Distributive Property of Equality

Given: m∠RST=m∠RSU+m∠UST; m∠RSU=3x-5; m∠UST=x+4; m∠RST=6x-7 Prove: x=3 Statements m∠RST=m∠RSU+m∠UST; m∠RSU=3x-5; m∠UST=x+4; m∠RST=6x-7 6x-7=(3x-5)+(x+4) G/(6x-7=4x-1) 2x-7=-1 2x=6 A/(x=3)

Given: m∠RST=m∠RSU+m∠UST; m∠RSU=3x-5; m∠UST=x+4; m∠RST=6x-7 Prove: x=3 Reasons Given D/(Substitution Property of Equality) Simplify Subtraction Property of Equality E/(Addition Property of Equality) Division Property of Equality

Given: 4(x-3)=2(x-2) Prove: x=4 Statements 4(x-3)=2(x-2) 4x-12=2x-4 2x=8 x=4

Given: 4(x-3)=2(x-2) Prove: x=4 Reasons Given Distributive Property Subtraction Property Division Property

Given: AB¯≅DB¯ and BE¯≅BC¯ Prove: AC¯≅DE¯ Statements AB¯≅DB¯ and BE¯≅BC¯ C/(AB=DB,BE=BC) AB+BC=AC,DB+BE=DE DB+BE=AC AC=DB+BE E/(AC=DE) AC¯≅DE¯

Given: AB¯≅DB¯ and BE¯≅BC¯ Prove: AC¯≅DE¯ Reasons Given Def. of ≅ line segments Segment Addition Postulate B/(Substitution) Symmetric Property of = Transitive Property of =7 D/(Def. of ≅ line segments)

Given: B is the midpoint of AC¯, and AB¯≅DE¯ . Prove: BC¯≅DE¯ Statements B is the midpoint of AC¯, and AB¯≅DE¯ B/(BC¯≅AB¯) D/(BC¯≅DE¯)

Given: B is the midpoint of AC¯, and AB¯≅DE¯ . Prove: BC¯≅DE¯ Reasons Given Def. of Midpoint Transitive Property of ≅

Given: CE=CD+DE;CD=(7x-5); DE=(2x+5); CE=(x+8) Prove: x=1 Statements CE=CD+DE;CD=(7x-5); DE=(2x+5); CE=(x+8) x+8=(7x-5)+(2x+5) x+8=9x E /(8=8x) 1=x x=1

Given: CE=CD+DE;CD=(7x-5); DE=(2x+5); CE=(x+8) Prove: x=1 Reasons Given F/(Substitution Property of Equality) Simplify Subtraction Property of Equality C/(Division Property of Equality) D/(Symmetric Property of Equality)

Given: ∠1 and ∠2 are right angles. Prove: ∠1≅∠2 Statements B/(∠1 and ∠2 are right angles) D/(m∠1=90∘ and m∠2=90∘) H/(m∠1=∠2) C/(∠1≅∠2)

Given: ∠1 and ∠2 are right angles. Prove: ∠1≅∠2 Reasons G/(Given) E/(Definition of right angles) A/(Transitive property of equality) F/(Definition of congruence)

Given: ∠1 and ∠2 are supplementary; ∠3 and ∠4 are supplementary; ∠2≅∠3 Prove: ∠1≅∠4 Statements ∠1 and ∠2 are supplementary;∠3 and ∠4 are supplementary; ∠2≅∠3 m∠1+m∠2=180∘,m∠3+m∠4=180∘ 2. 3. m∠2=m∠3 4. m∠2+m∠4=180∘ 5. m∠1+m∠2=m∠2+m∠4 B/(b. m∠2=m∠2) 7. m∠1=m∠4 D/(∠1≅∠4)

Given: ∠1 and ∠2 are supplementary; ∠3 and ∠4 are supplementary; ∠2≅∠3 Prove: ∠1≅∠4 Reasons Given Def. of supplementary ∠ s Def. of ≅∠ s C/() Substitution Reflexive Prop. of = E/() Def. of ≅∠ s

What is the converse of: If a=b and b=c , then a=c

If a=c, then a=b and b=c

What is the inverse of: If a=b and b=c , then a=c

If a≠b or b≠c, then a≠c

What is the contrapositive of: If a=b and b=c , then a=c

If a≠c then a≠b or b≠c

The _____ Theorem states that "If any two angles form a linear pair, then they are supplementary."

Linear Pair

If any two angles form a linear pair, then they are supplementary.

Linear Pair Theorem

Conjecture: The number 60 can be evenly divided by all numbers from 1 to 9. Is this conjecture true?

No, 60 cannot be evenly divided by 7, 8, or 9

Is this a valid argument? Given: If a geometric figure starts at a point and extends forever in one direction, then it is a ray. AB→ is a ray. Conclusion:AB→ starts at one point and extends forever in one direction.

No. This is an improper use of the Law of Detachment.

Is this a valid argument? Given: If a flame is applied to paper, then the paper will burn. If there are ashes, then the paper burned. Conclusion: A flame burned the paper to ashes.

No. This is an improper use of the Law of Syllogism.

Which property is illustrated? ∠A≅∠A

Reflexive Property of Congruence

Which is a proper biconditional statement formed from the statement:

The same expression is added to both sides of an equation if, and only if, both sides of the equation remain equal.

Which property is illustrated? If ΔA≅ΔB and ΔB≅ΔC, then ΔA≅ΔC

Transitive Property of Congruence

True or False: A compound logic statement made up of two statements joined with the word "and" is a conjunction.

True

True or False: A justification is a statement of the reason for each step in a proof.

True

True or False: A series of reasons that leads to a conclusion is a valid argument.

True

Conjecture: When one plane intersects another plane, the intersection of the two planes is a line. Is this conjecture true?

Yes.

Is this a valid argument? Given: If an object is in the shape of a sphere, then it rolls. This ball is in the shape of a sphere. Conclusion: This ball rolls.

Yes. This argument uses the Law of Detachment.

Is this a valid argument? Given: If it is July 4th, then we celebrate Independence Day. Today is July 4th. Conclusion: Today, we celebrate Independence Day.

Yes. This argument uses the Law of Detachment.

Is this a valid argument? Given: If two points lie on the same plane, then they are coplanar. Points X and Y both lie on plane R. Conclusion: Points X and Y are coplanar.

Yes. This argument uses the Law of Detachment.

Is this a valid argument? Given: If it is autumn, the leaves will fall off the trees. If the leaves fall off the trees, I will have to pick up leaves. Conclusion: If it is autumn, I will have to pick up leaves.

Yes. This argument uses the Law of Syllogism.

Which property is illustrated? For all expressions a , b , and c , if a=b and c≠0 , then a/c=b/c .

Division Property of Equality

True or False: A biconditional statement is a statement in which a conclusion is true if the conditions of a particular hypothesis are true.

False

True or False: the process of reasoning that a rule, condition, definition, property, or statement is true because specific cases have been observed to be true is deductive reasoning.

False

Which statement is a biconditional statement formed from: I will pass Geometry if I complete my homework correctly.

I will pass Geometry if, and only if, I complete my homework correctly.

Which statement is the converse of: If a figure is a triangle, then it has three sides.

If a figure has three sides, then it is a triangle.

Which property is illustrated? For all expressions a and b , if a=b , then b can be substituted for a in any expression.

Substitution Property of Equality

Which property is illustrated? ∠A≅∠B so ∠B≅∠A

Symmetric Property of Congruence

Which property is illustrated? For all expressions a and b , if a=b then b=a .

Symmetric Property of Equality

Which property is illustrated? If x=y , then y=x .

Symmetric Property of Equality

Which property of congruence is illustrated? For any geometric figures A , B , and C , if A≅B and B≅C , then A≅C .

Transitive

Which property is illustrated? 3×10=30 and 30=5×6 so 3×10=5×6

Transitive Property of Equality

Which property is illustrated? For all expressions a , b , and c , if a = b and b = c , then a = c .

Transitive Property of Equality

True or False: A biconditional statement is a logical statement formed by the combination of a conditional statement and its converse.

True

True or False: an example that proves a conjecture to be false is a counterexample.

True

_____ is a judgment about how true or false a conditional statement is.

Truth value

Which statement is a biconditional formed from: If two line segments have the same length, then they are congruent.

Two line segments have the same length if, and only if, they are congruent.

What words match this description? an application of deductive reasoning such that the reasoning is logically correct and undeniably true

Valid argument

Which statement is a proper biconditional statement formed from the conditional statement: If today is Friday, then we will have pizza for lunch.

We will have pizza for lunch today if, and only if, today is Friday.

Conjecture: According to current data, the star VY Canis Majoris with a size of about 2,000 solar diameters is the largest star in the known universe. Is this conjecture true?

Yes.

Is this a valid argument? Given: If an angle is acute, then it cannot be obtuse ∠A is acute. Conclusion:∠A cannot be obtuse.

Yes. This argument uses the Law of Detachment.

Is this a valid argument? Given: If the sum of the measure of two angles equals 90∘ , then the angles are complementary. The sum of the measures of ∠A and ∠B is 90∘ Conclusion:∠A and ∠B are complementary.

Yes. This argument uses the Law of Detachment.

Is this a valid argument? Given: If I drive over 60 miles an hour, then I am breaking the law. If I am breaking the law, then I could go to jail. Conclusion: If I drive over 60 miles an hour, then I could go to jail.

Yes. This argument uses the Law of Syllogism.

Is this a valid argument? Given: If the temperature is below 32∘F, then water will freeze. If water freezes, then it will turn into a solid. Conclusion: If the temperature is below 32∘F, water will turn into a solid.

Yes. This argument uses the Law of Syllogism.

Which property is illustrated? For all expressions a , b , and c , if a=b , then a+c=b+c .

Addition Property of Equality

Which statement is a biconditional formed from: If an angle is acute, then its measure is less than 90 degrees.

An angle is acute if, and only if, its measure is less than 90 degrees.

Which statement is a proper biconditional statement formed from the conditional statement: If an angle is bisected, then it is divided into two congruent angles.

An angle is bisected if, and only if, it is divided into two congruent angles.

What words match this description? a series of reasons that leads to a conclusion

Argument

Which property is illustrated? For all expressions a , b , and c , a+(b+c)=(a+b)+c

Associative Property of Addition

Which property is illustrated? (3x)y=3(xy)

Associative Property of Multiplication

Which property is illustrated? For all expressions a , b , and c , (ab)c=a(bc)

Associative Property of Multiplication

Average Depth of the River Per Day Sunday: 10 Monday: 12 Tuesday: 14 Wednesday: 16 Thursday: 14 Friday: 12 Saturday: __________

Average Depth of the River Per Day Saturday: 10

Choose the correct number to continue the pattern. 2, 4, 8, 16, 32, 64, _____

Choose the correct number to continue the pattern. 128

Choose the correct number to continue the pattern. 1, 3, 6, 10, 15, 21, 28, _____

Choose the correct number to continue the pattern. 36

Choose the correct shape to fill in the blank. (triangle),(oval),(diamond),(pentagon),(square),(triangle),(oval),(_________),(pentagon)

Choose the correct shape to fill in the blank. diamond

Which property is illustrated? For all expressions a and b , a+b=b+a

Commutative Property of Addition

Which property is illustrated? x+9=9+x

Commutative Property of Addition

Which property is illustrated? For all expressions a and b , ab=ba .

Commutative Property of Multiplication

What words match this description? a compound logic statement made up of two statements joined together with the word "and"

Conjunction

Which property is illustrated? 2(4+5)=2(4)+2(5)

Distributive Property of Equality

Which property is illustrated? For all expressions a , b , and c , a(b+c)=ab+ac

Distributive Property of Equality

Which statement is the inverse of: If today is Tuesday, then I have Geometry class.

If today is not Tuesday, then I do not have Geometry class.

Which statement is the contrapositive of: If two angles are vertical angles, then they are not adjacent angles.

If two angles are adjacent angles, then they are not vertical angles.

Which statement is the converse of the conditional statement: If two figures have the same shape and size, then they are congruent.

If two figures are congruent, then they have the same shape and size.

Which statement is the contrapositive of the conditional statement: If two figures have the same shape and size, then they are congruent.

If two figures are not congruent, then they do not have the same shape and size.

Which statement is the inverse of the conditional statement: If two figures have the same shape and size, then they are congruent.

If two figures do not have the same shape and size, then they are not congruent.

Choose the best statement about this definition. An animal that is a mammal is a whale.

It could not be rewritten as a true biconditional statement.

Choose the best statement about this definition. A postulate is a statement accepted as true without proof as a basis for reasoning.

It is a good definition

Choose the best statement about this definition. The measure of an obtuse angle is greater than 90 degrees but less than 180 degrees.

It is a good definition.

Choose the best statement about this definition. A fermion is any particle which obeys the Fermi-Durac statistics.

It is not written in terms that have been previously defined or are clearly understood.

Choose the best statement about this definition. A coordinate is a number.

It uses terminology that is not precise or accurate.

What words match this description? a law of deductive reasoning that states that if a conditional statement is true and its hypothesis is true, then the conclusion will also be true

Law of Detachment

What words match this description? a law of deductive reasoning that states that if two conditional statements are true, and if the conclusion of the first statement is the hypothesis of the second statement, then a conclusion based on the conditional statements will also be true

Law of Syllogism

Which property is illustrated? For all expressions a , b , and c , if a=b then ac=bc .

Multiplication Property of Equality

Which property is illustrated? x=y so 4x=4y

Multiplication Property of Equality

Conjecture: The combined populations of American Indian and Alaska Native with Asian is more than the population of "Some other race." Is this conjecture true?

No, The combined populations of American Indian and Alaska Native with Asian is 15,624,350, which is less than the category "Some other race." No, the population of Asian is only 13,201,056, which is less than the category "Some other race." No, the population of Hispanic or Latino is 45,476,934, which is more than the category "Some other race." Yes.

Conjecture: The population of White is 80% of the total population. Is this conjecture true?

No, the percentage of White is found by dividing 224,469,780 by 301,461,533 and gives 74.46%.

Conjecture: Intersecting lines form only vertical angles.

No, ∠AEB and ∠BEC are a counterexample.

Conjecture: The noncommon rays of all adjacent angles form straight lines. Is this conjecture true?

No, ∠IJK is a counterexample.

Conjecture: When it is completed, One World Trade Center in New York will stand 1,776 feet tall and be the tallest building in the world. Is this conjecture true?

No. Burj Khalifa and Abraj Al Bait Tower are both counterexamples.

Is this a valid argument? Given: All football wide receivers eat bacon. Carl eats bacon. Conclusion: Carl is a football wide receiver.

No. This is an improper use of the Law of Detachment.

Is this a valid argument? Given: If a triangle has one 30 degree and one 60 degree angle, then it is a right triangle. A triangle with a 30 degree and a 60 degree angle has a 90 degree angle. Conclusion: A right triangle has a 90 degree angle.

No. This is an improper use of the Law of Syllogism.

What words match this description? an argument that uses logic in the form of definitions, properties, and previously proved principles to show that a conclusion is true

Proof

Which property of congruence is illustrated? For any geometric figure A , A≅A .

Reflexive

Which property is illustrated? XY¯≅XY¯

Reflexive Property of Congruence

Which property is illustrated? For every expression a, a=a .

Reflexive Property of Equality

Purchase Orders Supply Requisitions Correspondence Records Attendance Records

blue - 1 red - 3 green - 2 purple - 4

A statement in which a conclusion is true if the conditions of a particular hypothesis are true is a _____.

conditional statement

A statement that you conclude to be true based on logical reasoning is a(n) ___.

conjecture

A related conditional statement resulting from the exchange and negation of both the hypothesis and conclusion of a conditional statement is the ___.

contrapositive of a conditional statement

The exchange of the hypothesis and conclusion of a conditional statement results in a related conditional statement called a(n) _____.

converse

A biconditional statement that is used to describe a geometric object or concept is a _____.

definition

A biconditional statement that is used to describe a geometric object or concept is called a _____.

definition

The _____ is the part of a conditional statement that expresses the conditions that must be met by the statement.

hypothesis

The part of a conditional statement that expresses the conditions that must be met by the statement is the _____.

hypothesis

The process of reasoning that a rule, condition, definition, property, or statement is true because specific cases have been observed to be true is called ___.

inductive reasoning

The negation of the hypothesis and conclusion of a conditional statement results in a related conditional statement called a(n) _____.

inverse

A related conditional statement called the ___ results from the negation of the hypothesis and conclusion of a conditional statement.

inverse of a conditional statement

The _____ is the negative form of any part of a conditional statement.

negation

The ___ is the negative form of a conditional statement.

negation of a conditional statement

The converse, inverse, and contrapositive of a conditional statement are called _____.

related conditional statements