1 geometry 1 unit 2: reasoning and proof. 2 geometry 1 unit 2 2.1 conditional statements

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Geometry 1

Unit 2: Reasoning and Proof

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Geometry 1 Unit 2

2.1 Conditional Statements

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Conditional Statements

Conditional Statement- A statement with two parts

If-then form A way of writing a conditional statement that clearly

showcases the hypothesis and conclusion Hypothesis-

The “if” part of a conditional Statement Conclusion

The “then” part of a conditional Statement

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Examples of Conditional Statements If today is Saturday, then tomorrow is Sunday. If it’s a triangle, then it has a right angle. If x2 = 4, then x = 2. If you clean your room, then you can go to the

mall. If p, then q.

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Example 1 Circle the hypothesis and underline the conclusion in

each conditional statement

If you are in Geometry 1, then you will learn about the building blocks of geometry

If two points lie on the same line, then they are collinear

If a figure is a plane, then it is defined by 3 distinct points

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Example 2 Rewrite each statement in if…then form

A line contains at least two points

When two planes intersect their intersection is a line

Two angles that add to 90° are complementary

If a figure is a line, then it contains at least two points

If two planes intersect, then their intersection is a line.

If two angles add to equal 90°, then they are complementary.

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CounterexampleAn example that proves that a given

statement is false Write a counterexample

If x2 = 9, then x = 3

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Example 3Determine if the following statements are true

or false. If false, give a counterexample.

If x + 1 = 0, then x = -1 If a polygon has six sides, then it is a decagon. If the angles are a linear pair, then the sum of the

measure of the angles is 90º.

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Negation In most cases you can form the negation of a

statement by either adding or deleting the word “not”.

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Examples of Negations Statement:

Negation :

Statement: John is not more than 6 feet tall. Negation: John is more than 6 feet tall

30Am

30Am

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Example 4 Write the negation of each statement.

Determine whether your new statement is true or false. Yuma is the largest city in Arizona. All triangles have three sides. Dairy cows are not purple. Some CGUHS students have brown hair.

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Converse Formed by switching the if and the then part.

Original If you like green, then you will love my new shirt.

Converse If you love my new shirt, then you like green.

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Inverse Formed by negating both the if and the then

part. Original

If you like green, then you will love my new shirt.

Inverse If you do not like green, then you will not love my new

shirt.

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Contrapositive Formed by switching and negating both the if

and then part. Original

If you like green, then you will love my new shirt.

Contrapositive If you do not love my new shirt, then you do not like

green.

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Write in if…then form. Write the converse, inverse and

contrapositive of each statement.

I will wash the dishes, if you dry them.

A square with side length 2 cm has an area of 4 cm2.

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Point-line postulate: Through any two points, there exists exactly

one line Point-line converse:

A line contains at least two points Intersecting lines postulate:

If two lines intersect, then their intersection is exactly one point

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Point-plane postulate: Through any three noncollinear points there exists

one plane Point-plane converse:

A plane contains at least three noncollinear points Line-plane postulate:

If two points lie in a plane, then the line containing them lies in the plane

Intersecting planes postulate: If two planes intersect, then their intersection is a line

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Geometry 1 Unit 2

2.2: Definitions and Biconditional Statements

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Definitions and Biconditional Statements Can be rewritten with “If and only if” Only occurs when the statement and the

converse of the statement are both true. A biconditional can be split into a

conditional and its converse.

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Definitions and Biconditional Statements Example 1

An angle is right if and only if its measure is 90º

A number is even if and only if it is divisible by two.

A point on a segment is the midpoint of the segment if and only if it bisects the segment.

You attend school if and only if it is a weekday.

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Definitions and Biconditional Statements Perpendicular lines

Two lines are perpendicular if they intersect to form a right angle

A line perpendicular to a plane A line that intersects the plane in a point and is

perpendicular to every line in the plane that intersects it

The symbol is read, “is perpendicular to.

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Write the definition of perpendicular as biconditional statement.

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Give a counterexample that demonstrates that the converse is false.

If two lines are perpendicular, then they intersect.

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The following statement is true. Write the converse and decide if it is true or false. If the converse is true, combine it with its original to form a biconditional.

If x2 = 4, then x = 2 or x = -2

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Consider the statement

x2 < 49 if and only if x < 7. Is this a biconditional? Is the statement true?

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Geometry 1 Unit 2

2.3 Deductive Reasoning

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Deductive Reasoning

Symbolic Logic Statements are replaced with variables, such

as p, q, r.Symbols are used to connect the statements.

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Deductive Reasoning

Symbol Meaning

~ not

Λ and

V or

→ if…then

↔ if and only if

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Deductive Reasoning

Example 1Let p be “the measure of two angles is 180º”

and Let q be “two angles are supplementary”.

What does p → q mean?

What does q → p mean?

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Deductive Reasoning

Example 2p: Jen cleaned her room.q: Jen is going to the mall.

What does p → q mean? What does q → p mean? What does ~q mean? What does p Λ q mean?

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Deductive Reasoning

Example 3 Given t and s, determine the meaning of the

statements below. t: Jeff has a math test today s: Jeff studied

t V s s → t ~s → t

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Deductive Reasoning

Deductive ReasoningDeductive reasoning uses facts, definitions,

and accepted properties in a logical order to write a logical argument.

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Deductive Reasoning

Law of Detachment When you have a true conditional statement

and you know the hypothesis is true, you can conclude the conclusion is true.

Given: p → q

Given: p

Conclusion: q

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Deductive Reasoning

Example 4Determine if the argument is valid.

If Jasmyn studies then she will ace her test.

Jasmyn studied.

Jasmyn aced her test.

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Deductive Reasoning


If Mike goes to work, then he will get home late.

Mike got home late.

Mike went to work

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Deductive Reasoning

Law of Syllogism Given two linked conditional statements you

can form one conditional statement.

Given: p → q

Given: q → r

Conclusion: p → r

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Deductive Reasoning


If today is your birthday, then Joe will bake a cake.If Joe bakes a cake, then everyone will celebrate.

If today is your birthday, then everyone will celebrate.

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Deductive Reasoning


If it is a square, then it has four sides.

If it has four sides, then it is a quadrilateral.

If it is a square, then it is a quadrilateral.

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Geometry 1 Unit 2

2.4 Reasoning with Properties from Algebra

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Reasoning with Properties from Algebra Objectives

Review of algebraic properties

Reasoning

Applications of properties in real life

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Reasoning with Properties from Algebra Addition property

If a = b, then a + c = b + c

Subtraction property If a = b, then a – c = b – c

Multiplication property If a = b, then ac = bc

Division property If a = b, then cbca

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Reasoning with Properties from Algebra Reflexive property

For any real number a, a = a Symmetric property

If a=b, then b = a Transitive Property

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

If a = b, then a can be substituted for b in any equation or expression

Distributive property 2(x + y) = 2x + 2y

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Reasoning with Properties from Algebra Example 1

Solve 6x – 5 = 2x + 3 and write a reason for each step

Statement Reason

6x – 5 = 2x + 3 Given

4x – 5 = 3

4x = 8

x = 2

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Reasoning with Properties from Algebra

Example 2 2(x – 3) = 6x + 6

1. Given

2.

3.

4.

5.

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Reasoning with Properties from Algebra Determine if the equations are valid or invalid.

(x + 2)(x + 2) = x2 + 4

x3x3 = x6

-(x + y) = x – y

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Reasoning with Properties from Algebra Geometric Properties of Equality

Reflexive property of equality For any segment AB, AB = AB

Symmetric property of equality If then

Transitive property of equality If AB = CD and CD = EF, then, AB = EF,BmAm AmBm

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Reasoning with Properties from Algebra

Statement Reason

AB = CD

AB + BC = BC + CD

AC = AB + BC

BD = BC + CD

AC = BD

A B C DExample 3

In the diagram, AB = CD. Show that AC = BD

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Geometry 1 Unit 2

2.5: Proving Statements about Segments

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Proving Statements about Segments Key Terms:

2-column proof A way of proving a statement using a numbered

column of statements and a numbered column of reasons for the statements

Theorem A true statement that is proven by other true

statements

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Proving Statements about Segments Properties of Segment Congruence

Reflexive For any segment AB,

Symmetric If , then

Transitive If and ,then

AB CD

AB AB

CD AB

AB CDAB EF CD EF

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Proving Statements about Segments Example 1

In triangle JKL,Given: LK = 5, JK = 5, JK = JLProve: LK = JL

Statement Reason

1. 1. Given

2. 2. Given

3. 3. Transitive property of equality

4. 4.

5. 5. Given

6. 6. Transitive property of congruence

J

K

L

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Proving Statements about Segments Duplicating a Segment Tools

Straight edge: Ruler or piece of wood or metal used for creating straight lines

Compass: Tool used to create arcs and circles

A B

C D

Steps1. Use a straight edge to

draw a segment longer than segment AB

2. Label point C on new segment

3. Set compass at length of segment AB

4. Place compass point at C and strike an arc on line segment

5. Label intersection of arc and segment point D

6. Segment CD is now congruent to segment AB

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Geometry 1 Unit 2

2.6: Proving Statements about Angles

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Proving Statements about Angles

Properties of Angle Congruence Reflexive

For any angle A,

Symmetric

Transitive

.A A

, .If A B then B A

, .If A Band B C then A C

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Right Angle Congruence TheoremAll right angles are congruent.

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Congruent Supplements Theorem If two angles are supplementary to the same angle,

then they are congruent.

1 2 180

2 3 180 ,

1 3.

If

m m and

m m

then

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3

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Congruent Complements Theorem If two angles are complementary to the same angle,

then the two angles are congruent.

4 5 90

5 6 90 ,

4 6.

If

m m and

m m

then

4

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Linear Pair Postulate If two angles form a linear pair, then they are

supplementary.

1 2

1 2 180m m

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Vertical Angles TheoremVertical angles are congruent

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1 3, 2 4

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Example 1 Given: Prove:

1 2, 3 4, 2 3.

1 4

A

12 4

3

C

B

Statement Reason

1. 1.

2. 2.

3. 3.

4. 4.

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Example 2 Given: Prove:

1 63 , 1 3, 3 4

4 63

m

m

1 2

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Statement Reason

1. 1.

2. 2.

3. 3.

4. 4.

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Example 3 Given: are right angles

Prove:

,DAB ABC

ABC BDC

DAB BDC

A

D C

BStatement Reason

1. 1.

2. 2.

3. 3.

4. 4.

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Proving Statements about Angles Example 4

Given:

m1 = 24º,

m3 = 24º

1 and 2 are complementary

3 and 4 are complementary

Prove: 2 4

Statement Reason

1. 1.

2. 2.

3. 3.

4. 4.1 2

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Example 5 In the diagram m1 = 60º and BFD is right.

Explain how to show m4 = 30º.

12 3

4A F E

D

C

B

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Example 6 Given: 1 and 2 are

a linear pair, 2 and 3 are a linear pair

Prove: 1 3

12

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Statement Reason

1. 1.

2. 2.

3. 3.

1 geometry 1 unit 2: reasoning and proof. 2 geometry 1 unit 2 2.1 conditional statements

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