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Lesson 10 - Segment & Angle Addition Proofs February 03, 2015

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Lesson 10

Segment & Angle Addition Proofs

Lesson 9 Review

Announcements:Tuesday, Feb. 10th - Unit 2 TEST"Little Piece of Random" Sign-up sheetCalculator ProjectSupply Bucket

Quiz 1.3

Student Led Conferences

Lesson 10 Continued

Proof Levels Poster

Lesson 9&10 Quiz

Lesson 9-10 Review

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HomeworkLesson 10 PRACTICE 1

Study Flash Cards!!!

1.2.


Lesson 10Angle Theorems

LearningTarget H:Write two-column proofs about1) Vertical Angles2) Right Angles3) Angles supplementary or complementary to congruent angles (or the same angle)4) Segment and Angle Addition


Vertical Angles Theorem (Theorem 2-1)Vertical angles are congruent.

12

34

∠1≅∠3∠2≅∠4

Given:

You can conclude:

Reason:

NOT Definition of vertical angles.

Definition: Two angles whose sides are opposite rays.


Given:

You can conclude:

Reason:

1 2

3 4

∠1 is supplementary to ∠2∠3 is supplementary to ∠4∠1≅∠3

∠2≅∠4

Supplements of congruent angles are congruent.

Congruent Supplements Theorem (2-2)


Given:

You can conclude:

Reason:

1 2

3 ∠1 is supplementary to ∠2∠3 is supplementary to ∠2

∠1≅∠3

Supplements of the same angle are congruent.

Congruent Supplements Theorem (2-2)


Given:

You can conclude:

Reason:

Complements of congruent angles arecongruent.

Congruent Complements Theorem (2-3)

1 2

3 4

∠1 is complementary to ∠2∠3 is complementary to ∠4∠1≅∠3

∠2≅∠4


Given:

You can conclude:

Reason: Complements of the same angle are congruent.

Congruent Complements Theorem (2-3)

∠2≅∠3

∠1 is complementary to ∠2∠1 is complementary to ∠31

2

3


Given:

You can conclude:

Reason: All right angles are congruent.

Theorem 2-4

∠1≅∠2

1 2

∠1 and ∠2 are right angles


Given:

You can conclude:

Reason: If two angles are congruent and supplementary,then each angle is a right angle.

Theorem 2-5

m∠1≅m∠2=90

1 2∠1 is supplementary to ∠2∠1≅∠2


PRACTICE


JUSTIFY

Vertical angles are congruent.


JUSTIFY

Definition of Perpendicular Lines


JUSTIFY

Definition of Angle Bisector


JUSTIFY

Definition of Complementary Angles


JUSTIFY

Definition of Supplementary Angles


JUSTIFY

Complements of the same angle are congruent.


JUSTIFY

Definition of Perpendicular Lines


PROOFS


TWO COLUMN PROOFS

Statements Reasons1. Given Stmt.2.3.Prove Stmt.

diagram Given: StatementProve: Statement

1. Given2.3.Reasons you can use in a proof:

1. Given information2. Definitions3. Postulates4. Properties of Algebra5. Theorems


R

S

QT

P

Given:

Prove:

statements reasons

EXAMPLE:

RS=PS and ST=SQ

2.3.4.

2.3.4.


Given:

Prove:

statements reasons

EXAMPLE:

RS=PS and ST=SQ

2.3.4.

2.3.4.

R

S

QT

P


Given:

Prove:

statements reasons

EXAMPLE:

of Equality

R

S

QT

P


EXAMPLE 2:

A

B C

D

OGiven:

Prove:statements reasons

2.3.4.5.

2.3.4.5.

12

3


EXAMPLE 2:

A

B C

D

OGiven:


2.3.4.5. m<1=m<3

2.3.4.5.

12

3


EXAMPLE 2:

A

B C

D

OGiven:


"note overlapping in the given statement, will lead to reflexive"

2.3.4.5. m<1=m<3

2.3.4.5.

12

3


EXAMPLE 2:

A

B C

D

OGiven:


"note overlapping in the given statement, will lead to reflexive"

12

3


1. m∠1=m∠3; m∠2=m∠4

2. m∠1+m∠2=m∠3+m∠4

3. m∠1+m∠2=m∠ABC; m∠3+m∠4=m∠DEF

4. m∠ABC=m∠DEF

1. Given

2. Addition Property of Equality

3. Angle Addition Postulate

4. Substitution Propertyof Equality

4.

GIVEN: m∠1=m∠3; PROVE: m∠ABC=m∠DEF m∠2=m∠41)


1. ST=RN

2. _______ = SI + IT;

_______ = RU + UN

3. SI + IT = RU + UN

4. IT = RU

5. _______________

1. Given

2. Segment AdditionPostulate

3. Substitution Property of Equality

4. Given

5. Subtraction Propertyof Equality

GIVEN: ST=RN; PROVE: SI=UN IT=RU2)


GIVEN: DiagramPROVE: m∠AOD=m∠1+m∠2+m∠3

1. m∠AOD=m∠AOC+m∠3

2. m∠AOC=m∠1+m∠2

3.

1. Angle AdditionPostulate

2. Angle Addition Postulate


3)


GIVEN: DW=ON PROVE: DO=WN

1. DW=ON

2. DW=DO+OW;

ON= _____ + _____

3.

4. OW=OW

5.

1. Given

2. Segment Addition Postulate

3. Substitution Prop.

4. Reflexive Property ofEquality

5. Subtraction Property ofEquality

4)


Thoughts about Proofs1. Sometimes it may take more than one try.

Walk away, come back and try again.2. Talk it out3. Make sure you have enough steps to allow the reader

to follow your argument.4. If you find yourself stuck in a proof, try working

backwards.5. Look in notebook/flash cards for hints.6. Understand what it takes to "prove" the prove.7. Use picture and colors to help "see" what is

happening in the picture.8. Retrace steps to see what you've done so far and

see where it can lead.


GIVEN: m∠1=m∠2; m∠3=m∠4 PROVE: m∠SRT=m∠STR

5)


GIVEN: m∠1 = m∠2; m∠3 = m∠4 PROVE: m∠SRT = m∠STR

1. m<1 = m<2 m<3 = m<4

2. m<1 + m<3 =m<2+m<4

3. m<1+m<3 = <SRT m<2+m<4 = m<STR

4. m<SRT = m<STR

1. Given

2. Addition Property of Equality

3. Angle AdditonPostulate


5)


GIVEN: RQ=TP; ZQ=ZP PROVE: RZ=TZ

6)


GIVEN: RQ=TP; ZQ=ZP PROVE: RZ=TZ

1. Given

2. Segment Addition Postulate


4. Subtraction Property of Equality

1. RQ=TP ZQ=ZP

2. RZ+ZQ=RQ TZ+ZP=TP

3. RZ+ZQ=TZ+ZP

4. RZ=TZ

6)

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