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Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
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Segment & Angle Addition Proofs
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Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
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Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
HomeworkLesson 10 PRACTICE 1
Study Flash Cards!!!
1.2.
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
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
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
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.
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
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)
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
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)
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
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
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
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
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
Given:
You can conclude:
Reason: All right angles are congruent.
Theorem 2-4
∠1≅∠2
1 2
∠1 and ∠2 are right angles
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
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
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
JUSTIFY
Vertical angles are congruent.
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
JUSTIFY
Definition of Perpendicular Lines
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
JUSTIFY
Definition of Complementary Angles
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
JUSTIFY
Definition of Supplementary Angles
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
JUSTIFY
Complements of the same angle are congruent.
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
JUSTIFY
Definition of Perpendicular Lines
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
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
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
R
S
QT
P
Given:
Prove:
statements reasons
EXAMPLE:
RS=PS and ST=SQ
2.3.4.
2.3.4.
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
R
S
QT
P
Given:
Prove:
statements reasons
EXAMPLE:
RS=PS and ST=SQ
2.3.4.
2.3.4.
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
Given:
Prove:
statements reasons
EXAMPLE:
RS=PS and ST=SQ
2.3.4.
2.3.4.
R
S
QT
P
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
Given:
Prove:
statements reasons
EXAMPLE:
of Equality
R
S
QT
P
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
EXAMPLE 2:
A
B C
D
OGiven:
Prove:statements reasons
2.3.4.5.
2.3.4.5.
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3
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
EXAMPLE 2:
A
B C
D
OGiven:
Prove:statements reasons
2.3.4.5. m<1=m<3
2.3.4.5.
12
3
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
EXAMPLE 2:
A
B C
D
OGiven:
Prove:statements reasons
"note overlapping in the given statement, will lead to reflexive"
2.3.4.5. m<1=m<3
2.3.4.5.
12
3
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
EXAMPLE 2:
A
B C
D
OGiven:
Prove:statements reasons
"note overlapping in the given statement, will lead to reflexive"
12
3
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
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)
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
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)
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
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. Substitution Property of Equality
3)
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
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)
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
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.
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
GIVEN: m∠1=m∠2; m∠3=m∠4 PROVE: m∠SRT=m∠STR
5)
Lesson 10 - Segment & Angle Addition Proofs February 03, 2015
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
4. Substitution Property of Equality
5)