Theorems 4 – 18 & more definitions, too!. Page 104, Chapter Summary: Concepts and Procedures After studying this CHAPTER, you should be able to... 2.1

Chapter 2 Overview: Basic Concepts and Proofs

Theorems 4 – 18 & more definitions, too!

Page 104, Chapter Summary: Concepts and Procedures

After studying this CHAPTER, you should be able to . . .

2.1 Recognize the need for clarity and concision in proofs2.1 Understand the concept of perpendicularity

2.2 Recognize complementary and supplementary angles

2.3 Follow a five-step procedure to draw logical conclusions

2.4 Prove angles congruent by means of four new theorems

2.5 Apply the addition properties of segments and angles2.5 Apply the subtraction properties of segments and angles

2.6 Apply the multiplication and division properties of segments and angles2.7 Apply the transitive properties of angles and segments

2.7 Apply the Substitution Property

2.8 Recognize opposite rays2.8 Recognize vertical angles

2

Chapter 2, Section 1: “Perpendicularity”

COORDINATES

ORIGIN

PERPENDICULAR

X-axis

Y-axis

After studying this SECTION, you should be able to . . .

Related Vocabulary

OBLIQUE LINES

3

2.1 Recognize the need for clarity and concision in proofs

2.1 Understand the concept of perpendicularity


Related Vocabulary


4



DEFINITIONS

PERPENDICULAR – lines, rays, or segments that INTERSECT at right angles

OBLIQUE LINES – when lines, rays, or segments INTERSECT and are NOT PERPENDICULAR


Related Vocabulary


5



PERPENDICULAR

If a right angle is created at the intersection of two rays, then the rays are perpendicular!CONDITIONAL

If two rays are perpendicular, then they create a right angle!CONVERSE

RIGHT ANGLENOT PERPENDICULAR

⊬H

KO

Given: OH OK

If OH OK ,

then ∡HOK is a Rt ∡and if ∡HOK is a Rt ∡,then m∡HOK = 90

SYMBOLS:

CHAIN REASONING


Related Vocabulary


6





H

KO

Given: m∡ HOK = 90

then OH OK

then ∡HOK is a Rt ∡and if ∡HOK is a Rt ∡,If m∡HOK = 90

CHAIN REASONING

90⁰

Right Angle

s

90⁰



7





H

KO

90⁰

90⁰

90⁰

Right ∡

Right ∡Perpendicularity, right angles, and

90⁰ measurements all go together!


Related Vocabulary


8

2.1 Recognize the need for clarity and concision in proofs2.1 Understand the concept of perpendicularity

COORDINATES

ORIGIN

x-axis 1 2 3 4 5

1

2

3

4

5

-1

-2

-3

-4

-5

-1-2-3-4-5 0

y-axis

A (3, 2)B (-3, 2)C (-3, -2)

E (0, 0)F (4, 0)

G (-4, 0)

H (0, 3)

J (0, -3)

D (3, -2)

COORDINATES

COORDINATES

COORDINATES

Remember: The x-axis is to the y-axis

H

FG

J

Can you name the lines?

Can you name the ‖ lines?‖ parallel

Couldany lines drawn be

“oblique lines”?

Find the area of rectangle PACE

9

1 2 3 4 5

1

2

3

4

5

-1

-2

-3

-4

-5

-1-2-3-4-5 0

Given: AP ‖ to the y-axis CE ‖ to the y-axis

C

P

A

E

2.1 Example

4

7

AreaRECT = (length)(width)

Width = |y – y|

Width = |2 – (-2)|

Width = |2 + 2|

Width = |4|

Length = |x – x|

Length = |3 – (-4)|

Length = |3 + 4|

Length = |7|

AreaRECT = (7 units)(width)

AreaRECT = 28 units2

(4 units)

Remember an important property of rectangles is

that BOTH pairs of opposite sides are congruent, and:

If two segments are congruent, then they

have the SAME measure!


COMPLEMENTCOMPLEMENTARY ANGLES

SUPPLEMENT

SUPPLEMENTARY ANGLES

Related Vocabulary

Chapter 2, Section 2: “Complementary and Supplementary Angles”

10


(NOT the same as: “You look very nice today!”)

(NOT THE SAME AS: “Did you take your vitamins today!”)


Related Vocabulary


11


COMPLEMENTCOMPLEMENTARY ANGLES

- the NAME given to each of the two angles whose sum equals 90⁰

- two angles whose sum equals a 90⁰ right angle

15⁰

75⁰

30⁰

60⁰57

41’20

”

⁰

3218

’40”

⁰

V

N V

NA A

V

N

QUESTION!

If two angles are COMPLEMENTARY

ANGLES,(then) are they also ADJACENT

ANGLES?


Related Vocabulary


12


SUPPLEMENTSUPPLEMENTARY ANGLES

- the NAME given to each of the two angles whose sum equals 180⁰

- two angles whose sum equals a 180⁰ straight angle

85⁰

95⁰

130⁰

50⁰67 41’20”⁰112 18’40”⁰

T

R

T AR

A

T

R

P

QUESTION!

If two angles are SUPPLEMENTARY

ANGLES,(then) are they also ADJACENT

ANGLES?


Related Vocabulary


x

2x + 15

x + 2x + 15 = 90

13


The measure of one of two complementary angles is 15 more than twice the other. Find the measure of each angle.

Write equation

THINK –

If two angles are complementary

angles,then their sum

equals 90! Simplify 3x + 15 =

90 Solve for x 3x = 75

x = 25 Substitute

25⁰

50 + 1575⁰

Is the answer reasonable?Is one of the

angles 15 more than twice the

other?

YES!

14

If a problem contains ONLY complements or ONLY supplements, use the previous method.

Begin by drawing a right angle for two complementary angles or a straight angle to model two supplementary angles,

and label them according to the information given in the problem!

HOWEVER, if a problem refers to BOTH the complement AND the supplement

in the same problem ,

use the NEXT method:

15

Chapter 2, Section 2: “Complementary and Supplementary Angles”After studying this SECTION, you should be able to . . .


Use the “Boxer” Method to write expressions for each type of angle:

Are you wondering, “what is the “Boxer Method”?”

Well, first make a “BOX,” and then let “the angle” equal x

THE ANGLE

COMPLEMENT

SUPPLEMENT

x⁰

(90 – x)⁰

(180 – x)⁰

x⁰

x⁰30⁰

30⁰

30⁰

60⁰

150⁰

60⁰

150⁰

Complements

Supplements

16

Chapter 2, Section 2: “Complementary and Supplementary Angles”Example 2.2 Recognize complementary and supplementary

anglesThe measure of the supplement of an angle is 60 less than 3 times the

complement of the angle.

Find the measure of the complement.The measure of the supplement of an angle is 60 less than 3 times the complement

ANGLE

COMP

SUPP

x

90 – x

180 – x Complement

Supplement

“the angle”x

90 – x

180 – x

(180 – x) = 3(90 – x) - 60

180 – x = 270 -3x -60 180 + 2x = 210

2x = 30

x = 15

15⁰

75⁰

165⁰

√

15

15

15180 – 15

90 – 15

x

x

75⁰

Related Vocabulary


Chapter 2, Section 3: “Drawing Conclusions”

No NEW vocabulary!

17


See very important TABLE on page 72!



NOTE: The “If . . .” part of the reason should match the GIVEN information!

5-STEP Procedure for Drawing Conclusions:

• 1. MEMORIZE theorems, definitions, and postulates

• 2. Look for KEY WORDS and SYMBOLS in the “givens”

• 3. Think of all the theorems, definitions, and postulates that involve those keys.

• 4. Decide which theorem, definition, or postulate allows you to draw a conclusion

• 5. DRAW A CONCLUSION, and give a reason to justify it.

18


AND the “then . . .” part matches the CONCLUSION being justified!CAUTION! Be sure not to reverse that order!!!



1) If B bisects AC, then ____?______

3) If ∡ABC ≅ ∡CBD ≅ ∡DBE, then ____?____.

2) If AB AC, then _____?_______.

D

E

BA C

19

PRACTICE EXAMPLES

B

B

A

AC

C

then . . . AB ≅ BC

then ∡BAC is a Rt ∡

then . . . BC and BD trisect ∡ABE

Key info: a point, bisect, and seg

Key info: ,, and

Key info: ∡ ≅ ∡ ≅ ∡



1) If B bisects AC, then ____?______

3) If ∡ABC ≅ ∡CBD ≅ ∡DBE,

then ____?____.

2) If AB AC, then _____?_______.

BA C

20

JUSTIFY your CONCLUSIONS!

B

A C

then . . . AB ≅ BC

then ∡BAC is a Rt ∡

then . . . BC and BD trisect ∡ABE

D

EB

AC

REASON: If a seg is bisected by a point, then the seg is divided into two congruent segs

REASON: If two rays are perpendicular, then they form a right angle

REASON: If an angle has been divided into 3 congruent angles,

then it was trisected by two rays.

Related Vocabulary


THEOREM #5

THEOREM #4

THEOREM #6

THEOREM #7

21


Chapter 2, Section 4: “Congruent Supplements and Complements”

No NEW vocabulary!BUT . . .

THEOREM #4


If angles are supplementary to the same angle,

then they are congruent

22



120⁰ 1G

2

∡1 is supplementary to ∡G

∡2 is also supplementary to ∡G What can we conclude about ∡1 and ∡2?

60⁰ 60⁰

=


If angles are supplementary to congruent angles,


23



G

∡G is supplementary to ∡E

∡O is supplementary to ∡M

What can we conclude about ∡G and ∡M?

∡E ≅ ∡OM

EO

50⁰ 50⁰130⁰

130⁰

THEOREM #5


If angles are complementary to congruent angles,


24



If angles are complementary to the same angle,

then they are congruentWhat can we conclude?

What can we conclude? THEOREM #7

THEOREM #6

The only difference is the sum! (90 versus 180)

Complete a Proof!


Given:

PROVE:


∡1 is comp to ∡4

25

R

S

V

∡2 is comp to ∡3RT bisects ∡SRVTR bisects ∡STV

1

2

3

4T

?

?

4) ∡3 ≅ ∡41) ∡1 is comp to ∡42) ∡2 is comp to ∡33) RT bisects ∡SRV

1) Given2) Given3) Given4) If a ray bis an ∡, it div it into 2 ≅ ∡s 5) ∡1 ≅ ∡2 5) If ∡’s comp ≅ ∡s, then they are ≅

Statements Reasons

6) TR bisects ∡STV 6) If an ∡ is div into 2 ≅ ∡s, then it was bisected by a ray!

2.5 Apply the addition properties of segments and angles

Related Vocabulary


Chapter 2, Section 5: “Addition and Subtraction Properties”

26

2.5 Apply the subtraction properties of segments and angles

7cm3cm7 cmA B C D

AC = BD, because

AC BDAB + BC = BC + CD,

If two segments have the same measure, they are congruent!

(7) + (7)(3) = (3) +(Commutative Property of Addition!)

If a segment is added to two congruent segments, the sums are

congruent. (Addition Property)

Note that we first need to know that two segments are congruent, and then that we

are adding the SAME segment to both of them.


Related Vocabulary



27


If two angles have the same measure, they are congruent!

(Commutative Property of Addition!)

mDBE = 50.03

mABC = 50.03A

B

E

C

D

ABD CBEm

∡ABC

= 5

0⁰

m∡DBE = 50⁰

50 + ∡CBD = ∡CBD + 50

m∡ABC + m∡CBD = m∡CBD + m∡DBE

m∡ABD = m∡CBE, so

If an angle is added to two congruent angles,

then the sums are congruent. (Addition Property)

Note that we first need to know that two angles are congruent, and then that we are

adding the SAME angle to both of them.

ÐABC @ ÐDBE


Related Vocabulary



28


If two angles have the same measure, they are congruent!

mDBE = 50.03

mABC = 50.03A

B

E

C

D

m∡ABD = 80⁰

m∡CBE = 80⁰

80 - ∡CBD = ∡CBD

80 -

m∡ABD - m∡CBD = m∡CBE - m∡CBD

m∡ABC = m∡DBE, so

If an angle is subtracted from two congruent angles, the differences

are congruent. (Subtraction Property)

Note that we first need to know that two angles are congruent, and then that we are

subtracting the SAME angle from both of them.

D E

GC

H

F




CF + FG = DE + EH

CG = DH, so

CG DH≅

If congruent segments are added to congruent segments, the sums are

congruent. (Addition Property)

Note that first we need 2 congruent segments, then we need 2 different

congruent segments to ADD.

30

J

I K

L

JIK JKI




m∡JIL + m∡LIK = m∡LKI + m∡JKLIf congruent angles are added to

congruent angles, the sums are congruent.

(Addition Property)

Note that first we need 2 congruent angles, then we need to add two

different congruent angles

10

10

Q RB AQR - BR = BA - BR




QB RA≅

If a segment (or angle) is subtracted from congruent segments (or angles),

the differences are congruent. (Subtraction Property)

Note that we need to start with congruent angles or segments and

then subtract the same angle or segment from both.

If a segment (or angle) is subtracted from congruent segments (or angles), the

differences are congruent. (Subtraction Property)

Note that we need to start with congruent angles or segments and then subtract the same angle or

segment from both. mABD - mCBD = mCBE - mCBDmABD = 78

mCBE = 78D

C

E

B

A

ABC DBE




mSTV - mWTV = mUVT - mWVT

W mSTV = mUVT = 130mWTV =mWVT = 30T V

US

2.5 Apply the subtraction properties of segments and angles2.5 Apply the addition properties of segments and angles



∡STW UVW≅ ∡

If congruent segments (or angles) are subtracted from congruent segments (or angles), the differences are congruent.

(Subtraction Property)Note that we start with congruent segments or

angles, and then subtract congruent segments or angles.

Using the Addition and Subtraction Properties

An addition property is used when the segments or angles in the conclusion are greater than those in the given information

A subtraction property is used when the segments or angles in the conclusion are smaller than those in the given information.

Theorem: If a segment is added to two congruent segments, the sums are congruent. (Addition Property)

Given:

Conclusion: PR QS

RQP S

5. If two segments have the same measure then they are congruent

5.

4. Addition of Segments4. PR = QS

3. Additive Property of Equality3. PQ + QR = RS + QR

2. If two segments are congruent, then they have the same measure

2. PQ = RS

1. Given1.

ReasonsStatements

PQ RS

PQ @ RS

PR @ QS

Given:

Conclusion:

GJ HK

GH JK G K

M

H J

Statements Reasons

1.

2.

GJ HK

GH JK

How to use this theorem in a proof:

2. If a segment is subtracted from congruent segments, then the resulting segments are congruent. (Subtraction)

1. Given

? ?

Multiplication Property

If segments (or angles) are congruent, then their like multiples are congruent.

Example: If B, C, F, and G are trisection points and

then by the Multiplication

Property.

A B C D E F G H

,EFAB

EHAD

Division Property

If segments (or angles) are congruent, then their like divisions are congruent.

C

SA

T

D

ZO

G

If ∡CAT ≅ ∡DOG, andthen, ∡CAS ≅ ∡DOZ by the division property

AS and OZ are angle bisectors

Using the Multiplication and Division Properties in Proofs

Look for the DOUBLE USE of the words midpoint, trisects, or bisects in the “Givens.”

Use MULTIPLICATION if what is Given is less than the Conclusion

Use DIVISION if what is Given is greater than the Conclusion

Example

Given: O is the midpoint of

R is the midpoint of

Prove: Statements Reasons

NSMPMPNS

NRMO

M O P

N R S

1. MP NS≅2. O is mdpt of MP3. MO OP≅4. R is mdpt of NS5. NR RS≅5. MO NR≅

1. Given2. Given3. A mdpt divides a seg into 2 segs≅4. Given4. Same as #35. If segs are , then their like divisions are ≅ ≅

(DIVISION PROPERTY)

2.7 Apply the Transitive Property of angles and segments


Related Vocabulary


SUBSTITUTE

SUBSTITUTION

Chapter 2, Section 7: “Transitive and Substitution Properties”

41

Theorems

Theorem 16

Theorem 17


THEOREM:

CONCLUSION?

AB ≅ BC

42




BC ≅ CD

BA

C D

AB ≅ CD

If segments are congruent to the SAME segment,

then they are congruent to each other.


THEOREM:

CONCLUSION?

If angles are congruent to the SAME angle,


∡1 ≅ ∡2

43




∡2 ≅ ∡3 2

1

3

∡1 ≅ ∡3


THEOREM:

CONCLUSION?

AB ≅ NM

44




QR ≅ MPBA Q

PAB ≅ CD

If segments are congruent to congruent segments,


N M

R

NM ≅ MP


THEOREM:

CONCLUSION?

If angles are congruent to congruent angles,


∡7 ≅ ∡5

45




∡6 ≅ ∡8 57

6

∡7 ≅ ∡8∡5 ≅ ∡68

46

Chapter 2, Section 7: “Transitive and Substitution Properties”After studying this SECTION, you should be able to . . .



12 3

Given:∡1 comps ∡2∡2 ≅ ∡3 m∡1 + m∡2 = 90m∡2 ≅ m∡3∴ m∡1 + m∡3 = 90By Substitution Property!

2.8 Recognize opposite rays

2.8 Recognize Vertical Angles

Related Vocabulary


Vertical Angles-

THEOREM 18

Chapter 2, Section 8: “Vertical Angles”

Opposite Rays - (definition) – collinear rays that share a common endpoint

(definition) – two angles whose sides are formed by opposite rays.

Vertical angles are CONGURENT!

47

and extend in opposite directions



Related Vocabulary



Opposite Rays - (definition) – collinear rays that share a common endpoint

48

and extend in opposite directions

Name the opposite rays:

1)A B C

2) 3)

F

HD

GE

KI

LH

J

BA

and

BC

EH

and

EG

ED

and

EF

IL

and

IJ



Related Vocabulary



49

4) Which numbered angle is vertical with ∡1?5) Which numbered angle is vertical with ∡4?6) If m∡1 = 65, find the measure of the numbered angles. F

HD

G

Vertical Angles-


1 2

34

∡3∡2 65

°65°

115°115°



Related Vocabulary



50

7) If m∡3 = 55, which other numbered angle must be 55°?

Vertical Angles-


∡6∡4 55

°

7) If m∡1 = 40, which other numbered angle must be 40°?

1

65

4

3240

°

51

A

F

E

C G H B

D

Conclusion: CE ≅ DB

E and D are the midpoints of AC and AB,

and AC ≅ AB

1. Given:

Because? AdditionSubtractionMultiplicationDivisionTransitiveSubstitution

Like divisions of ≅ segs are ≅

Self-Check Properties Quiz Questions

52

A

F

E

C G H B

D

Conclusion: CD ≅ EB

FE FD, and≅

FC ≅ FB

2. Given:


≅SEGS added to ≅SEGS are ≅ SEGS


53

A

F

E

C G H B

D

Conclusion: ACD ABE∡ ≅ ∡

CD bisects ACB,∡

BE bisects ABC, ∡

3. Given:

Because?AdditionSubtractionMultiplicationDivisionTransitiveSubstitution

Like divisions of ≅ ∡s are ≅

and ACB ABC ∡ ≅


54

A

F

E

C G H B

D

Conclusion: CG BH≅

CG GH,≅

BH GH, ≅

4. Given:


If segs are ≅ to SAME seg, then ≅ to each other


55

A

F

E

C G H B

D

Conclusion: ACD ABE∡ ≅ ∡

∡BCD CBE ,≅ ∡

5. Given:


If ≅ ∡s are subtracted from ≅ ∡s,then the like diffs are ≅

and ACB ABC ∡ ≅


56

A

F

E

C G H B

D

Conclusion: FD + FB = EB

EF = FD, and

EF + FB = EB

6. Given:


One seg measure can be substituted for the other in the EQUATION!

= !


57

A

F

E

C G H B

D

Conclusion: CG ≅ BH

CH BG≅

7. Given:


If the SAME seg is subtracted from ≅SEGS , the like diffs are ≅


58

A

F

E

C G H B

D

Conclusion: 2( ABC) + CAB = ∡ ∡180°

∡CAB + ACB + ABC = ∡ ∡180°

8. Given:


and ACB ABC ∡ ≅

One ANGLE measure can be substituted for the other in the EQUATION!


59

A

F

E

C G H B

D

Conclusion: ACB ABC∡ ≅ ∡

CD bisects ACB,∡

BE bisects ABC, ∡

9. Given:


Like multiples of ≅ ∡s are ≅

and ACD ABE ∡ ≅


60

A

F

E

C G H B

D

Conclusion: AFC AFB∡ ≅ ∡

∡AFD AFE ,≅ ∡

10. Given:


If ≅ ∡s are added to ≅ ∡s,then the like sums are ≅

and DFB EFC ∡ ≅


Documents

Theorems 4 – 18 & more definitions, too!. Page 104, Chapter Summary: Concepts and Procedures After studying this CHAPTER, you should be able to... 2.1