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1

Unit 4

Algebraic and

Geometric Proof

Math 2

Spring 2017

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2

Table of Contents Introduction to Algebraic and Geometric Proof ................................................................ 3

Properties of Equality for Real Numbers .................................................................. 3

Algebra Proofs ......................................................................................................... 4

Introduction to Lines & Angle Proofs ............................................................................... 6

Types of Angle Pairs ................................................................................................ 7

Practice and Closure ................................................................................................ 8

Parallel Lines and Transversals with Algebra ................................................................ 10

Practice and Closure .............................................................................................. 12

Adding and Subtracting Line Segments and Angles ..................................................... 15


Formal Flow Proof ......................................................................................................... 22

Flow Proofs Continued .................................................................................................. 25


Math 2 Unit 4 Review Sheet .......................................................................................... 29

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3

Introduction to Algebraic and Geometric Proof

Properties of Equality for Real Numbers Using the word bank, write each property next to its corresponding definition.

Word Bank:

Transitive Property Reflexive Property Symmetric Property Distributive Property

Substitution Property (Simplifying Property)

Addition/Subtraction Properties Multiplication/Division Properties

If two things are equal, then you can add/subtract the

same thing on both sides of the equal sign (e.g., if a = b,

then a + c = b + c and a – c = b – c).

If two things are equal, then you can multiply/divide the

same thing on both sides of the equal sign (e.g.,

if a = b, then cb=ca and c

b

c

a .

If a = b, then a may be replaced by b in any equation or

expression.

a (b + c) = ab + ac.

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

Everything is equal to itself (e.g., 3 = 3).

If two things are equal, you can write that equality “either

way” (e.g. if a = b, then b = a).

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4

Algebra Proofs

Proof – a set of statements (and reasons) that lead to a logical conclusion.

Algebraic Proof – an algebra equation that is solved using the “two-column proof” format.

Ex. 1) Given: 1532x =+

Prove: 6=x

Statements Reasons .

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

Ex 2) Given: 3x835x2 +=

Prove: 2=x


1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

7. 7.

8. 8.

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5

Ex.3. Given: 483x167x +=+

Prove: 8=x


1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

7. 7.

Ex.4. Given: 93

132x

=

Prove: 7=x


1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

7. 7.

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6

Introduction to Lines & Angle Proofs

Linear Pair Angles – Angles that are ______________ and ________________.

Vertical Angles – Two _____________________ formed by a pair of intersecting lines.

Vertical Angles Conjecture -

Vertical angles are ________________.

Intersecting Lines Conjecture -

Congruent Angles Supplementary Angles

Lines and Transversals

Interior –

Exterior –

Transversal –

Interior Angles Exterior Angles

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Types of Angle Pairs

Corresponding Angles – ___________________________________

___________________________________

___________________________________

Same-Side Interior Angles – aka ( )

___________________________________

___________________________________

Same-Side Exterior Angles – aka ( )

___________________________________

__________________________________

Alternate Interior Angles – ___________________________________

___________________________________

___________________________________

Alternate Exterior Angles – __________________________________

__________________________________

__________________________________

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Practice and Closure

Using the diagram to the right, identify all pairs of angles which match the given angle pair name.

1. Same-Side Interior ________________________

2. Alternate Exterior _________________________

3. Alternate Interior ________________________

4. Same-Side Exterior _______________________

5. Corresponding ___________________________

6. Vertical ________________________________

7. Linear Pair __________________________________________________________

Name each angle pair and the transversal used. (Note: Do not use “congruent” or “supplementary”

as angle pair names.)

15. 6 and 14

Angle pair name: _____________________

Transversal: ________________

16. 4 and 7

Angle pair name: _____________________

Transversal: ________________

17. 3 and 10

Angle pair name: _____________________

Transversal: ________________

1 2

3 4 5 6

7 8

9 10

11 12

13 14

15 16

l

m

r t

1 5

\ 2 6

3 7

4 8

l

m

t

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9

For problems 8 – 14, write the Special Name for each angle pair. (Note: Do not use “congruent” or

“supplementary” as angle pair names.)

8. 1 and 9 _______________________

9. 3 and 10 _______________________

10. 7 and 13 _______________________

11. 6 and 16 _______________________

12. 11 and 14 _______________________

13. 2 and 3 ________________________

14. 7 and 8 ________________________

Name each angle pair and the transversal used. (Note: Do not use “congruent” or “supplementary”

as angle pair names.)

18. 11 and 14

Angle pair name: _____________________

Transversal: ________________

19. 13 and 16

Angle pair name: _____________________

Transversal: ________________

20. 3 and 8

Angle pair name: _____________________

Transversal: ________________

21. 2 and 4

Angle pair name: _____________________

Transversal: ________________

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Parallel Lines and Transversals with Algebra

Steps To Solve for x: 1. Determine the types of angle pair or pairs

a. Corresponding, Atl. Int., Alt. Ext., Same-Side Interior, Vertical, Linear Pair 2. Set-up the problem using the appropriate algebraic relationship

a. Corresponding, Atl. Int., Alt. Ext., , Vertical are congruent b. Same-Side Interior, Linear Pair are supplementary (add to 180°)

3. Solve for x 4. Check your answer

Example 1: Solve for x ___ Example 2: Example 3:

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Example 4: Example 5:

You Try: Solve the following example for x

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For problems 1 – 7, identify the special name for each angle pair.

1. 1 and 4 _______________________

2. 13 and 10 _______________________

3. 5 and 13 _______________________

4. 12 and 16 _______________________

5. 11 and 14 _______________________

6. 2 and 7 ________________________

7. 7 and 8 ________________________

For problems 8 – 13, the figure shows l || m . Find the measures of each angle and list the

angle pair name. Treat each problem independently.

8. If m1 = 120°, find m5 = ________

pair name: __________________

9. If m6 = 72°, find m4 = ________

pair name: __________________

10. If m2 = 64°, find m8 = ________

pair name: __________________

11. If m4 = 112°, find m5 = _______

pair name: __________________

12. If m2 = 82°, find m7 = ________

pair name: __________________

13. If m2 = 80°, find m5 = ________

pair name: __________________

1 2

3 4

5 6

7 8

l

m

t

1 2

3 4 5 6

7 8

9 10

11 12

13 14

15 16

l

m

r t

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For problems 14 – 15, the figures show p || q .

14. m1 = 3x – 15 and m2 = 2x + 7, find x and m 1.

x = ______

m1 = ______

15. m3 = 7x – 12 and m4 = 12x + 2, find x and m 4.

x = ______

m4= ______

For problems 13 – 14, find the values of x, y and z in each figure.

16. 17.

q

t

1

2

p

q

t

3

4

p

x (3z + 18)

3y

72°

(y+12)

(y-18)

z x

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Class work

For problems 1 – 7, the figure at the right shows p ||q, m1 = 78° and m2 = 47°. Find the

measures of the following angles.

1. m1 = 78o

2. m2 = 47 o

3. m3 = _____

4. m4 = _____

5. m5 = _____

6. m6 = _____

7. m7 = _____

8. m8 = _____

9. m9 = _____

For problems 10 – 12, find the values of x and y in each figure.

10. 11.

12.

1

2

3

4 5

6

7 8 9

p

q

(6x-14)

(3x+5)

(y+8)

5x 9y

(8x + 40)

(3y – 10)

7y

6x

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Adding and Subtracting Line Segments and Angles Everybody knows you can add and subtract numbers: 7 + 3 = 10 makes perfect sense. However,

adding and subtracting objects is different. It is nonsense to say that an apple + banana = banapple

Line segments are somewhere in between. In general, you can’t add or subtract just any two

random line segments and get another segment. But sometimes it makes sense. Your job is to

understand when.

IMPORTANT:

1) AB BC AC only makes sense when A, B, and C are collinear and B is between A and C. In

other words, to add segments, they must be collinear and the second one must start where the

first one ends.

AB BC AC AB BC nonsense AB CD nonsense

AC BD nonsense

2) AC BC AB and AC AB BC only make sense when A, B, and C are collinear and B is

between A and C. In other words, to subtract segments, the one being subtracted must be

part of the one being subtracted from and they must share an endpoint.

AC BC AB AC BC nonsense AD BC nonsense

AC AB BC AC BC nonsense AC BD nonsense

6. Based on the diagram at right, tell if each of the following is True or

False. Remember the difference between AB and AB.

a. AB + BC = CP b. AB BC CP

c. AB + BC = AC d. AB BC AC

e. AC BC = AB f. AC BC AB

g. PC PB = CD h. PC PB CD

7. In the diagram at right, FLAG . For each of the following, either fill in the appropriate

line segment or write “nonsense.”

a. LA AG ______ b. FL LP ______ c. FA LG ______

d. FL AG ______ e. FL LG ______ f. FL LA AG ______

g. FP FL ______ h. FA LA ______ i. FA LA ______

j. FP FL ______. k. FG FL ______ l. FG LA ______

A

F

L

P

G

A B C A B

C

A B C D

A B C A B

C

A B C D

A

P

B C D 1 3

5 4

2

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1. Use the diagram at right to answer the following.

a. How many angles in the diagram have their vertex at A?

b. How many angles in the diagram have their vertex at B?

c. What angle (number) is named BDC?

d. Name two adjacent angles in the diagram.

e. Are ADC and BDC adjacent?

f. Give three alternate names for 4.

g. Explain why we should not refer to D in the diagram. (Yes, you may lose points for sloppy

notation on quizzes and tests.)

h. Name one acute angle on the diagram.

i. Name one obtuse angle on the diagram.

j. Which angle on the diagram appears to be closest to a right angle?

2. In the diagram at right, which angle has a larger measure,

PAQ or RAS?

3. In the diagram at right, NOP , OR OQ , and m POQ = 40. Find m NOR.

4. The measures of two supplementary angles are in the ratio 5:7.

Find the measure of the smaller angle.

5. The measure of the complement of an angle is 18 less than twice the measure of the angle.

What is the numerical measure of the angle?

6. If ET bisects BEG, mBET = x2 and mGET = 5x + 14, find the numerical measure of BEG.

7. If OY bisects BOT, mBOY = 3x + 8 and mBOT= 8x – 2, find the numerical measure of

TOY.

D

A

C

B 1 2

3

4 5 6

N

R P

Q O

A Q

S

P

R

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READ: : Numbers can always be added and subtracted. It makes no sense to add or subtract

people. Line segments can sometimes be added or. Angles are like segments. They can

sometimes be added and subtracted. Remember, ABC represents an actual angle (a geometric

object); mABC is a number that represents the degree measure of ABC.

1) Adding two angles only makes sense if they are adjacent: they share a vertex and one side but

have no interior points in common (one is not “inside” the other).

APB +BPC = APC APB +BPC = nonsense APB +CPD = nonsense

APC +BPD = nonsense

2) Subtracting two angles only makes sense if they share a vertex and one side and the second

side of the smaller angle is on the interior of the larger angle (the smaller angle is part of the

larger angle).

APC BPC = APB BPC APC = nonsense APC BPD = nonsense

APC APB = BPC APD BPC = nonsense

8. Based on the diagram at right, tell if each of the following is True or False. Remember the

difference between A and mA.

a. mCAD + mABC = mBCA

b. CAD + ABC = BCA

c. mCAD + mDAB = mCAB

d. CAD + DAB = CAB

e. mDBA mDAC = mBAD

f. DBA DAC = BAD

g. mBAC mBAD = mDAC

h. BAC BAD = DAC

9. Use the diagram at right to fill in an appropriate angle for each of the

following or write “nonsense.”

a. NAG + LAG = ________ b. SEG + AEL = ________

c. ANS + NSE = ________ d. LGS – EGS = ________

e. NSE – ESG = ________ f. ALG – ALE = ________

g. LGS + EGS = ________ h. LSN – LEA = ________

G

A L

E

N S

A

B

15

D

C

40

55

70

C

A

P B

C

A

P B

B

C

P A

B

C

P A

B

A

P

C

D

B

A

P

C

D

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You Try:

In problems #1 - 5, for each given, state a valid conclusion and a reason based on the definitions we

have covered. (Note: some of these have more than one correct answer.)

1. Given: AB CD

Conclusion:

Reason:

2. Given: X is the midpoint of PQ .

Conclusion:

Reason:

3. Given: BD bisects ABC.

Conclusion:

Reason:

4. Given: BD bisects AC at E.

Conclusion:

Reason:

5. Given: AB AC

Conclusion:

Reason:

6. In the diagram at right, BD bisects ABC, mABD = 66 – 2x and

mCBD = 3x – 24. Find the numerical value (a number, not just an

algebraic expression) of mABC.

C B

A

C

E

B

D A

A

D C

B

P X Q . . .

A

D

C

B

A

B

D

3x –

24 C

66 –

2x

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1. If the angles of ABC have the following measures: mA = 3x + 2, mB = 5x – 3, mC = 6x – 1, list the sides of ABC from Longest to Shortest.

For problems 14 – 22, find the values of the given variables in each of the figures below. 14] 15] 16] 17] 18] 19] 20] 21] 22]

x

108

142 x 33

53

x 30

40

x

42

105

x

x

26

19

44 73

x x x

72°

x

5x-32

8x+4

6y

42˚ 7x-8 4x+19

2y

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20

For #1 - 4, name the postulate that justifies the conclusion.

1. Given: FT AT , AT RT

Conclusion: FT RT

Reason:

2. Given: (Diagram at right)

Conclusion: mDBE = m4 + m2 + m5

Reason:

3. Given: (Diagram at right)

Conclusion: AT AT

Reason:

4. Given: m1 + m2 = 180°, m2 = m3 (Diagram at right)

Conclusion: m1 + m3 = 180

Reason:

5. Given: m1 + m2 = 180; m3 = m1.

Conclusion:

Reason:

6. Given: QAbisects UAD.

Conclusion:

Reason:

7. Given: mAOB = 90.

Statement: mAOB = mAOX + mXOB

Conclusion:

Reason:

F T

R

A

1

3

2

1

5 2

3

4

A

D B E

C

3 2

1

Q

D

A

U

O

B

X

A

F T

R

A

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Class work

If two line segments are added or subtracted, the result is another line segment. (See diagram

below.)

Ex: a. AC CD AD b. AC AB BC

c. AB CD nothing (why?) d. BC AB nothing (why?)

e. AC BD nothing (why?) f. BD AC nothing (why?)

g. AC CE nothing (why?)

If two angles are added or subtracted, the result is another angle. (Same diagram.)

Ex: a. FCE + ECD =FCD b. ABF + DCF = nothing (why?)

c. BCE – FCE =BCF d. ABF – FBC = nothing (why?)

1. Use the diagram at right to answer the following:

a. BP PC b. AS SD .

c. AS RD d. AQ QD .

e. BD BQ f. AD AS .

g. AD SR h. AR RD .

2. Use the same diagram to answer the following:

a. ABD + DBC = .

b. AQR + DQR = .

c. RDQ + RSQ = .

d. BQC – BQP = .

e. CQS – CQD = .

f. DCQ – PCQ = .

3. If M is the midpoint of AY , AM = x + 8 and AY = 3x2, find the numerical length of AY .

4. If HOT is the perpendicular bisector of DOG , HO = 2x + 1, OT = 3x – 2,

DO = 4x – 5, and OG = 2x + 3, find the numerical length of HOT .

A

B C

D

P

R

Q

S

A

B C

D

P

R

Q

S

.

F

A

E

C D

B

ABCD

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Formal Flow Proof

Today we will begin formal flow proofs. Just like a graphic organizer, flow proofs can be used to

organize your work into a picture that is easy to read and understand.

_____________________________________________________________________________

1) Given: 𝑊𝑇̅̅ ̅̅ ̅ ≅ 𝑆𝑈̅̅̅̅ ; ∡𝑊 ≅ ∡𝑈; 𝑉𝑊̅̅ ̅̅ ̅ ≅ 𝑂𝑈̅̅ ̅̅

Prove: ∆𝑂𝑆𝑈 ≅ ∆𝑉𝑇𝑊

_____________________________________________________________________________

2) Given: ∡𝐴 ≅ ∡𝑆; ∡𝑈 ≅ ∡𝐵; 𝐴𝐵̅̅ ̅̅ ≅ 𝑆𝑈̅̅̅̅

Prove: ∆𝑀𝑆𝑈 ≅ ∆𝐶𝐴𝐵

O

U S

T W

V

A C

B

M S

U

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23

3) Given: ∡1 ≅ ∡2; 𝐿𝑂̅̅̅̅ ≅ 𝑂𝐸̅̅ ̅̅

Prove: ∆𝐿𝑂𝑉 ≅ ∆𝐸𝑂𝑉

_____________________________________________________________________________

4) Given: ∡1 ≅ ∡2; ∡3 ≅ ∡4

Prove: ∆𝐸𝐶𝑈 ≅ ∆𝑈𝑆𝐸

_____________________________________________________________________________

5) Given: ∡𝐴 ≅ ∡𝐶; 𝑀𝑇̅̅̅̅̅ ≅ 𝑇𝐻̅̅ ̅̅

Prove: ∆𝑀𝐴𝑇 ≅ ∆𝐻𝐶𝑇

L

O

V E

2 1

C E

U S

C

M A

T

H

1 3

2 4

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24

6) Given: 𝐶𝑅̅̅ ̅̅ ≅ 𝑅𝑊̅̅ ̅̅ ̅; 𝐸 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐷𝑊̅̅ ̅̅ ̅

Prove: ∆𝑅𝐸𝐶 ≅ ∆𝑅𝐸𝑊

_____________________________________________________________________________

7) Given: ∡𝐵 ≅ ∡𝑆; 𝐶 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐵𝑆̅̅̅̅

Prove: ∆𝐵𝑈𝐶 ≅ ∆𝑆𝐾𝐶

C E

R

W

K

B U

C

S

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25

Flow Proofs Continued

8) Given: 𝐵𝐶̅̅ ̅̅ ≅ 𝐷𝑂̅̅ ̅̅ ; ∡𝐷 ≅ ∡𝐶; ∡𝐵 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒; ; ∡𝑂 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒

Prove: ∆𝐷𝑂𝐺 ≅ ∆𝐶𝐵𝐴

_____________________________________________________________________________

9) Given: 𝐽𝑈̅̅ ̅ ≅ 𝑇𝑁̅̅ ̅̅ ; 𝐽�̅� ≅ 𝑇𝐼̅̅̅; ∡𝐽 ≅ ∡𝑇

Prove: ; ∡𝑈 ≅ ∡𝑁

A

C B D

G

O

S

T

N

J

I U

26

26

10) Given: ∡𝐽 ≅ ∡𝑆; 𝐽𝑁̅̅̅̅ ≅ 𝑁𝑆̅̅ ̅̅

Prove: ∡𝑂 ≅ ∡𝐴

_____________________________________________________________________________

11) Given: 𝐽𝑁̅̅̅̅ ≅ 𝑁𝑆̅̅ ̅̅ ; 𝐵𝐶̅̅ ̅̅ 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∡𝐴𝐵𝐷

Prove: ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐵𝐶

A J

O S

N

A

B

D

C

27

27

12) Given: 𝐴𝐵̅̅ ̅̅ ≅ 𝐵𝐶̅̅ ̅̅ ; 𝐴𝑂̅̅ ̅̅ ≅ 𝑈𝐶̅̅ ̅̅

Prove: ∡1 ≅ ∡2

_____________________________________________________________________________

13) Given: 𝐴𝐵̅̅ ̅̅ ≅ 𝐵𝐶̅̅ ̅̅ ; 𝐴𝑂̅̅ ̅̅ ≅ 𝑈𝐶̅̅ ̅̅

Prove: ∡1 ≅ ∡2

A

B

O C

U

2 1

E

S

O

M

H

2

1

28

28


1) Given: 𝑂𝑆̅̅̅̅ ⊥ 𝑆𝑈̅̅̅̅ ; 𝐴𝐵̅̅ ̅̅ ⊥ 𝐵𝐶̅̅ ̅̅ ; 𝐵𝐶̅̅ ̅̅ ≅ 𝑈𝑆̅̅ ̅̅ ; ∡𝐶 ≅ ∡𝑈

Prove: ∡𝐴 ≅ ∡𝑂

2) Given: ∡𝑇 ≅ ∡𝑅; 𝐴 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝑆𝐸̅̅̅̅

Prove: 𝑅𝐸̅̅ ̅̅ ≅ 𝑆𝑇̅̅̅̅

3) Given: 𝑂𝐴̅̅ ̅̅ 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∡𝐺𝑂𝐿; 𝐺𝑂̅̅ ̅̅ ≅ 𝑂𝐿̅̅̅̅

Prove: ∡𝐺 ≅ ∡𝐿

A

C B U

O

S

R

T

A

E

S

L

O

V E

29

29

B

A C

D

E

FH

I

G

Math 2 Unit 4 Review Sheet

Solve the given proof. Show all possible steps.

1. Given: ∠1 ≅ ∠2

AD̅̅ ̅̅ ≅ DC̅̅ ̅̅

Prove: ΔABD≅ΔCBD

2. Given: G is the midpoint of FH

𝐸𝐹̅̅ ̅̅ ≅ 𝐿𝐻̅̅ ̅̅

𝐸𝐺̅̅ ̅̅ ≅ 𝐿𝐺̅̅̅̅

Prove: ∠E≅∠L

3. Given: 𝐶𝐷̅̅ ̅̅ bisects ∠𝐴𝐶𝐵

∠𝐴 ≅ ∠𝐵

Prove: 𝐴𝐷̅̅ ̅̅ ≅ 𝐷𝐵̅̅ ̅̅

D

C

A B

1 2

1 2

30

30

4. Given: G is the midpoint of FI̅

∡F ≅ ∡I

Prove: 𝐸𝐹̅̅ ̅̅ ≅ 𝐼𝐻̅̅̅̅

5. Given: 𝐴𝐵̅̅ ̅̅ ⊥ 𝐵𝐶;̅̅ ̅̅ ̅ 𝐷𝐸̅̅ ̅̅ ⊥ 𝐸𝐹;̅̅ ̅̅ ̅

∡𝐶 ≅ ∡𝐷; 𝐵𝐶̅̅ ̅̅ ≅ 𝐷𝐸̅̅ ̅̅

Prove: ∡𝐴 ≅ ∡𝐹


6.1 and 9 _______________________

7. 3 and 10 _______________________

8. 7 and 13 _______________________

9. 6 and 16 _______________________

10. 11 and 14 _______________________

11. 2 and 3 ________________________

12. 7 and 8 ________________________

G

F

E

H

I

A

B C

D E

F

1 2

3 4 5 6

7 8

9 10

11 12

13 14

15 16

l

m

r t

31

31


13. If m1 = 120°, find m5 = ________

14. If m6 = 72°, find m4 = ________

15. If m4 = 112°, find m5 = _______

16. If m2 = 82°, find m7 = ________

17. If m2 = 80°, find m5 = ________

18. Solve for x 19. Solve for x

20. 21.

x = ________, y = ________ x = ________, y = _______

1 2

3 4

5 6

7 8

l

m

t

q

t

3x – 15

2x + 7

p

q

t

7x – 12

12x + 2

p

3y + 12

2x + 40 y

y 65°

3x + 5

1 unit 4 - weebly · 2018-08-31 · apc bpc = apb bpc apc = nonsense apc bpd = nonsense apc apb =...

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