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!A P E X H I G H S C H O O L

1501 L A U R A D U N C A N R O A D

A P E X , N C 27502

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Common Core Math 3

Unit 2B - Proofs

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Getting Blood From a Stone

The rules of writing a mathematical proof are very similar to the rules of "getting blood

from a stone". The old saying "You can't get blood from a stone” means that nobody can

give you anything that they, themselves, do not have.

The "rules of the game", and how these rules relate to mathematical proof:

1) You must start with the word STONE and end up with the word BLOOD by a series of

logical steps (much like the steps in a mathematical proof).

2) You must change one letter in each step. Each step must follow directly from the

previous step. (In a proof, you must address one concept at a time. Each step in a proof

must follow directly from the step before it - including from the 'Given', and lead to the

next.)

3) The word in each step must be a real word, and in the dictionary. (In a proof, each step

must be true: a definition, postulate, or theorem). If you use an unusual word, you must

write the definition.

4) In this game, some people can do this in 10 steps; others may take 15, and both can be

correct. There is more than one route, and no one way is necessarily better than another.

One may be longer, but the best one is the one is the one that the student "sees" when

trying to do the problem! (This is very true of proof; there are dozens of proofs of the

Pythagorean Theorem, for example, and even in classroom assignments there is often

more than one method.)

5) In this game, some people like to work backwards, or even from both ends toward the

middle. (This works very well with proofs, too!)

Example:

Can you get a DOG from a CAT? (Can you go from the word CAT to the word DOG?)

Here are some possible solutions to this problem:

Solution 1: CAT - BAT - BAG - BOG - DOG

Solution 2: CAT - COT - DOT - DOG

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Algebraic PROPERTIES used in geometry

Property Example(s)

Distributive 3(n + 5) = 3n + 15

-(x – 8) = -x + 8

Substitution if 5x + 12 - 2x = 18, then 3x + 12 = 18 (combining like terms, simplifying)

If 7x + y = 9 and y = -3, then 7x – 3 = 9 (replacing y with -3)

Reflexive -3 + x = -3 + x ONE EQUATION

AB = AB

Symmetric if y = 7 then 7 = y TWO EQUATIONS

If

AB @ CD then

CD @ AB

Transitive if a = b and b = c then a = c THREE EQUATIONS If

AB @ BC and

BC @ CD then

AB @ CD

Properties of Equality:

Addition prop. of = IF 3x – 8 = 10, then 3x = 18 *add an equal amount to both sides

If AB = DE then AB + BC = DE + BC

Multiplication prop. of = if *multiply both sides by an equal amount

Subtraction prop. of = if x + 5 = -6, then x = -11 *subtract both sides by an equal amount

If AB + BC = CD + BC, then AB = CD

Division prop. of = if 3x = 12, then x = 4 *divide both sides by an equal amount

If 3AB = 6EF, then AB = 2EF

Algebraic Properties Worksheet

I. Name the property that justifies each statement.

1. If m

– A = m

– B, then m

– B = m

– A. 2. If x + 3 = 17, then x = 14

3. xy = xy 4. If 7x = 42, then x = 6

5. If XY – YZ = XM, then XY = XM + YZ 6. 2(x + 4) = 2x + 8

7. If m

– A + m

– B = 90 , and m

– A = 30, 8. If x = y + 3 and y + 3 = 10, then x = 10.

then 30 + m

– B = 90.

II. Complete the reasons in each algebraic proof.

9. Prove that if 2(x – 3) = 8, then x = 7.

Given: 2(x – 3) = 8

Prove: x = 7

Statements Reasons

a) 2 (x – 3) = 8 a)________________________________

b) 2x – 6 = 8 b)________________________________

c) 2x = 14 c) _______________________________

d) x = 7 d) _______________________________

10. Prove that if 3x – 4 = x + 6 , then x = 4.

Given: 3x – 4 = x + 6

Prove: x = 4

Statements Reasons

a) 3x – 4 = x + 6 a) ________________________________

b) 2(3x – 4) = 2( x + 6) b) ________________________________

c) 6x – 8 = x + 12 c) ________________________________

d) 5x – 8 = 12 d) ________________________________

e) 5x = 20 e) ________________________________

f) x = 4 f) ________________________________

Name the property that justifies each statement.

____________________________1. If 3x = 120, then x = 40.

____________________________2. If 12 = AB, then AB = 12.

____________________________3. If AB = BC and BC = CD, then AB = CD.

____________________________4. If y = 75 and y = m

– A, then m

– A = 75.

____________________________5. If 5 = 3x – 4 , then 3x – 4 = 5.

____________________________6. If

3 x -5

3

Ê Ë

ˆ ¯

= 1, then 3x – 5 = 1.

____________________________7. If m

– 1 = 90 and m

– 2 = 90, then m

– 1 = m

– 2.

____________________________8. For XY, XY = XY.

____________________________9. If EF = GH and GH = JK, then EF = JK.

____________________________10. If m

– 1 + 30 = 90, then m

– 1 = 60.

Choose the number of reason in the right column that best matches each statement in the left column.

Statements Reasons

_______A. If x – 7 = 12, then x = 19 1. Distributive Property

_______B. If MK = NJ and BG = NJ, then MK = BG 2. Addition Property of Equality

_______C. If y = m

– 5 – 30 and m

– 5 = 90, then y = 90 – 30 3. Symmetric Property

_______D. If ST = UV, then UV = ST. 4. Substitution Property

_______E. If x = -3(2x – 4), then x = -6x + 12 5. Transitive Property

Name the property that justifies each statement.

______________________________________11. If AB + BC = DE + BC, then AB = DE.

______________________________________12. m

– ABC = m

– ABC

______________________________________13. If XY = PQ and XY = RS, then PQ = RS.

______________________________________14. If , then x = 15.

______________________________________15. If 2x = 9, then x = .

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Writing Reasons

Fill in the blank with a reason for the statement. Write your answer in “if-then” form if it is a definition.

STATEMENTS REASONS

1. a) a) Given

b) AB = CD b) ___________________________________

2. a) M is the midpoint of . a) Given

b) b) ___________________________________

3. a) bisects at point X. a) Given

b) X is the midpoint of . b) ___________________________________

4. a) and a) Given

b) b) ___________________________________

5. a) M is between A and B, a) Given

b) M is the midpoint of b) ___________________________________

6. a) M is between C and D a) Given

b) CM + MD = CD b) ___________________________________

7. a) MN = RS a) Given

b) b) ___________________________________

8. a) a) Given

b) b) ___________________________________

9. a) AB = CD and BC = BC a) Given

b) AB + BC = CD + BC b) ___________________________________

A proof is a logical argument that establishes

the truth of a statement.

Presenting a proof:

A proof is a written account of the complete thought process that is used

to reach a conclusion. Each step of the process is supported by a theorem,

postulate or definition verifying why the step is possible. Proofs are developed

such that each step in the argument is in proper chronological order in relation to

earlier steps. No steps can be left out.

There are three classic styles for presenting proofs:

Method 1: The Paragraph Proof…is a detailed paragraph explaining the

proof process. The paragraph is lengthy and contains steps and reasons that lead

to the final conclusion. Be careful when using this method -- it is easy to leave out

critical steps (or supporting reasons) if you are not careful.

Method 2: The Flowchart Proof … is a concept map that shows the

statements and reasons needed for a proof using boxes and connecting arrows.

Statements, written in the logical order, are placed in the boxes. The reason for

each statement is placed under that box.

Method 3: The Two-Column Proof … consists of two columns where

the first column contains a numbered chronological list of steps ("statements")

leading to the desired conclusion. The second column contains a list of "reasons"

which support each step in the proof. These reasons are properties, theorems,

postulates and definitions.

A formal 2-column proof contains the following elements:

! Statement of the original problem

! Diagram, marked with "Given" information

! Re-statement of the "Given" information in the proof

! Complete supporting reasons for each step in the proof

! The "Prove" statement as the last statement

EXAMPLE:

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Method 1 - Paragraph Proof:

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;+.2/75<!/745!"%!$!#&'!!!(+1845+!#!-5!=+/>++.!"!8.*!%<!/7+!#+26+./!?**-/-0.!

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?**-/-0.!@05/4;8/+!/+;;5!45!/7+!#&!$!#%!A!%&'!!B+!18.!/745!54=5/-/4/+!"#!A!#%!,03!"%<!

8.*!#%!A!%&!,03!#&!-./0!/7+!+:48;-/)!"%$#&!/0!2-9+!"#!A!#%!$!#%!A!%&'!!!#%!$!#%!=)!

/7+!3+,;+C-9+!D30D+3/)'!!&5-.2!/7+!54=/381/-0.!D30D+3/)!0,!+:48;-/)<!>+!18.!/745!

54=/381/!#%!,306!=0/7!5-*+5!0,!/7+!+:48;-/)!!"#!A!#%!$!#%!A!%&!/0!D309+!/78/!"#!$!%&'!

!

Method 2 - Flowchart Proof:

!

Method 3 - Two-Column Proof:

Statements !"#$%&$'

1.

RT @ SU ,'(")#

-.##/0#1#23# 45#*67#8"&9")*8#$%"#:7)&%;")*<#*+")#

*+"'%#=")&*+8#$%"#">;$=.#

?.##/0#1#/2#@#20# 2"&9")*#ABB'*'7)#C78*;=$*"#

D.##23#1#20#@#03# 2"&9")*#ABB'*'7)#C78*;=$*"#

E.##/2#@#20#1#20#@#03# 2;F8*'*;*'7)#C%7G"%*H##

I.##20#1#20# /"5="J'("#C%7G"%*H#

K.##/2#1#03# 2;F*%$:*'7)#G%7G"%*H#75#L>;$='*H#

!!"! #! %! &!

!!!!!!

"%!$!"#A#%!

#&!$!#%!A!%& !

Segment Addition Postulate !

RT @ SU !

Given !

"%!$!#& !

If 2 segments are congruent

their lengths are =.

!

"#!A!#%!$!#%!A!%& !

Substitution Property

!

Subtraction Property of

Equality

!

#%!$!#% !

"%!$!#& !

Reflexive Property

!

Definition Proofs

EXAMPLE:

Given: XY = YZ

Prove: Y is the midpoint of

STATEMENT REASON

1. XY = YZ 1. Given

2. 2. If 2 segments have = lengths, then they are

@.

3. Y is the midpoint of 3. If a point divides a segment into 2

@ segments,

then it is the midpoint

1. Given: Line m bisects

Prove: XP = YP

STATEMENT REASON

1. line m bisects .

2. P is the midpoint of .

3.

4. XP = YP

2. Given: B is the midpoint of

D is the midpoint of

Prove: AB + CD = BC + DE

STATEMENT REASON

1. B is the midpoint of

D is the midpoint of

2. 2.

3. AB = BC

CD = DE

4. AB + CD = BC + DE

! X Y Z

! !

A B C D E ! ! ! ! !

X

Y

P

!

!

! m

AB @ BC

CD @ DE

3. Given: PQ = MN

MN = QR

Prove: Q is the midpoint of

STATEMENT REASON

4. Given: line k bisects

line k bisects

Prove: AB + DE = BC + EF

STATEMENT REASON

5. Given: C is the midpoint of

Prove: AB + BC = CE

STATEMENT REASON

A B C

D

E

F

P N ! !

M !

Q R ! !

D

E

B

C

A !

!

! F !

!

!

k

Practice with Segment Proofs

Fill in the reason for each statement below.

Then rewrite the proof as a FLOWCHART PROOF.

1.

Given: DE = EF, AD = BF

Prove: AE = BE

Statements Reasons

1. AD = BF 1.

DE = EF

2. AD + DE = BF + FE 2.

3. AD + DE = AE 3.

BE = BF + FE

4. AE = BE 4.

FLOWCHART PROOF:

2.

Given: AX = BY, XC = YC

Prove:

Statements Reasons

1. AX = BY 1.

XC = YC

2. AX + XC = BY + YC 2.

3. AX + XC = AC 3.

BY + YC = BC

4. AC = BC 4.

5. 5.

FLOWCHART PROOF:

A D E F B

! !

B A

C

X Y ! !

! ! !

Write 2-column proofs (from scratch)

3. Given: AE = FD

Prove: AF = ED

Statements Reasons

4. Given: AB = BE

BC = BD

Prove: AC = ED

Statements Reasons

5. Given: AF = EC

Prove: AE = FC

Statements Reasons

A F E D

B

C

B

D C

A E

B

A F

E

D

C

6. Given: AC = BD

BE = EC

Prove: AE = DE

Statements Reasons

7. Given: AB = DE

BC = CD

Prove:

Statements Reasons

B A

C D

E

C

B D

A E

!"#$%&'!"#$!%&'(&!()!*!&(&+,(--'&$./!/.01!2'"#!.!,(33(&!$&45('&"!

6'7$!.&!$8.35-$!(/!4/.2!.!4'.9/.3!()!$.,#!.&9-$!(/!.&9-$!5.'/:!! !

! (%)*"*+*,"' -./01$%',2'(*/#2/0'

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'1!;!!<=>!

!

.4?.,$&"!.&9-$1!

"2(!.&9-$1!"#."!1#./$!.!

7$/"$8!.&4!.!/.0!.&4!&(!

'&"$/'(/!5('&"1!

!

,(35-$3$&"./0!

.&9-$1!

"2(!.&9-$1!./$!

,(35-$3$&"./0!')!"#$!

1%3!()!"#$'/!3$.1%/$1!'1!

<=>!

!

-'&$./!5.'/!

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2#(1$!&(&+,(33(&!

1'4$1!)(/3!.!-'&$!

!

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'1!A!!<=>!

!

/'9#"!.&9-$!.&!.&9-$!2#(1$!3$.1%/$!

'1!B!!<=>!

!

1%55-$3$&"./0!

.&9-$1!

"2(!.&9-$1!./$!

1%55-$3$&"./0!')!"#$!

1%3!()!"#$'/!3$.1%/$1!'1!

CD=>!

!

7$/"',.-!.&9-$1!

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.&9-$1!)(/3$4!@0!"#$!

'&"$/1$,"'(&!()!"2(!-'&$1!

!

! ! ! ! ! ! !

!

'

!"#$%&!''()(*"&+*,)-$.)%/&!

!

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!

&

0-44$%5%")&+*,)-$.)%"!!!

!

&&

67%*2%5,/&

&

!

8%9("()(*",:&+*,)-$.)%,:&."'&67%*2%5,&

!;<=>0/&

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! 45!6!&'()*+!.&7*!*/0&)!1*&+02*8!-.*'!-.*9!&2*!:%'(20*'-3!

45!

m–A = m–B !-.*'!

–A @ –B !

! 45!6!&'()*+!&2*!:%'(20*'-8!-.*'!-.*9!.&7*!*/0&)!1*&+02*3!;<%'7*2+*=!

45!!

–A @ –B-.*'!

m–A = m–B !!

>>>?*1*1@*2!

–A !2*5*2+!-%!-.*!&'()*8!$.,)*!

m–A !,+!&!'01*2,:!7&)0*8!$.,:.!

,+!-.*!1*&+02*!%5!-.*!&'()*3!

"

–A @ –B !!!!!!!!!,+!:%22*:-8!&'()*+!:&'!@*!:%'(20*'-3

"

m–A = m–B ,+!:%22*:-8!-.*!1*&+02*+!%5!&'()*+!:&'!@*!*/0&)3!

@A6……

#

–A = –B ,+!,':%22*:-8!6!&'()*+!&2*!:%'(20*'-!@0-!AB#!*/0&)3!

#

m–A @ m–B ,+!,':%22*:-8!6!&'()*!1*&+02*+!&2*!*/0&)8!AB#!!

! ! ! ! :%'(20*'-3!&

!"#$%&@(,%3)*2/&C!),'*8!2&98!%2!+*(1*'-!$.,:.!:0-+!&'!&'()*!,'-%!-$%!*/0&)!.&)7*+3!

! 45!&!2&9!@,+*:-+!&'!&'()*8!-.*'!,-!D,7,D*+!-.*!&'()*!,'-%!6!:%'(20*'-!

&'()*+3!

&

Given:

– 1 and

–2 are supplementary

– 2

@

– 3

Prove:

m

–1 + m

–3 = 180° !

EXAMPLE:

Method 1 - Paragraph Proof:

We are given that

– 1 and

– 2 are supplementary and that

–2

@

–3. By definition the

measures of supplementary angles add to 180° so m

– 1 + m

– 2 = 180°. By definition

congruent angles have equal measures so m

– 2 = m

–3. We can use the substitution

property of equality to substitute m

–3 for m

– 2 in the equation m

– 1 + m

– 2 = 180°

and prove that m

– 1 + m

– 3 = 180°.

Method 2 - Flowchart Proof:

Method 3 - Two-Column Proof:

Statements Reasons

1.

– 1 and

– 2 are supplementary Given

2.

– 2

@

– 3 Given

3. m

– 1 + m

– 2 = 180° Definition of supplementary angles

4. m

– 2 = m

– 3 Definition of congruent angles

5. m

– 1 + m

– 3 = 180° Substitution property of equality

!

Given !

– 1 and

– 2 are

supplementary

!

Given !

– 2

@

– 3

!

Definition of

supplementary angles

!

m

– 1 + m

– 2 = 180°!

Definition of

congruent angles

!

m

–2 = m

–3!

m

– 1 + m

– 3 = 180°

!

Substitution property

of equality

!

!" #"

Prove Angle Theorems

!"#$%&#'()*+,#'-+#-.)/$#'()*+,0#$/+(#$/+1#'-+#2&()-3+($4##5-&6+#$/.,#$/+&-+74

1. Given: A and B are right angles

Prove: A B

!

!"#$%&#'()*+,#'-+#6+-$.2'*#'()*+,0#$/+(#$/+1#'-+#2&()-3+($4##5-&6+#$/.,#$/+&-+74

2. Given: Lines k and m intersect

Prove: 1 3

"

"

"

"

"

"

"

"

!"#$%&#'()*+,#/'6+#$/+#,'7+#,388*+7+($0#$/+(#$/+1#'-+#2&()-3+($4##5-&6+#$/.,#$/+&-+74!

$%""&'()*+" !",*-" #",.)"/0112)3)*4,.5",*62)/"

" """ #",*-" 7",.)"/0112)3)*4,.5",*62)/"

"""""8.9()+" !" " 7""

"

""

#"

$"%"

&"

'"

ANGLES Worksheet

ANGLE ADDITION POSTULATE: If point B lies in the interior of

– AOC, then _______ + ______ = _______

If two angles form a linear pair then they are _________________________.

All right angles are ____________. All vertical angles are _________________.

If two angles are

@ and supplementary, then each is a _________ angle.

If two angles are

@, then their _________________ are

@ .

If two angles are

@ , then their _________________ are

@

If two angles have the same complement (supplement), then they are ___________.

If two angles are supplements (complements) of

@ angles, then the two angles are ____________.

Fill in the blank with Always, Sometimes, or Never.

________________1. Vertical angles ________ have a common vertex.

________________2. Two right angles are _________ complementary.

________________3. Right angles are __________ vertical angles.

________________4. Angles A, B, and C are __________ supplementary.

________________5. Vertical angles __________ have a common supplement.

Solve for the given variables.

1. 2.

3. 4.

5. bisects

–KAT

m

– 1 = x + 12

m

– 3 = 6x – 20

_______________6. If m

– C = 40 – x , and

– C and

– D are complementary, what is the m

– D?

_______________7. If m

– A = x + 80 and

– A and

– B form a linear pair, what is the m

– B?

3x – 5

6x – 23 x + 16 2x – 16

3x – 8 x

2y – 17

3x – y 50

x

2x – 16

1

2

3

S A T

L

K

Use the figure to the right to answer #8 - 15

8. Name all angles shown.

9. Name

– 1 in 2 other ways.

10. Name all angles which have side .

11. Name a point in the interior of

– GEF.

12. Name a point in the exterior of

– 2.

13. Name 2 pairs of adjacent angles.

14. What is the union of and ?

15. What is the union of and ?

State whether the numbered angles shown in 16 - 23 are adjacent or not. If NOT, explain.

16. 17. 18. 19.

– ABC,

– ABD

29. 21. 22. 23.

Answers:

16. ____________________________________________________________________________

17. ____________________________________________________________________________

18. ____________________________________________________________________________

19. ____________________________________________________________________________

20. ____________________________________________________________________________

21. ____________________________________________________________________________

22. ____________________________________________________________________________

23. ____________________________________________________________________________

D E F

G H

1

2 1

2 1 2

A

B

C

D

1 2 1 2 1

2

1 2

1 2 3

1

2 3

B

E

D

H I

N

Angle Proofs Worksheet

1. Given: bisects

– NDH

Prove:

– 1

@

– 3

Statements Reasons

2. Given:

– 1

@

– 2

Prove:

– 1

@

– 3

Statements Reasons

3. Given:

– 1 is a supplement of

– 2

Prove:

– 1

@

– 3

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

1 2

3

1

3 2

4. Given:

– ADC

@

– CEA

Prove:

– CDB

@

– AEB

Statements Reasons

5. Given:

– 1

@

– 4

Prove:

– 2

@

– 3

Statements Reasons

B

D E

A C

1 2 3 4

6. Given:

– DMH

@

– DHM

– RMH

@

– RHM

Prove:

– DMR

@

– DHR

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

7. Given:

Prove:

– ABD and

– DBC are complementary

Statements Reasons

!

!

!

!

!

D

M H

R

A B

C

D

8. Given: ;

– 1 is a complement of

– 3

Prove:

– 2

@

– 3

Statements Reasons

!" #

is the complement of

1.

2.

m–BAC = 90!#2.

3. 3.

4.

m–1+ m–2 = 90! #4.

5.

m–1+ m–3 = 90! #5.

6. #6.

7. #7.

8. # 8.

9. Given: ;

– 2

@

– 3

Prove:

– 1

@

– 4

Statements Reasons

# # # # #

#

A O E

C

B D

1 2 3

4

C

A

B

1 2

3

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ANGLES TYPE ! , SUPPL., OR NONE 1. " 1 and " 14

2. " 2 and " 15

3. " 7 and " 9

4. " 9 and " 16

5. " 10 and " 17

6. " 16 and " 14

7. " 9 and " 14

8. " 18 and " 19

9. " 1 and " 16

10. " 3 and " 8

11. " 6 and " 9

12. " 12 and " 13

13. " 7 and " 11

14. " 6 and " 8

15. " 4 and " 13

2

5

1 6

7

8

3 4

9

10 12

11 13

14 15

16

17

18 19

b

x

y a

Parallel line Proofs

1. Given: l // m ; s // t

Prove:

– 2

@

– 4

2. Given: l // m ; s // t

Prove:

– 1

@

– 5

3. Given: l // m;

–1

@

–4

Prove: s // t

4. Given: l // m;

–2

@

–5

Prove: s // t

5. Given: ;

Prove:

–B and

– E are supplementary

A

B

D

P C

E F

l

m

s t

1

2

3 5

4

Diagram for #1-4

6. Given: ;

Prove:

– B

@

– E

7. Given: g // h; g // j

Prove:

–2

@

–3

8. Given: ;

Prove:

–B

@

–D

A

B

D

P

E

C

F

1

2

3

g

h

j

A

B

C

D

E

!"#$%$&$'%()*+'(&,-.&"()*.%/*01"'2"3(*

0456789:;<*

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*

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*

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*

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)

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Triangle Proofs Practice

If two sides of a triangle are congruent, the angles opposite those sides

are congruent. (Isosceles Triangle). Prove this theorem.

1. Given:

AC @ BC

Prove:

–CAD @ –CBD

Statements Reasons

1.

AC @ BC

2. Draw

CD bisecting

–ACB

3.

–1@ –2

4.

CD @ CD

5.

DACD @ DBCD

6.

–CAD @ –CBD

2. Given:

BD bisects

AC .

BD is perpendicular to

AC .

Prove: ! ABC is isosceles

Statements Reasons

1.

BD bisects

AC

2.

BD is perpendicular to

AC

3.

AD @ CD

4.

BD @ BD

5.

–ADB and

–BDC are

right angles

6.

–ADB

–BDC

7. "ABD !CBD

8.

AB @ CB

9. ! ABC is isosceles

!"

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3. Given:

AB @ DE

BC @ EF

–B @ –E Prove:

DABC @ DDEF

4. Given: E is the midpoint of

BD

AE @ EC

Prove:

DAEB @ DCED

5. Given:

BA @ DC

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AC !

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AC !"#$%&'$!

BD !

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DABE @ DCDE

6. Given:

–BAC @ –DAE

AE @ AC A is the midpoint of

BD

Prove:

DBEA @ DDCA

Statements Reasons

7. Given:

–A @ –E

AB @ BE

Prove:

AD @ EC

Statements Reasons

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Compasses are a drawing

instrument used for

drawing circles and arcs. It

has two legs, one with a

point and the other with a

pencil or lead. You can

adjust the distance between

the point and the pencil and

that setting will remain

until you change it.

%

A straightedge is simply a guide for

the pencil when drawing straight

lines. In most cases you will use a

ruler for this, since it is the most

likely to be available, but you must

not use the markings on the ruler

during constructions. If possible,

turn the ruler over so you cannot see

them.

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MATH OPEN REFERENCEwww.mathopenref.com NAME:

Constructing a copy of a given line segment with compass and straightedge

(For assistance see www.mathopenref.com/constcopysegment.html)

1. Construct a copy of the line segment below.

2. Construct a copy of the line segment AB from the triangle. The copy should have one endpoint at P.

Worksheet - Construct a copy of a given line segment http://www.mathopenref.com/wscopysegment.html

1 of 1 8/20/14 11:05 PM


Constructing a copy of a given angle with compass and straightedge

(For assistance see www.mathopenref.com/constcopyangle.html)

1. Construct a copy of the angle below.

2. Using the fewest arcs and lines, construct a copy of the shape below. (The record: 5 arcs, 2 lines )

Worksheet - Construct a copy of a given angle http://www.mathopenref.com/wscopyangle.html

1 of 1 8/20/14 10:37 PM

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Constructing the perpendicular bisector of a line segment with compass and straightedge

(For assistance see www.mathopenref.com/constbisectline.html)

1. Draw the perpendicular bisector of both of the two lines below

2. Construct the four perpendicular bisectors of the sides of the rectangle below,

using the fewest arcs and lines. (The record: 3 arcs, 2 lines)

Worksheet - Perpendicular line bisector http://www.mathopenref.com/wsbisectline.html

1 of 1 8/20/14 10:13 PM


Bisecting an angle with compass and straightedge

(For assistance see www.mathopenref.com/constbisectangle.html)

1. Bisect the two angles below.

2. Challenge Problem: Bisect all 4 angles, using the fewest arcs and lines. (record: 4 arcs 2 lines)

Worksheet - Bisect an angle http://www.mathopenref.com/wsbisectangle.html

1 of 1 8/20/14 10:18 PM

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Construct a line parallel to a given line through a given point with compass and straightedge

(For assistance see www.mathopenref.com/constparallel.html)

1. Construct a line parallel to the one below that passes through the point P

2. (a) Construct a line parallel to AB through Q, and another line parallel to CD also through Q

(b) What is the name of the resulting 4-sided shape?

Worksheet - Construct a parallel through a point http://www.mathopenref.com/wsparallel.html

1 of 1 8/20/14 10:43 PM


Construct a line perpendicular to a given line from a point on the line, with compass and straightedge

(For assistance see www.mathopenref.com/constperplinepoint.html)

1. Construct a line perpendicular to the ones below that passes through the point P

Worksheet - Construct a perpendicular from a point on the line http://www.mathopenref.com/wsperplinepoint.html

1 of 1 8/20/14 10:48 PM

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Construct a line perpendicular to a line that passes through a point, with compass and straightedge

(For assistance see www.mathopenref.com/constperpextpoint.html)

1. Construct a line perpendicular to the one below that passes through the point P

2. (a) Construct a line perpendicular to AB through P, and another line perpendicular to CD also

through P

(b) What is the name of the resulting 4-sided shape? Measure its side lengths with a ruler and

calculate its area.

Worksheet - Construct a perpendicular through a point http://www.mathopenref.com/wsperpextpoint.html

1 of 1 8/20/14 10:54 PM


Divide a given line segment into a N equal segments with compass and straightedge

(For assistance see www.mathopenref.com/constdividesegment.html)

1. Divide the line segment below into 3 equal parts

2. Divide the line segment below into 7 equal parts.

Worksheet - Divide a line segment into N parts http://www.mathopenref.com/wsdividesegment.html

1 of 1 8/20/14 10:29 PM

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