Download Assignments Intro Proofs

Geometry Assignments: Introduction to Geometry Proofs
Day
Topics
Homework
1
Lines and segments
HW IP - 1
2
Angles
HW IP - 2
3
Definitions; drawing conclusions
HW IP - 3
4
Basic postulates **QUIZ**
HW IP - 4
5
Addition & subtraction postulates
HW IP - 5
6
Multiplication & Division postulates **QUIZ**
HW IP - 6
7
Statement-Reason proofs
HW IP - 7
8
Simple angle theorems
HW IP - 8
9
Practice **QUIZ**
HW IP - 9
10
More practice
HW IP - 10
11
Review **QUIZ**
HW IP - Review
HW
Grade
Quiz
Grade
***TEST***
Note: Assignments 3 – 8 may be done on the assignment sheet and handed in. The rest of the
assignments need to be done on SEPARATE PAPER.
TEST CORRECTIONS
To raise your test grade, you may* do a test correction as follows:
On separate paper (not on the original test), for each problem you got wrong:
a. Tell what your mistake was. Be brief but informative. (Do not just write “I put -4 but the right
answer was 17/3.” I want to know what you did wrong that made you get -4.)
b. Correct the problem and get the right answer. Show work. (“The right answer is 17/3 because
that’s what Norman put and he got it right” does not get credit. I want to see that you know
how/why 17/3 is the right answer.) For multiple choice or true/false questions, explain why the
right answer is correct.
GOOD test corrections turned in within 10 class days of getting the test back are worth ¼ of the points
you lost. (Ex: If you got a 70, then you lost 30 points. A good test correction will get you
30  4 = 7.5 or 8 points back for a new grade of 78.)
*Test corrections are optional.
Geometry Homework: Intro Geo Proofs - 1
1. Draw a single diagram to illustrate the following givens: HAT , CAP .
Notes: 1) Since they are written separately, you should not assume that all the points are collinear.
2) There cannot be two different points A in the same problem.
2. If M is the midpoint of AB , AM = x2 + 24 and MB = 10x, find the length of AB .
3. PR bisects ST at Q. PQ = 4x + 12, QR = 9x – 13, SQ = 6x – 5 and QT = 3x + 16. Find the length of PR .
4. Given: MATH , A is the midpoint of MT , MH = 21 and AH = 15. Find TH.
5. In RST , RS = 7x – 1, ST = 2x + 3 and RT = 12x – 7. Find the numerical value of RT.
READ: Adding and Subtracting Line Segments
Everybody knows you can add and subtract numbers: 7 + 3 = 10 and 7 – 3 = 4 make perfect sense.
However, adding and subtracting people (not numbers of people but actual persons) is meaningless. It is
nonsense to say Devin + Bree = Ken or Devin – Bree = Thor.
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.
C
A
B
C
A
B
AB  BC  AC
AB  BC  nonsense
AC  BC  AB
AC  AB  BC
AC  BC  nonsense
AC  BC  nonsense
A
B
C
D
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.
C
C
B C
D
A
B
A
A
B
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
(This assignment is continued on the next page.)
AD  BC  nonsense
AC  BD  nonsense
P
5
4
A 2 B
3
C1D
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  ______
F
P
L
A
G
Geometry Homework: Intro Geo Proofs - 2
1. Use the diagram at right to answer the following.
A
a. How many angles in the diagram have their vertex at A?
1
B
b. How many angles in the diagram have their vertex at B?
2
3
c. What angle (number) is named BDC?
d. Name two adjacent angles in the diagram.
6
4
5
e. Are ADC and BDC adjacent?
C
D
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.
R
i. Name one obtuse angle on the diagram.
j. Which angle on the diagram appears to be closest to a right angle?
P
2. In the diagram at right, which angle has a larger measure,
PAQ or RAS?
A
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.
S
Q
N
Q
O
R
P
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, mBET = x2 and mGET = 5x + 14, find the numerical measure of BEG.
7. If OY bisects BOT, mBOY = 3x + 8 and mBOT= 8x – 2, find the numerical measure of TOY.
(This assignment is continued on the next page.)
READ: Remember from the last assignment: Numbers can always be added and subtracted. It makes no sense
to add or subtract people. Line segments can sometimes be added or subtracted (if you don’t remember when,
review the note after homework IP – 1 #5). Angles are like segments. They can sometimes be added and
subtracted. Remember, ABC represents an actual angle (a geometric object); mABC is a number that
represents the degree measure of ABC.
Your answers to #8 should have been “Yes” for all except parts b and f.
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).
A
C
A
B
B
C
C
P
A
APB +BPC = APC
P
B
APB +BPC = nonsense
D
P
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).
A
C
A
B
B
C
C
P
A
APC BPC = APB
APC APB = BPC
8. Based on the diagram at right, tell if each of the following is True or
False. Remember the difference between A and mA.
a. mCAD + mABC = mBCA
b. CAD + ABC = BCA
c. mCAD + mDAB = mCAB
d. CAD + DAB = CAB
e. mDBA  mDAC = mBAD
f. DBA  DAC = BAD
g. mBAC  mBAD = mDAC
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 = ________
D
P
P
B
BPC APC = nonsense
APC BPD = nonsense
APD BPC = nonsense
B
55
D
40
70
15
A
N
S
G
E
A
C
L
Name
Geometry Homework: Intro Geo Proofs - 3
Rewrite each definition in the form of two conditionals:
1. Perpendicular lines form right angles.
a. If two lines
b. If two lines
2. An angle bisector is a line (or segment) that divides an angle into two congruent parts.
a. If a line (or segment)
b. If a line (or segment)
In problems #3 - 12, 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.)
3. Given: AB  CD
D
B
Conclusion:
Reason:
C
A
4. Given: X is the midpoint of PQ .
Conclusion:
P
.
.
.
Q
X
Reason:
A
B
5. Given: BD bisects ABC.
Conclusion:
D
Reason:
6. Given: BD bisects AC at E.
C
A
D
Conclusion:
E
Reason:
B
C
7. Given: AB  AC
A
Conclusion:
Reason:
B
C
8. Given: AC  BC .
A
1st Conclusion:
Reason:
C
2nd Conclusion:
B
Reason:
9. Given: RST and RS  ST .
R
.
.
.
S
T
Conclusion:
Reason:
K
10. Given: JL divides KM into two congruent parts.
L
J
Conclusion:
Reason:
M
S
11. Given: A is the vertex of isosceles triangle SAM
Conclusion:
A
M
Reason:
A
12. Given: FAT  RAT
Conclusion:
Reason:
F
R
T
13. Given LINE , N is the midpoint of IE , LE = 30 and NE is three less than LI. Find the numerical length of
LI.
A
14. In the diagram at right, BD bisects ABC, mABD = 66 – 2x and
mCBD = 3x – 24. Find the numerical value (a number, not just an
algebraic expression) of mABC.
66 – 2x
B
D
3x – 24
C
Name
Geometry HW: Intro Geo Proofs - 4
A
For #1 - 4, name the postulate that justifies the conclusion.
1. Given: FT  AT , AT  RT
Conclusion: FT  RT
F
R
T
Reason:
B
D
2. Given: (Diagram at right)
Conclusion: mDBE = m4 + m2 + m5
4
1
A
Reason:
E
5
2
3
C
A
3. Given: (Diagram at right)
Conclusion: AT  AT
Reason:
F
R
T
4. Given: m1 + m2 = 180°, m2 = m3 (Diagram at right)
Conclusion: m1 + m3 = 180
1
Reason:
2
3
For the following, give a valid conclusion and a reason.
2
5. Given: m1 + m2 = 180; m3 = m1.
3
1
Conclusion:
Reason:
U
6. Given: QA bisects UAD.
Conclusion:
Q
A
Reason:
D
7. Given: mAOB = 90.
Statement: mAOB = mAOX + mXOB
B
Conclusion:
Reason:
Conclusion:
Reason:
X
O
A
You should already know the following from previous assignments but read it anyway.
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
F
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?)
.
A
C
B
If two angles are added or subtracted, the result is another angle. (Same diagram.)
ABCD
Ex: a. FCE + ECD =FCD
b. ABF + DCF = nothing (why?)
c. BCE – FCE =BCF
d. ABF – FBC = nothing (why?)
E
D
8. 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 
.
P
B
C
Q
A
R
S
D
9. 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 =
.
P
B
C
Q
A
S
R
10. If M is the midpoint of AY , AM = x + 8 and AY = 3x2, find the numerical length of AY .
11. 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 .
D
Name
Geometry HW: Intro Geo Proofs - 5
For each of the following givens, state a valid conclusion based on the postulates we have covered and tell what
postulate was used.
1.
A
Given: AB  AC , AC  AD .
Conclusion:
Reason:
D
B
C
A
2.
Given: ADB , AEC , AD  AE , DB  EC .
Conclusion:
Reason:
3.
E
D
B
C
F
A
Given: ABCACB, ABDACD
Conclusion:
D
Reason:
C
B
4.
Given: ABECDE, CBEADE
A
D
Conclusion:
E
Reason:
5.
B
C
Given: AEB , DFC , AB  CD , AE  CF .
E
A
B
D
Conclusion:
Reason:
D
E
6.
C
F
Given: BAD  CAD, BAD  FAE
F
A
Conclusion:
Reason:
B
D
D
C
Probems #7 – 9 are simple “statement-reason” geometry proofs. For each one, fill in the missing reasons with
appropriate postulates.
7.
8.
9.
Given: mKJL + mLJM = 90, mKJL = mMJN
Prove: mMJN + mLJM = 90
Statement
Reason
1. mKJL + mLJM = 90 1. Given
2. mKJL = mMJN
2. Given
3. mMJN + mLJM = 90 3.
Given: ABCD , AB  CD
Prove: AC  BD
Statement
1. ABCD
2. AB  CD
3. BC  BC
4. AB + BC  CD + BC
or AC  BD
J
K
.
.
A
.
B
.
C
D
Reason
1. Given
2. Given
3.
4.
Given: KJM  NJL
Prove: KJL  MJN
Statement
1. KJM  NJL
2. LJM  LJM
3. KJL  MJN
N
C
M
L
B
D
J
Reason
1. Given
2.
3.
K
M
L
B
D
10. In the diagram at right, AB  BC , mABD = 3x + 17 and mCBD = 5x – 3.
Find the value of x.
N
C
A
D
B
C
11. What is the measure of the supplement of an angle that measures x degrees?
Name
Geometry HW: Intro Geo Proofs - 6
For each problem, use the definitions and postulates we have covered to state a valid conclusion for each set of
givens and give a reason for your conclusion. Good conclusions should use all the information in the givens.
The reason should be either a brief statement of the definition used or the name of the postulate used. For
problems #1 - 8, use the figure below. Treat each problem as separate (the givens for one problem do not
apply to the following problems). You may assume BTR , BGS , and RAS for all eight problems.
1.
Given: AB bisects RBS.
B
D
Conclusion/Reason:
2.
Conclusion/Reason:
3.
Given: BATBAG,
RATSAG
Conclusion/Reason:
4.
Given: BR  BS
Conclusion/Reason:
5.
Given: BR  BS ,
TR  GS .
Conclusion/Reason:
6.
Given: RAT  ATR, ATR  TRA
Conclusion/Reason:
7.
Given: BAR is a right angle.
Conclusion/Reason:
8.
Note: draw TG on the diagram and label its intersection with AB as point M.
Given: AB bisects TG at M.
Conclusion/Reason:
G
T
Given: RA  AS .
R
A
S
The following are simple “statement-reason” geometry proofs. For each one, fill in the missing reasons with
appropriate definitions or postulates.
9.
Given: A is supplementary to Z
B is supplementary to Z
Prove: AB
Statement
1. A is supplementary to Z
B is supplementary to Z
Reason
1. Given
2. mA + mZ = 180
2.
3. mB + mZ = 180
3. (same as #2)
4. mA + mZ = mB + mZ
4.
5.
5.
mZ =
mZ
6. mA = mB or AB
6.
R
10.
11.
Given: OR  ON
Prove: ROT is complementary to NOT
T
Statement
1. OR  ON
Reason
1. Given
2. NOR is a right angle
2.
3. mRON = 90
3.
4. mRON = mROT + mNOT
4.
5. mROT + mNOT = 90
5.
6. ROT is complementary to NOT
6.
O
In the diagram at right, AOD , and OC  BOE ,
mDOC = x2 + 15 and mAOB = 20x  81.
a.
C
B
Find mBOC.
A
b.
N
Find the value of x.
D
O
E
c.
Find mDOE.
d.
Find mAOE.
Name
Geometry HW: Intro Geo Proofs - 7
C
1. Fill in appropriate reasons in the proof below.
Given: AFE  BFD.
E
D
Prove: AFD  BFE
A
Statement
F
Reason
1.
AFE  BFD
1.
2.
DFE  DFE
2.
3. AFE – DFE  BFD – DFE
B
3.
or AFD  BFE
2. Write a complete “statement-reason” proof .
C
D
F
Given: AEFC , AE  CF .
E
Prove: AF  EC
Statement
A
B
D
Reason
B
D
3. Fill in appropriate reasons in the proof below.
Given: BD is an angle bisector of ABC, DBC  DCB
Prove: DBA  DCB
A
Statement
Reason
1. BD is an angle bisector of ABC
1.
2. DBA  DBC
2.
3. DBC  DCB
3.
4. DBA  DCB
4.
D
C
4. Write a complete “statement-reason” proof .
A
Given: E is the midpoint of BD , DE  AB
Prove: ABE is isosceles
Statement
B
D
E
D
Reason
C
5. Given: A is a right angle; B is a right angle
A
B
a. Write a brief explanation of why . Your explanation should refer to at least one postulate.
b. Think. Does the logic of your proof only work for the two right angles A and B shown above or will it
work for other right angles? Are there right angles for which the logic would not apply?
You have (hopefully) proven the following simple but very important and useful theorem:
Theorem: All right angles are congruent.
Abbreviation: All rt. s are .
Memorize.
Geometry HW: Intro Geo Proofs - 8
Do this homework neatly on SEPARATE PAPER.
1. Based on the diagrams, tell whether the given
angles are vertical angles.
a. 1 and 3
b. 1 and 4
c. 2 and 4
d. 5 and 7
2.
4
3
1
5
2
7
We wish to prove the following theorem: Vertical angles are congruent.
Given: AEB and CED
Prove: AEC  BED
a. Draw a diagram.
b. Outline a proof of the theorem. (There is more than one way to do this. The easiest way is to consider
how AEC and BED are related to CEB and then use theorems covered in today’s notes.)
Write a complete statement-reason geometry proof for each of #1 – 4.
D
E
F
G
A
6
B
C
D
Problem #3
C
D
T
P
E
G
I
A
B
F
Problem #4
3. Given: ABCD , ABG  DCG
Prove: CBG  BCG
W
N
Problem #5
4. Given: AB  AC , AE  AF
Prove: BAE  FAC
5. Given: PIW , GIN , IT bisects PIG
Prove: NIT  WIT
The following are algebraic exercises; not proofs.
6. If AEB intersects CED at E, mBEC = 5x – 25, and mDEA = 7x – 65, find the numerical values of the
measures of all four angles.
7. If AEB intersects CED at E, mAEC = 5(x + 15), and mAED = 7x – 75, find the numerical values of the
measures of all four angles.
Geometry HW: Intro Geo Proofs - 9
Do this homework neatly on SEPARATE PAPER.
Determine if each conclusion and reason is True or False. If false, change the conclusion and/or the reason
(not the given).
B
1. Given: BD bisects ABC
Conclusion: BAD  BCD because a bisector divides an
angle into two congruent parts
A
2. Given: m1 + m2 = 90 (No diagram for this problem.)
m3 + m4 = 90
Conclusion: m1 + m2 = m3 + m4 by the Addition Post.
C
D
A
3. Given: AB intersects CD at E
Conclusion: CE  ED because a bisector divides a segment
into 2  parts
C
E
D
B
Write a complete geometry proof for each of #4 - 6:
4. Given: ABCDE , B is the midpoint of AC , AB  DE
Prove: BD  CE
(Draw your own diagram.)
C
E
5. Given: ABC with right ACB, CD  AB , ACD EDC.
Prove: ECD  EDB
A
B
D
6. Given: BAD  FAD, BAE , FAC
Prove: DA bisects CAE
D
E
C
A
B
F
Geometry HW: Intro Geo Proofs - 10
Write complete geometry proofs for each of the following.
P
1. Given: ABMCD , M is the midpoint of BC , PM bisects AD
Prove: AB  CD
A
B
2. Given: AP  CA , AN  RA
AT bisects PAN.
Prove: CAT  RAT
M
D
C
T
N
P
J
C
3. Given: EAL , NAY , PEA is a right angle, PA  NY , NEA  NAE
Prove: PEN  PAL
E
A
D
R
N
N
A
L
P
Y
4. Two vertical angles are complementary. What is the measure of each?
5. Given: MATH , A is the midpoint of MT , MH = 21 and AH = 15. Find the value of TH.
6. Given line l and ma:mb:mc = 2:3:4, find the numerical value of ma.
a b c
l
7. The measure of an angle is 24 degrees less than twice the measure of its supplement. Find the measure of
the angle.
Geometry HW: Intro Geometry Proofs - Review
E
1. Given that AEB  CED, which is not a valid conclusion?
(1) mAEB = mCED
(2) AEC  BED
(3) mAEC = mBED
(4) AE  ED
2. If A, B, and C are collinear and ABE is complementary to CBD, then
mEBD
(1) is less than 90
(2) equals 90
(3) is greater than 90
(4) can not be determined.
3. Give a suitable reason for step 2:
(No diagram for this problem.)
Statement
1. AB  BC
2. ABC is a right angle
A
a
C
B
E L
B
D
A
C
B
Reason
1. Given
2.
Using the diagram below, draw a valid conclusion for each set of givens and give a reason.
4. Given: BE bisects ABC
B
5. Given: BAE  DCF; DAE  BCF
E
6. Given: AEFC , AE  EF ; EF  FC
7. Given: mABE + mCBE = 120; mADF = mCBE
D
N
D C
A
C
F
D
8. Given: FD bisects EC
Using the same diagram as above, write complete proofs for the following. (Note: each problem is independent
of the others.)
9. Given: BE  AE and DF  CF
Prove: AEB  CFD
10. Given: ABC  CDA; ABE  CDF
Prove: CBE  ADF
11. Given: AEFC , AE  FC
Prove: AF  EC 
12. Given: mBAE + mABE = mAEB; mAEB = 90
Prove: BAE and ABE are complementary.
Problems #13 - 15 are arithmetic/algebraic problems, not proofs.
13. In the diagram at right, L is the midpoint of HP and E is
the midpoint of HL . If EL = 12, find the length of EP.
14. In the diagram at right AOD , BOE and OC  BOE .
Find the numerical measure of AOE.
.
.
H
B
.
E
.
L
P
C
15x  59
A
x2 – 5
D
O
15. If BD bisects ABC, mABD = 2x + 5 and mABC = 5x – 6, find mCBD. (No diagram.)
E
Write a “statement-reason” geometry proof for each of the following.
16.
17. Given: RID , MIP , IR bisects BIM, IG  RID
Prove: BIG  PIG
G
I
Given: PENS , PN  IG , IG  ES
Prove: PE  NS
P
E
N
S
G
M
R
D
I
B
P
STUFF YOU SHOULD KNOW:
Vocabulary
Postulate
Theorem
Corollary
Given
Prove/proof
Statement/reason
Point
Line
Plane
Distance/length
Between
Collinear
Ray
Segment
Angle
Straight angle
Obtuse angle
Right angle
Acute angle
Congruent
Complementary
Supplementary
Adjacent
Interior/exterior
Intersect
Midpoint
Bisect/bisector
Perpendicular
Postulates
Two points determine a unique line
Every segment has exactly one midpoint
Every angle has exactly one bisector
Reflexive Postulate
Transitive Postulate
Substitution Postulate
Partition Postulate
Addition/Subtraction Postulates
Multiplication/Division Postulates
Theorems
All right angles are congruent.
If two adjacent angles form a straight angle, they are supplementary.
If two adjacent angles for a right angle, they are complementary,
If two angles are congruent, their supplements (or complements) are congruent.
If two angles are supplementary (or complementary) to the same angle, they are congruent.
Vertical angles are congruent.
If two supplementary angles are congruent, they are both right angles.
How to:
Draw simple conclusions from givens
Write a complete proof from givens to the desired conclusion
Geometry: Intro Proofs Answers
HW IGP – 1
2. AB = 80 or AB = 120
5a. Yes
b. No
6a. Yes
b. No
g. Yes
h. No
7a. LG
b. nonsense
h. nonsense i. FL
3. 90
c. Yes
c. Yes
i. Yes
c. nonsense
j. nonsense
4.
d.
d.
j.
d.
k.
9
No
Yes
Yes
nonsense
LG
e. No
f. No
e. FG
l. nonsense
f. FG
8. 29
g. nonsense
HW IGP – 2
1a. One
b. Three
c. 5
d. 5 and 6 or 2 and 3
e. No
f. C, DCB, BCD
g. 1, BAD, DAB
h. “D” could refer to ADB or BDC or ADC
h. Any angle except A, ABC or DBC
i. A or ABC
j. DBC
2. They are the same angle. 3. 130
4. 75
5. 36
6. 8 or 98
7. 35
9a. NAL
b. nonsense c. nonsense d. LGE
e. nonsense f. ELG
g. nonsense
h. nonsense
HW IGP – 3
1a. are perpendicular, then they form right angles.
b. form right angles, then they are perpendicular.
2a. is an angle bisector then it divides the angle into two congruent parts.
b. divides an angle into two congruent parts, then it is an angle bisector.
3. ACD (and/or BCD) is a right angle b/c perpendicular segments form right angles.
4. PX  XQ b/c a midpoint divides a segment into two congruent parts.
5. ABD = CBD b/c a bisector divides an angle into two congruent parts.
6. AE  EC b/c a bisector divides a segment into two congruent parts.
7. ABC is isosceles b/c it has two congruent sides.
8. C is a right angle b/c perpendicular segments form right angles
ABC is a right triangle because it contains a right angle.
9. S is the midpoint of RT b/c S divides RT into two congruent parts.
10. JL bisects KM b/c JL divides KM into two congruent parts.
11. AS  AM b/c an isosceles triangle has two congruent sides which meet at the vertex.
12. AT bisects FAR b/c AT divides FAR into two congruent pieces.
13. 12
14. 60
Review Answers
1. (4)
2. (2)
3.  segments form rt. s
4. ABE  CBE (A bisector divides an  into 2  parts.)
5. BAD  BCD (Additon Post.); DAE  BCF
6. AE  FC (Transitive Post.) or AF  EC (Addition Post.)
7. mABE + mADF = 120 (Substitution Post.)
8. EF  FC (A bisector divides a segment into 2  parts.)
9. Statement
1. BE  AE and DF  CF
2. AEB and CFD are rt. s
3. AEB  CFD
Reason
Given
 segments form rt. s
All rt. s are 
10. Statement
1. ABC  CDA
2. ABE  CDF
3. CBE  ADF
Reason
Given
Given
Subtraction Post. (1, 2)
11. Statement
1. AEFC
2. AE  FC
3. EF  EF
4. AF  EC
Reason
Given
Given
Reflexive Post.
Addition Post. (1, 2)
12. Statement
1. mBAE + mABE = mAEB
2. mAEB = 90
3. mBAE + mABE = 90
4. BAE and ABE are complementary.
13. 36
14. 134
15. 37
16. Statement
1. PENS
2. PN  IG
3.
4.
5.
6.
Reason
Given
Given
Substitution Post (1, 2)
Two s that sum to 90 are complementary
IG  ES
PN  ES
EN  EN
PN – EN  ES – EN
PE  NS
or
Reason
1. Given
2. Given
3.
4.
5.
6.
17. Statement
1. RID , MIP
2. DIP and RIM are vert. s
3. DIP  RIM
4. IR bisects BIM
5. RIM  RIB
6. RIB  DIP
7. IG  RID
8. RIG and DIG are rt. s
9. RIG  DIG
10. RIB + RIG  DIP + DIG

or BIG  PIG
Given
Transitive Post. (2, 3)
Reflexive Post.
Subtraction Post. (4, 5)
Reason
1. Given
2. Intersecting segments RID and MIP form vert. s
3. Vert. s are 
4. Given
5. A bisector divides an  into 2  parts
6. Transitive Post. (3, 5)
7. Given
8.  segments form rt. s
9. All rt. s are 
10. Addition Post. (6, 9)