Math B Assignments: Introduction to Proofs



Geometry Assignments: Introduction to Geometry Proofs

|Day |Topics |Homework |HW Grade |Quiz Grade |

|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 | | |

| |***TEST*** | | | |

Geometry Homework: Intro Geo Proofs - 1

Name

1. Construct equilateral (JKL having [pic]as one side.

2. Let A be one vertex of a rhombus with two sides lying along rays

AY and AZ. Let the length of each side be congruent to [pic].

Construct this rhombus. (Recall: A rhombus has all four sides

congruent.

3. Triangle DOG is the image of (FOX after a dilation by

a scale factor of 3 centered at O. This means that point

O will not move but that all sides of (FOX will become

3 times as long. Construct (DOG.

4. Draw a single diagram to illustrate the following givens: [pic], [pic].

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.

5. If M is the midpoint of [pic], AM = x2 + 24 and MB = 10x, find the length of [pic].

6. [pic] bisects [pic] at Q. PQ = 4x + 12, QR = 9x – 13, SQ = 6x – 5 and QT = 3x + 16. Find the length of [pic].

7. Given: [pic], A is the midpoint of [pic], MH = 21 and AH = 15. Find TH.

8. In [pic], 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) [pic] 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.

[pic] [pic] nonsense [pic]nonsense

[pic]nonsense

2) [pic]and [pic] 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.

[pic] [pic] nonsense [pic] nonsense

[pic] [pic] nonsense [pic] nonsense

9. Based on the diagram at right, tell if each of the following is True or False. Remember the difference between [pic]and AB.

a. AB + BC = CP b. [pic]

c. AB + BC = AC d. [pic]

e. AC ( BC = AB f. [pic]

g. PC ( PB = CD h. [pic]

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

line segment or write “nonsense.”

a. [pic] ______ b. [pic] ______ c. [pic] ______

d. [pic] ______ e. [pic]______ f. [pic]______

g. [pic]______ h. [pic]______ i. [pic]______

j. [pic] ______. k. [pic] ______ l. [pic] ______

8. READ and LEARN the definitions on the first page and a quarter of Notes IGP – 2.

Geometry Homework: Intro Geo Proofs - 2

Name

1. Construct ΔABC on [pic]such that (B ( (A.

2. In ΔDEF, construct the bisector of (E.

3. A median of a triangle is a segment joining one vertex of the triangle to the midpoint of the opposite side. In ΔGHI, locate the midpoint of side HI by construction and then draw the median from G.

4. 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.

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?

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

(PAQ or (RAS?

6. In the diagram at right, [pic], [pic]⊥ [pic], and m∠POQ = 40. Find m∠NOR.

7. The measures of two supplementary angles are in the ratio 5:7. Find the measure of the smaller angle.

8. 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?

9. If [pic] bisects (BEG, m(BET = x2 and m(GET = 5x + 14, find the numerical measure of (BEG.

10. If [pic] bisects (BOT, m(BOY = 3x + 8 and m(BOT= 8x – 2, find the numerical measure of (TOY.

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.

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

11. 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

12. 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 = ________

Geometry Homework: Intro Geo Proofs - 3

Name

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: [pic] ⊥ [pic]

Conclusion:

Reason:

4. Given: X is the midpoint of [pic].

Conclusion:

Reason:

5. Given: [pic] bisects ∠ABC.

Conclusion:

Reason:

6. Given: [pic] bisects [pic]at E.

Conclusion:

Reason:

7. Given: [pic] ” [pic]

Conclusion:

Reason:

8. Given: [pic] ⊥ [pic].

1st Conclusion:

Reason:

2nd Conclusion:

Reason:

9. Given: [pic]and [pic] ” [pic].

Conclusion:

Reason:

10. Given: [pic] divides [pic]into two congruent parts.

Conclusion:

Reason:

11. Given: A is the vertex of isosceles triangle SAM

Conclusion:

Reason:

12. Given: ∠FAT ” ∠RAT

Conclusion:

Reason:

13. Given [pic], N is the midpoint of [pic], LE = 30 and NE is three less than LI. Find the numerical length of LI.

14. In the diagram at right, [pic] 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.

Geometry HW: Intro Geo Proofs - 4

Name

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

1. Given: [pic] ” [pic], [pic] ” [pic]

Conclusion: [pic] ” [pic]

Reason:

2. Given: (Diagram at right)

Conclusion: m∠DBE = m∠4 + m∠2 + m∠5

Reason:

3. Given: (Diagram at right)

Conclusion: [pic] ” [pic]

Reason:

4. Given: m∠1 + m∠2 = 180°, m∠2 = m∠3 (Diagram at right)

Conclusion: m∠1 + m∠3 = 180

Reason:

For the following, give a valid conclusion and a reason.

5. Given: m(1 + m(2 = 180; m(3 = m(1.

Conclusion:

Reason:

6. Given: [pic]bisects ∠UAD.

Conclusion:

Reason:

7. Given: m∠AOB = 90.

Statement: m(AOB = m(AOX + m(XOB

Conclusion:

Reason:

Conclusion:

Reason:

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. [pic] b. [pic]

c. [pic]nothing (why?) d. [pic]nothing (why?)

e. [pic] nothing (why?) f. [pic]nothing (why?)

g. [pic]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?)

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

a. [pic] b. [pic] .

c. [pic] d. [pic] .

e. [pic] f. [pic] .

g. [pic] h. [pic] .

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 = .

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

11. If [pic]is the perpendicular bisector of [pic], HO = 2x + 1, OT = 3x – 2,

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

Geometry HW: Intro Geo Proofs - 5

Name

For each of the following givens, state a valid conclusion based on the postulates we have covered and tell what postulate was used.

1. Given: [pic] ” [pic], [pic] ” [pic].

Conclusion:

Reason:

2. Given: [pic], [pic], [pic] ” [pic], [pic] ” [pic].

Conclusion:

Reason:

3. Given: ∠ABC ” ∠ACB, ∠ABD ” ∠ACD

Conclusion:

Reason:

4. Given: ∠ABE ” ∠CDE, ∠CBE ” ∠ADE

Conclusion:

Reason:

5. Given: [pic], [pic], [pic] ” [pic], [pic] ” [pic].

Conclusion:

Reason:

6. Given: ∠BAD ” ∠CAD, ∠BAD ” ∠FAE

Conclusion:

Reason:

Probems #7 – 9 are simple “statement-reason” geometry proofs. For each one, fill in the missing reasons with appropriate postulates.

7. 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.

8. Given: [pic], [pic] ” [pic]

Prove: [pic] ” [pic]

Statement Reason

1. [pic] 1. Given

2. [pic] ” [pic] 2. Given

3. [pic]” [pic] 3.

4. [pic]+ [pic]” [pic]+ [pic] 4.

or [pic] ” [pic]

9. Given: ∠KJM ” ∠NJL

Prove: ∠KJL ” ∠MJN

Statement Reason

1. ∠KJM ” ∠NJL 1. Given

2. ∠LJM ” ∠LJM 2.

3. ∠KJL ” ∠MJN 3.

10. In the diagram at right, [pic], m(ABD = 3x + 17 and m(CBD = 5x – 3.

Find the value of x.

11. What is the measure of the supplement of an angle that measures x degrees?

Geometry HW: Intro Geo Proofs - 6

Name

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 [pic], [pic], and [pic]for all eight problems.

1. Given: [pic]bisects ∠RBS.

Conclusion/Reason:

2. Given: [pic]” [pic].

Conclusion/Reason:

3. Given: ∠BAT ” ∠BAG,

∠RAT ” ∠SAG

Conclusion/Reason:

4. Given: [pic] ” [pic]

Conclusion/Reason:

5. Given: [pic] ” [pic],

[pic] ” [pic].

Conclusion/Reason:

6. Given: ∠RAT ” ∠ATR, ∠ATR ” ∠TRA

Conclusion/Reason:

7. Given: ∠BAR is a right angle.

Conclusion/Reason:

8. Note: draw [pic]on the diagram and label its intersection with [pic]as point M.

Given: [pic]bisects [pic]at M.

Conclusion/Reason:

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 Reason

1. ∠A is supplementary to ∠Z 1. Given

∠B is supplementary to ∠Z

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. m∠Z = m∠Z 5.

6. m∠A = m∠B or ∠A ” ∠B 6.

10. Given: [pic]⊥ [pic]

Prove: ∠ROT is complementary to ∠NOT

Statement Reason

1. [pic]⊥ [pic] 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.

11. In the diagram at right, [pic], and [pic]⊥ [pic],

m(DOC = x2 + 15 and m(AOB = 20x ( 81.

a. Find m(BOC.

b. Find the value of x.

c. Find m(DOE.

d. Find m(AOE.

Geometry HW: Intro Geo Proofs - 7

Name

1. Fill in appropriate reasons in the proof below.

Given: ∠AFE ” ∠BFD.

Prove: ∠AFD ” ∠BFE

Statement Reason

1. ∠AFE ” ∠BFD 1.

2. ∠DFE ” ∠DFE 2.

3. ∠AFE – ∠DFE ” ∠BFD – ∠DFE 3.

or ∠AFD ” ∠BFE

2. Write a complete “statement-reason” proof .

Given: [pic], [pic] ” [pic].

Prove: [pic] ” [pic]

Statement Reason

3. Fill in appropriate reasons in the proof below.

Given: [pic] is an angle bisector of ΔABC, ∠DBC ” ∠DCB

Prove: ∠DBA ” ∠DCB

Statement Reason

1. [pic] is an angle bisector of ΔABC 1.

2. ∠DBA ” ∠DBC 2.

3. ∠DBC ” ∠DCB 3.

4. ∠DBA ” ∠DCB 4.

4. Write a complete “statement-reason” proof .

Given: E is the midpoint of [pic], [pic]” [pic]

Prove: ΔABE is isosceles

Statement Reason

5. Given: ∠A is a right angle; ∠B is a right angle

a. Write a brief explanation of why ∠“ ” ∠Β. 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 (.

Geometry HW: Intro Geo Proofs - 8

Name

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. We wish to prove the following theorem: Vertical angles are congruent.

Given: [pic] and [pic]

Prove: (AEC ( (BED

a. Draw a diagram.

b. Write a proof of the theorem in paragraph form. (There is more than one way to do this proof. 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 #3 – 5.

3. Given: [pic], ∠ABG ” ∠DCG

Prove: ∠CBG ” ∠BCG

4. Given: [pic] ⊥ [pic], [pic] ⊥ [pic]

Prove: ∠BAE ” ∠FAC

5. Given: [pic], [pic], [pic] bisects ∠PIG

Prove: (NIT ( (WIT

The following are algebraic exercises; not proofs.

6. If [pic]intersects [pic]at E, m∠BEC = 5x – 25, and m∠DEA = 7x – 65,

find the numerical values of the measures of all four angles.

7. If [pic]intersects [pic]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

Name

Determine if each conclusion and reason is True or False. If false, change the conclusion and/or the reason (not the given).

1. Given: [pic] bisects ∠ABC

Conclusion: ∠BAD ” ∠BCD because a bisector divides an

angle into two congruent parts

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.

3. Given: [pic] intersects [pic] at E

Conclusion: [pic] ” [pic] because a bisector divides a segment

into 2 ” parts

Write a complete geometry proof for each of #4 - 6:

4. Given: [pic], B is the midpoint of [pic], [pic] ” [pic] (Draw your own diagram.)

Prove: [pic]

5. Given: ΔABC with right ∠ACB, [pic]⊥ [pic], (ACD ((EDC.

Prove: ∠ECD ” ∠EDB

6. Given: (BAD ( (FAD, [pic], [pic]

Prove: [pic] bisects (CAE

Geometry HW: Intro Geo Proofs - 10

Name

4. a. Carefully construct rhombus ABCD on [pic] below. Let side AD be along the ray shown at A.

b. Carefully construct the angle bisector of (A. (If you do a good job, your angle bisector should pass

through point C.)

c. Carefully construct the perpendicular bisector of diagonal AC. (If you do a good job, the perpendicular bisector should be diagonal BD.)

d. List three things these constructions show about the diagonals of a rhombus.

Write complete geometry proofs for each of the following.

2. Given: [pic], M is the midpoint of [pic], [pic] bisects [pic]

Prove: [pic]

3. Given: [pic] ⊥ [pic], [pic] ⊥ [pic]

[pic] bisects ∠PAN.

Prove: ∠CAT ” ∠RAT

4. Given: [pic], [pic], (PEA is a right angle, [pic], (NEA ( (NAE

Prove: (PEN ( (PAL

Geometry HW: Intro Geometry Proofs - Review

Name

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) [pic] ” [pic]

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: Statement Reason

(No diagram for this problem.) 1. [pic] ⊥ [pic] 1. Given

2. ∠ABC is a right angle 2.

Using the diagram below, draw a valid conclusion for each set of givens and give a reason.

4. Given: [pic] bisects ∠ABC

5. Given: ∠BAE ” ∠DCF; ∠DAE ” ∠BCF

6. Given: [pic], [pic] ” [pic]; [pic] ” [pic]

7. Given: m∠ABE + m∠CBE = 120; m∠ADF = m∠CBE

8. Given: [pic] bisects [pic]

Using the same diagram as above, write complete proofs for the following.

(Note: each problem is independent of the others.)

9. Given: [pic] ⊥ [pic]and [pic] ⊥ [pic]

Prove: ∠AEB ” ∠CFD

10. Given: ∠ABC ” ∠CDA; ∠ABE ” ∠CDF

Prove: ∠CBE ” ∠ADF

11. Given: [pic], [pic] ” [pic]

Prove: [pic] ” [pic]

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 [pic]and E is

the midpoint of [pic]. If EL = 12, find the length of EP.

14. In the diagram at right [pic], [pic] and [pic] ⊥ [pic].

Find the numerical measure of ∠AOE.

15. If [pic] bisects ∠ABC, m∠ABD = 2x + 5 and m∠ABC = 5x – 6, find m∠CBD. (No diagram.)

Write a “statement-reason” geometry proof for each of the following.

16. Given: [pic], [pic] ” [pic], [pic] ” [pic]

Prove: [pic] ” [pic]

17. Given: [pic], [pic], [pic] bisects ∠BIM, [pic]⊥ [pic]

Prove: ∠BIG ” ∠PIG

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 3. 90 4. 9 5. 29

6a. Yes b. No c. Yes d. Yes e. Yes f. Yes g. Yes h. No

g. Yes h. No i. Yes j. Yes

7a. [pic] b. nonsense c. nonsense d. nonsense e. [pic] f. [pic] g. nonsense

h. nonsense i. [pic] j. nonsense k. [pic] l. 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. “(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

8a. T b. F c. T d. T e. T f. F g. T h. T

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. [pic] 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. [pic] 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 [pic] b/c S divides [pic] into two congruent parts.

10. [pic] bisects [pic] b/c [pic] divides [pic] into two congruent parts.

11. [pic] b/c an isosceles triangle has two congruent sides which meet at the vertex.

12. [pic] bisects (FAR b/c [pic] 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. [pic] ” [pic] (Transitive Post.) or [pic] ” [pic] (Addition Post.)

7. m∠ABE + m∠ADF = 120 (Substitution Post.)

8. [pic] ” [pic] (A bisector divides a segment into 2 ” parts.)

9. Statement Reason

1. [pic] ⊥ [pic]and [pic] ⊥ [pic] Given

2. ∠AEB and ∠CFD are rt. ∠s ⊥ segments form rt. ∠s

3. ∠AEB ” ∠CFD All rt. ∠s are ”

10. Statement Reason

1. ∠ABC ” ∠CDA Given

2. ∠ABE ” ∠CDF Given

3. ∠CBE ” ∠ADF Subtraction Post. (1, 2)

11. Statement Reason

1. [pic] Given

2. [pic] ” [pic] Given

3. [pic] ” [pic] Reflexive Post.

4. [pic] ” [pic] Addition Post. (1, 2)

12. Statement Reason

1. m∠BAE + m∠ABE = m∠AEB Given

2. m∠AEB = 90 Given

3. m∠BAE + m∠ABE = 90 Substitution Post (1, 2)

4. ∠BAE and ∠ABE are complementary. Two (s that sum to 90 are complementary

13. 36 14. 134 15. 37

16. Statement Reason

1. [pic] 1. Given

2. [pic] ” 2. Given

3. [pic] ” [pic] 3. Given

4. [pic] ” [pic] 4. Transitive Post. (2, 3)

5. [pic] ” [pic] 5. Reflexive Post.

6. [pic] – [pic]” [pic] – [pic] 6. Subtraction Post. (4, 5)

or [pic] ” [pic]

17. Statement Reason

1. [pic], [pic] 1. Given

2. ∠DIP and ∠RIM are vert. ∠s 2. Intersecting segments [pic]and [pic]form vert. ∠s

3. ∠DIP ” ∠RIM 3. Vert. ∠s are 2

4. [pic]bisects ∠BIM 4. Given

5. ∠RIM ” ∠RIB 5. A bisector divides an ∠ into 2 ” parts

6. ∠RIB ” ∠DIP 6. Transitive Post. (3, 5)

7. [pic] ⊥ [pic] 7. Given

8. ∠RIG and ∠DIG are rt. ∠s 8. ⊥ segments form rt. ∠s

9. ∠RIG ” ∠DIG 9. All rt. ∠s are ”

10. ∠RIB + ∠RIG ” ∠DIP + DIG 10. Addition Post. (6, 9)

or ∠BIG ” ∠PIG

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