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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 mPOQ = 40. Find mNOR. 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, 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. (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); mABC 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 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 = ________ 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, mABD = 66 – 2x and mCBD = 3x – 24. Find the numerical value (a number, not just an algebraic expression) of mABC. 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: mDBE = m4 + m2 + m5 4 1 A Reason: E 5 2 3 C A 3. Given: (Diagram at right) Conclusion: AT AT Reason: F R T 4. Given: m1 + m2 = 180°, m2 = m3 (Diagram at right) Conclusion: m1 + m3 = 180 1 Reason: 2 3 For the following, give a valid conclusion and a reason. 2 5. Given: m1 + m2 = 180; m3 = m1. 3 1 Conclusion: Reason: U 6. Given: QA bisects UAD. Conclusion: Q A Reason: D 7. Given: mAOB = 90. Statement: mAOB = mAOX + mXOB 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: ABCACB, ABDACD Conclusion: D Reason: C B 4. Given: ABECDE, CBEADE 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: mKJL + mLJM = 90, mKJL = mMJN Prove: mMJN + mLJM = 90 Statement Reason 1. mKJL + mLJM = 90 1. Given 2. mKJL = mMJN 2. Given 3. mMJN + mLJM = 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 , mABD = 3x + 17 and mCBD = 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: BATBAG, RATSAG 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: AB Statement 1. A is supplementary to Z B is supplementary to Z Reason 1. Given 2. mA + mZ = 180 2. 3. mB + mZ = 180 3. (same as #2) 4. mA + mZ = mB + mZ 4. 5. 5. mZ = mZ 6. mA = mB or AB 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. mRON = 90 3. 4. mRON = mROT + mNOT 4. 5. mROT + mNOT = 90 5. 6. ROT is complementary to NOT 6. O In the diagram at right, AOD , and OC BOE , mDOC = x2 + 15 and mAOB = 20x 81. a. C B Find mBOC. A b. N Find the value of x. D O E c. Find mDOE. d. Find mAOE. 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, mBEC = 5x – 25, and mDEA = 7x – 65, find the numerical values of the measures of all four angles. 7. If AEB intersects CED at E, mAEC = 5(x + 15), and mAED = 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: m1 + m2 = 90 (No diagram for this problem.) m3 + m4 = 90 Conclusion: m1 + m2 = m3 + m4 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 ma:mb:mc = 2:3:4, find the numerical value of ma. 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) mAEB = mCED (2) AEC BED (3) mAEC = mBED (4) AE ED 2. If A, B, and C are collinear and ABE is complementary to CBD, then mEBD (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: mABE + mCBE = 120; mADF = mCBE 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: mBAE + mABE = mAEB; mAEB = 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, mABD = 2x + 5 and mABC = 5x – 6, find mCBD. (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. mABE + mADF = 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. mBAE + mABE = mAEB 2. mAEB = 90 3. mBAE + mABE = 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)