Chapter 1 Points, Lines, Planes and Angles

GEO 9 CH1-2.2 1CH 1-2.2 ASSIGNMENT SHEETGEOMETRY 9DAY SECTION NAME PAGE ASSIGNMENT1 Algebra Review/Assignment #1 Handout2 Algebra Review/Assignment #2 Handout3 1.2 Points, Lines, Planes p. 7-8 1-10 all,15,16,17,21,25Supplementary Problems CH 1-2.2 # 1 - 63 1.3 Segments Rays Distances p.15-16 3,5-22all, 32,36,37,39Supplementary Problems CH 1-2.2 #7 - 94 1.4 Angles p.21-22 15,17,29-34,36Supplementary Problems CH 1-2.2 # 105 1.5 Postulates & Theorems p.25 3-8all, 13-162.1 "If - Then" p.35 1-4, 17,19,22BRING IN FLASHCARDS TO CLASS SECTION 2.26 2.2 Properties from Algebra p. 40 classroom 11, 12p. 41 written 6 - 10 all8 2.2 Properties from Algebra Packet # 6 - 109 REVIEW PacketSupplementary Problems CH 2.2-2.6 #1, 2, 310TEST CH 1.2 - 2.2

Geo 9 Ch 1-2.2 2

Geo 9 Ch 1-2.2 3Chapter 1 Points, Lines, Planes and AnglesIn Geometry we must start with the basics and build our ideas using definitions, postulates and theorems.We may not assume anything and prove every idea to be true.A. Undefined terms: points, lines, planesB. Definitions - State the meaning of a concept. Definintions are reversible. Definitions contain theleast possible amount of information.Ex. If a point cuts a segment into two equal lengths, then it is called a midpoint.If a midpoint is on a segment, then it cuts the segment into 2 equal lengths.C. Postulates or axioms are things we accept to be true. Not necessarily reversible.Ex.A BAB + BC = ACCD. Theorems – things we prove to be trueEx. If you have a rectangle, then the diagonals are the same length.

Geo 9 Ch 1-2.2 4Lesson 1.2 - Points, Lines and PlanesPoints, lines and planes are intuitive ideas that are accepted without definition. Theseterms are then used in the definitions of other terms.Point - Graph (-4, 0) labelLine - Graph (6,0) on the same graph -2 pts determine a linePlane - Graph ( 1, 5 )3 points determine a planeAdd point ( 7, 4 ) Create 2 intersecting lines4 Ways to Determine a plane

Geo 9 Ch 1-2.2 7Get into groups and find the following :PEx 1) The ray opposite to EG is _______2) The length of MG is ________.3) The distance between R and E is _______G-2E-1O0M1TR2 3Y44) The midpoint of GY is _____________5) The coordinate of midpoint of GY is ___________6) You are told that segment AB, notation AB, is 10 cm, segment AC is 3 cm,how long is segment BC ? You might have to think about this a little. A picture woulddefinitely help.7) Draw two segments, AB and CD for which the intersection of segments AB and CD is theempty set, but the intersection of lines AB and CD is exactly one point.8) Draw two segments, PQ and RS so that their intersection is the empty set but thelines, PQ and RS are the same.

Geo 9 Ch 1-2.2 8A. Segment Addition Postulate _____ _____ _____If B is between A and C, thenDraw a picture:AB +BC = AC1) If AB = 6 andBC = 8, then AC =2) If AC = 12, AB = 15, BC = 3, which point is between the other two?3) If L is between P and Q, and PL = 6x-5, LQ = 2x + 3, and PQ = 30? What is x?4) Given the following points, A, B, C, find the distances AB, BC and AC.A ( -2, -10 ), B ( 2, 2 ) C ( 4, 8 )

Geo 9 Ch 1-2.2 9B. Angle Addition Postulate ______ ______ _____If ray OB is between ray OA and ray OC , then the m

6)Geo 9 Ch 1-2.2 10RECTANGLE TSRPTSOPRa) If TPO=60 , how large is RPO? a) ___________b) If PTO=70 , how large is STO? b) ___________c) If TOP=50 , how large is POR? c) ___________7) If CBD DBE and BD bisects CBE, find m A ( CAB) 7) ___________CDx+10Ax+560BE8) 1 2; m 1 = x+14; m 2 = x 2 - 4x 8) ___________Solve for x.219) m ABD = 3x; m DBC = x; find m ABD. 9) ___________DA B C

Geo 9 Ch 1-2.2 1110) m FGJ = 3x - 5; m JGH = x + 27; GJ bisects FGH. Find m FGJ. 10) ___________FJGH11) m ABC = 90 ; m 1 = 2x + 10; m 2 =x+20; m 3=3x 11) ___________C123BA12) Has ABC been trisected?

Geo 9 Ch 1-2.2 121.5 Postulates and theorems relating to points, lines and planes.Group tables and go over homework. Then move tables to an oval.Postulate 5 : A line contains at least _______ points; a plane contains at least ________ points notall in one line; space contains at least ________ points not all in one plane.Postulate 6: Through any ______ points there is exactly one line.Postulate 7: Through any ______ points there is at least one plane, and through any ______ pointsthere is exactly one plane.Postulate 8: If two points are in a plane, then the ___________that contains the points is in thatplane.Postulate 9: If two planes intersect, then their intersection is a _______________________.

Geo 9 Ch 1-2.2 13Theorem 1-1: If two lines intersect, then they intersect in exactly ______________Theorem 1-2: Through_a line and a point not in the line there is exactly _______________________Theorem 1-3: If two lines intersect, then exactly ____________________ contains the lines.Theorem 1.4:If 2 lines are parallel, then exactly ___________contains them.

Geo 9 Ch 1-2.2 15Ch 12.1 Conditional StatementsObjectives:1) Recognize the hypothesis and the conclusion of an if-then statement.2) State the converse of an if-then statement.3) Understand the meaning of if-and-only-if.Conditional Statements :hypothesisconclusionIf _____________________, then _______________________.A conditional statement is one that states an assertion, usually called the hypothesis, based on agiven condition. It is usually in the form “if (given/ hypothesis)…., then (prove/conclusion).”, butcan take on other forms.hypothesisconclusiongiven or understood informationformed from the given informationex. “ If I live in Martinsville, then I live in New Jersey.”hypothesisconclusionex. “ If two angles sum is 180 , then they are supplementary.”ex. “ An angle is called a right angle if its measure is 90 .We take information that is given to us and then make conclusion upon conclusion until we get towhere we are going.ex. If I live in Martinsville, then I live in Somerset County.If I live in Somerset County , then I live in New Jersey.If I live in NJ, then I live in the United StatesIf I live in the United States, then I live in North Americaex. If the figure is a parallelogramthen the diagonals bisect each other.Converse: Is formed by interchanging the hypothesis and the conclusionex. If I live in New Jersey, then I live in Martinsville.Notice,the converse is not necessarily true!ex. If a figure is a square, then it is a quadrilateral.If a figure is a quadrilateral, then it is a square.Biconditional: “ If and only if”. They are reversible.ex. If a polygon is a quadrilateral, then it has four sides.If a polygon has four sides, then it is a quadrilateralGroups

Geo 9 Ch 1-2.2 16ALL DEFINITIONS ARE BICONDITIONAL, NOT ALL THEOREMS!2.1 IF ---> THEN statements. Complete the following and finish for homework if necessary.1. If 1=90 , then 1 is __________________________________________________.2. If two angles have the same degree measure, then __________________________3. State the converse of #1 and #2__________________________________________________________________________________________________________________________________4. Turn this statement into a conditional statement and then it’s converse.“ All right angles are congruent.”______________________________________________________________________________________________________________________________________________

Geo 9 Ch 1-2.2 172-2 Proof Properties Memorize SOON!!!!!Properties of EqualityMake file cards1. ADDITION PROPERTYIf a = b and c = d, then __________________________________________________2. SUBTRACTION PROPIf a = b and c = d, then ___________________________________________________3. MULTIPLICATION PROPIf a = b, and c exists, then ________________________________________________4. DIVISION PROPIf a = b, and c5. SUBSTITUTION0, then _________________________________________________If a = b, then either may replace the other in any equation.6. REFLEXIVEa = a7. SYMMETRIC PROPIf a = b, then __________________________________________________________8. TRANSITIVE PROPIf a = b, and b = c, then _________________________________________________Properties of Congruence1. REFLEXIVE PROP: DE DE

Geo 9 Ch 1-2.2 18In Algebra, you have learned so solve an equation by “balancing” while solving for x.Give reasons, using your past or present text, for the following steps in solving the algebraic equation.1) 2( x+ 1) = 5x – 3 1) Given2) 2x + 2 = 5x – 3 2) _____________________________3) 2x + 2 + (-2) = 5x – 3 + (-2) 3) _____________________________4) 2x = 5x – 5 4) _____________________________5) 2x – 5x = 5x – 5x – 5 5) ______________________________6) -3x = -5 6) ______________________________7) (- 31 )(- 3x) = (- 31 ) (– 5) 7) ______________________________8) x = 358) ______________________________

Geo 9 Ch 1-2.2 19Geo 2.2 Properties from Algebra*Elements of Two-Column ProofsRSPGiven: RS = PS; ST = SQProve: RTPQQTSTATEMENTSREASONS1) RS = PS; ST = SQ 1) Given2) RS + ST = QS + SP 2) __________________________________3) RS + ST = RT 3) __________________________________QS + SP = QP*4) RT = QP 4) ___________________________________5) RT PQ 5) ___________________________________

Geo 9 Ch 1-2.2 20Lets try a geometry proof: The first step is ALWAYS to mark your drawing according to the giveninformation. For instance, if segments are given congruent, MARK them congruent with tic marks!!A B C DEFGiven:Prove:AB DE, BC EF***(WARNING!)AC DFStatementsReasons*1. AB DE, BC EF 1.(what allows me to make this statement?)2. AB = DE; 2.BC = EF (why did I line it up like this?)3. AB + BC = DE + EF (why can I say this?) 3.4. AB + BC = AC; DE + EF = DF 4.( Uh oh, where did this come from?)5. AC = DF (so this is the same as…?) 5. (have I proved what is asked for?)6. AC DF6.Another, slightly different problem.Given: ABPr ove : ACCDBDA B CDStatementReasons1. AB CD1.*2. AB = CD 2.3. BC = BC 3. (isn’t this obvious?)4. AB + BC = BC + CD (here we go again!) 4.5. AB + BC = AC; BC + CD = BD 5.6. AC = BD 6.7. AC BD7.

How am I going to go from #1 to #7?Geo 9 Ch 1-2.2 21Now, lets try the reverse:Given:ACBDA B CDPr ove : ABCD(WARNING!)StatementsReasons1. AC BD1.*2. AC = BD (candy bar) 2.3. AB + BC = AC; BC + CD = BD 3.(breaking into pieces)4. AB + BC = BC + CD 4.(why do I need to put this in?)4. BC = BC 5.*6. AB = CD 6.7. AB CD7.The pattern for adding is: ( Small to large )1) ______________________________2) ______________________________3) ______________________________The pattern for subtracting is: ( Large to small )1) ______________________________2) ______________________________3) ______________________________Use definition of congruence on either end of the proof if needed.Everything for Add= and Sub = must be in the equality sign!

Geo 9 Ch 1-2.2 22Geo2-2Proofs in Groups/HW1) Given: GJ HKProve: GHJK*WARNING!MStatementsG H J KReasonsuse cards to recognize reasons1. GJ HK 1. Given2. GJ = HK ( large or small?) 2.3. GJ = GH + HJ 3.HK = HJ + JK4. GH + HJ = HJ + JK 4.5. HJ = HJ 5.6. GH = JK 6.7. GH JK 7.pattern2)HGsame proof except with angles.. E FGiven GHF HGEFHE EGFProve GHE HGFWARNING!StatementsReasons1. 1. Given2. m GHF = m HGE, m EHF = m FGE 23. 3. Addn Prop of =4. 4. AAP5. 5. Substitution6. EHG FGH6.

Geo 9 Ch 1-2.2 23A B C3)D E FGiven AC DFAB = DEProve BC EFStatementsReasons1. 1. Given2. 2. SAP3. AB + BC =DE + EF 3.4. 4. Sub Prop of =4.SRP3 41 2QGiven 1 23 4TThink about the big idea here. What is the pattern?StatementsReasons1. 1. GivenProve SRT STR 2. 2.3. 3.4. 4.5. 5.*6. SRT STR 6. Def of

Geo 9 Ch 1-2.2 24Geo 2.25. Statements ReasonsSThis is the same diagram. Am I doing the same thing?PQR3 41 2TGivenRPPSTQQSProveRSTS6. Statements ReasonsSPZQR3 41 2TGiven RQTPZQ = ZPProve RZTZ7. Statements ReasonsSPQR3 41 2TGivenSRT3 4STRProve 1 2

Geo 9 Ch 1-2.2 258. Statements Reasons321Given m 1+m 2 = 902 3Prove m1 m 3 909.B C DAFEGivenABDABFDEADECProveFBCCEAStatementsReasonsGeo 2.2

10.Geo 9 Ch 1-2.2 26EA B C DGiven : AEB DECProve: AEC DEBStatementsReasons

Geo 9 Ch 1-2.2 27WorksheetPoints, Lines and Planes(1) Refer to the diagram:DCABHGEFa) Name 2 planes that intersect in HG . ____________b) Are the points A, B, C and D collinear? ____________c) Are the points A, B, C and D coplanar? ____________d) Name 2 planes that do not intersect. ____________e) Name 3 lines that intersect at C. ____________(2.)J K L M N-4 1 3 5 7a) The ray opposite to KN is ____________ b) Another name for LM is ___________c) LN= ____________ d) The coordinate of the midpoint of JM is ____________(3)S T E P-9 4a) If TE = .5x and EP = x then x =________.b) The coordinate of E = _____________c) If T is the midpoint of SP, find the coordinate of S . _____________

Geo 9 Ch 1-2.2 28(4) Make a sketch showing the relative position of the four points mentioned in thefollowingstatement:a) Line XZ contains the points Y and V, but segment XY contains neither Y or V.b) V lies on ray XZ, but Y does not. YZ + ZV = YV.(5) If A, B and C are three points on a line such that AC + BC = AB, what is the intersection of;a) ray CB and ray BA ?b) ray AC and ray AB ?c) ray CA and ray CB ?BE6040(6) a) An angle adjacent to ADB is _________. A30DCb) Are A, B, and E collinear? _________c) Can you conclude from the diagram that BE BD? _______d) What postulate allows you to say m ABD + m DBC = m ABC? ___________________e) m CBE = _______.f) m BCD = _______.g) m BDA = _______.PQ(7) Refer to the diagram. Ray OR is a bisector of QOSa) If m 1=2x+15 and m 2=5x-8 then x=O321RSb) If m 1=x+7 and m 3=2x then x=

Geo 9 Ch 1-2.2 29(8) Name the definition or postulate that justifies each statement, given the markings on thediagram.RQTa) m RSQ + m QST = m RST. __________________________________________b) SQ bisects RT __________________________________________c) Q is the midpoint of RT __________________________________________d) RT = RQ + QT __________________________________________e) Are R, Q and T collinear? __________________________________________Use sometimes, always or never.(9) a) Adjacent angles are ___________ congruent.b) Two intersecting lines ___________ lie in exactly one plane.c) A line and a point not on the line ___________ lie in more than one plane.SEB and EC trisect ADA(10)AB = 7x + 3AC = 11y 7CD = 8x + 2y 10Find ADEBCD(11) In the figure, is < QPS acute, obtuse or right?Justify your anwer.EP2X+10X+25Q3XR5X+20S

Geo 9 Ch 1-2.2 30Ch 1-2.1 Geometry WorksheetFERefer to the figure to the right.Given:

Geo 9 Ch 1-2.2 31Ch 1-2.2 Geometry Review WorksheetAA(3) (4)BEBEFCDCGiven: AB = AE Given: m 1 = m 3AC = AD m 2 = m 41234DProve: BC = DE Prove: m ACD = m ADC(6)A B C D EProve: AB + BC + CD + DE = AE

Geo 9 Ch 1-2.2 32CH 1 – 2.2DEFINITIONS POSTULATES PROPERTIESDefined terms: DAY 11. collinear __________________________________________________________________________________________________2. non-collinear __________________________________________________________________________________________________3. coplanar __________________________________________________________________________________________________4. segment __________________________________________________________________________________________________5. ray _________________________________________________opposite rays_________________________________________________6. distance _________________________________________________7. congruent _________________________________________________8. midpoint _________________________________________________9. bisector _________________________________________________segmentangle__________________________________________________________________________________________________10. angle vertex _________________________________________________obtuse angleright angleacute angle___________________________________________________________________________________________________________________________________________________straight angle_________________________________________________

Geo 9 Ch 1-2.2 3311. adjacent angles _________________________________________________12. supplementary _________________________________________________13. complementary _________________________________________________14. vertical _________________________________________________15. perpendicular _________________________________________________16. congruent _________________________________________________17. congruent segments _______________________________________________18. SAP _________________________________________________19. AAP __________________________________________________20) Add = __________________________________________________21) Sub = __________________________________________________22) Div = __________________________________________________23) Mult = __________________________________________________24) Reflexive __________________________________________________25) Transitive __________________________________________________

Geo 9 Ch 1-2.2 34SUPPLEMENTARY HOMEWORK: CH 1-2.2Do your HW in a graph paper notebook1) The distance from (0,0) to (8,6) is exactly 10. Find other examples of points that are exactly 10units from (0,0). Using a graph will help. How do you think you can use points to find a distancebetween them? What if you moved the triangle so the points are (2,1) and (10, 7)?2) What do you think is the difference between the perpendicular bisector of a segment and abisector of a segment. Draw a diagram to show the difference.3) Find a way to show that points A = ( -4, -1 ), B = ( 4, 3 ), and C = ( 8, 5 ) are collinear.4) You are reading a geometry book and come across something called a “straight” angle. Withoutlooking it up, what do you think this is? Draw a picture5) Draw a picture of two angles that would be referred to as “adjacent’. What do you think thismeans?6) Several angles have the same vertex at O. Angle AOB is 100 degrees. Angle BOC is 40degrees. How big is angle AOC? Again, you might want to draw a picture.7) Given 3 non-collinear points, A, B, and C, is it possible for AB + BC > AC? If “yes” then give anexample. If “no” then explain why not.8) Given rays OA, OB and OC, with no three points collinear. Are the following statements T or F?If false, show why.(a) m

Geo 9 Ch 1-2.2 35For each of the following questions, fill in the blank with always true (A) , never true (N), orsometimes true (S). Please write a few sentences explaining your choice. Think of a plane as apiece of paper.FILL IN THE BLANKS. YOU MAY USE YOUR BOOK.(10) a) Two skew lines are ____________parallel.b) Two parallel lines are __________coplanar.c) Two lines that are not coplanar ___________intersect.d) A line in the plane of the ceiling and a line in the plane of the floor are _______parallel.e) Two lines in the plane of the floor are _______skew.f) If a line is parallel to a plane, a plane containing that line is _____parallel to the givenplane.g) Two lines parallel to the same plane are _________parallel to each other.h) Two lines parallel to a third line are _________parallel to each other.i) Two lines skew to a third line are ____________skew to each other.j) Two lines perpendicular to a third line are _________perpendicular to each other.k) Two planes parallel to the same line are ______ parallel to each other.l) Two planes parallel to the same plane are _______parallel to each other.