GEO 9 CH Chapter 1 Points, Lines, Planes and Angles

Transcription

1 GEO 9 CH Chapter 1 Points, Lines, Planes and ngles What you have done is 1) look for a pattern 2) made a conjecture 3) used logical reasoning to verify your conjecture That is what we do in geometry using definitions, postulates, properties and theorems to verify our conjectures.. Undefined terms: points, lines, planes RED BOUT THEM PG. 5 B. Definitions State the meaning of a concept. Definintions are reversible. Definitions contain the least possible amount of information. C. Postulates or axioms are things we accept to be true. Not necessarily reversible. D. Theorems things we prove to be true

2 Geo 9 Ch Segments, Rays and Distance SKETCHPD Get into groups and find the following : Ex 1) The ray opposite to OG is G 2 E 1 O 0 M 1 T R 2 3 Y 4 2) The length of OG is. 3) The distance between R and E is 4) The midpoint of GY is 5) The coordinate of midpoint of ET is 6) Give me a rule for finding the coordinate of a midpoint 7) Give me a rule for finding the distance between 2 points 8) You are told that segment B, notation B, is 10 cm, segment C is 3 cm, how long is segment BC? You might have to think about this a little. picture would definitely help.

3 Geo 9 Ch Segment ddition Postulate If B is between and C, then Draw a picture: B +BC = C Ex.. If B = 6 andbc = 8, then C = Ex. If C = 12, B = 15, BC = 3, which point is between the other two? Ex. If L is between P and Q, and PL = 6x 5, LQ = 2x + 3, and PQ = 30? What is x?

4 1.4 ngles Geo 9 Ch ngle ddition Postulate ( Used two ways). If point B lies in the interior of <OC, then m<ob + m<boc = m<oc (RENMING PRTS TO WHOLE OR V.V.) B O C B. If <OC is a straight angle and B in any point not on C, then m<ob + m<boc = 180 (EXPRESSING STRIGHT NGLE S SUM OF TWO NGLES) B O C K Ex. M T L L bisects < KT. Find the value of x. 1) m<1 = 3x, m<2 = 4x 15 2) m<2 = x 6, m<3 = 4x

5 Geo 9 Ch T RECTNGLE TSRP S POWERPOINT p9 O P R 3a) 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) 2. If CBD DBE and BD bisects CBE, find m ( CB) 2) C D x+10 x+5 B 60 E ; m 1=x+14; m 2= x 2 4x 3) Solve for x m BD=3x; m DBC=x; find m BD. 4) D B C

6 Geo 9 Ch m FGJ=3x 5; m JGH=x+27; GJ bisects FGH. Find m FGJ. 5) F J G H 6. m BC=90 ; m 1=2x+10; m 2=x+20; m 3=3x 6) C B Has BC been trisected?

7 Geo 9 Ch SUPPLEMENTRY HOMEWORK 1.5 Postulates and theorems relating to points, lines and planes. Group tables and go over homework. Then move tables to an oval. Postulate 5 : line contains at least points; a plane contains at least points not all 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 points there is exactly one plane. Postulate 8: If two points are in a plane, then the that contains the points is in that plane. Postulate 9: If two planes intersect, then their intersection is a.

8 Geo 9 Ch Theorem 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.

9 Geo 9 Ch Fill in the correct notation for the lines, segments, rays. Is TW on plane m? re TSW coplanar? re RWY coplanar? Where does XY intersect plane m? T x S m How many lines contain point T and S? How many planes contain T, S and X? R o W y Where do planes R & S intersect? R s B Name 3 lines that intersect E? Name 2 planes that intersect at FG? Name 2 planes that don t R intersect? re points RSGC coplanar? E F D H S C G B

10 Geo 9 Ch Ch Conditional Statements Objectives: 1) Recognize the hypothesis and the conclusion of an if then statement. 2) State the converse of an if then statement. 3) Use a counterexample to disprove an if then statement. 4) Understand the meaning of if and only if. Conditional Statements : If, hypothesis then. conclusion conditional statement is one that states an assertion, usually called the hypothesis, based on a given condition. It is usually in the form if (condition/ hypothesis)., then (conclusion)., but can take on other forms. hypothesis conclusion given or understood information formed from the given information ex. If I live in Martinsville, then I live in New Jersey. hypothesis ex. If two angles sum is conclusion 180, then they are supplementary. ex. n angle is called a right angle if its measure is We take information that is given to us and then make conclusion upon conclusion until we get to where we are going. 90. 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 States If I live in the United States, then I live in North merica ex. If the figure is a parallelogram then the diagonals bisect each other. Converse: Is formed by interchanging the hypothesis and the conclusion ex. 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.

11 Geo 9 Ch ex. If a polygon is a quadrilateral, then it has four sides. If a polygon has four sides, then it is a quadrilateral Groups LL DEFINITIONS RE BICONDITIONL, NOT LL 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. ll right angles are congruent. 5. There are 2 pairs of postulates that are converses of each other regarding lines and points. Name them and explain what they say. 6. (a) Write the converse of the statement If point P is equidistant from the coordinate axes, then point P is on the line y = x. (b) Give an example of a true statement whose converse is false. (c) Give an example of a true statement whose converse is also true. 7. In lgebra, 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. 2( x+ 1) = 5x 3 2x + 2 = 5x 3 2x ( 2) = 5x 3 + ( 2) 2x = 5x 5 2x 5x = 5x 5x 5 3x = ( )( 3x) = ( ) ( 5) x = 3

12 Geo 9 Ch Proof Properties Memorize SOON!!!!! Properties of Equality Make file cards 1. DDITION PROPERTY If a = b and c = d, then 2. SUBTRCTION PROP If a = b and c = d, then 3. MULTIPLICTION PROP If a = b, and c exists, then 4. DIVISION PROP If a = b, and c 0, then 5. SUBSTITUTION If a = b, then either may replace the other in any equation. 6. REFLEXIVE a = a 7. SYMMETRIC PROP If a = b, then 8. TRNSITIVE PROPŎŎ If a = b, and b = c, then Properties of Congruence 1. REFLEXIVE PROP: DE DE <D < D 2. SYMMETRIC PROP: If DE FE, then If < D < E, then 3. TRNSITIVE PROP: If DE FG and FG JK, then If < D < E, and <E < F, then SKETCHPD *WHICH ONES RE USED FOR EQULITY ND CONGRUENCE? DEFINITION OF CONGRUENCE: If B CD then B = CD *( WRNING ) If < <B then m< = m<b (watch use of = and ) Use this definition to convert congruence to equality and visa versa.

13 Geo 9 Ch Geo 2.2 Properties from lgebra *Elements of Two Column Proofs. Two Column Proof Examples 1. Example #1: 2. Example #2: Given: RS = PS; ST = SQ Prove: RT PQ R S P STTEMENTS RESONS Q T 1) RS = PS; ST = SQ 1) Given 2) RS + ST = QS + SP 2) 3) RS + ST = RT 3) QS + SP = QP *4) RT = QP 4) 5) RT PQ 5)

14 Geo 9 Ch Lets try a geometry proof: The first step is LWYS to mark your drawing according to the given information. For instance, if segments are given congruent, MRK them congruent with tic marks!! B C D E F Given: Prove: B C DF DE, BC EF ***(WRNING!) Statements Reasons *1. B DE, BC EF 1. (what allows me to make this statement?) 2. B = DE; 2. BC = EF (why did I line it up like this?) 3. B + BC = DE + EF (why can I say this?) B + BC = C; DE + EF = DF 4. ( Uh oh, where did this come from?) 5. C = DF (so this is the same as?) 5. (have I proved what is asked for?) 6. C DF 6. NOTES: Notice: Did we go from smaller pieces to larger pieces? That involves nother, slightly different problem. Given : Pr ove : B CD C BD B C D Statement Reasons 1. B CD 1. *2. B = CD BC = BC 3. (isn t this obvious?) 4. B + BC = BC + CD (here we go again!) B + BC = C; BC + CD = BD C = BD C BD 7.

15 Geo 9 Ch Notes: We went from to which means Now, lets try the reverse: Given : C BD Pr ove : B CD (WRNING!) Statements B C Reasons D How am I going to go from #1 to #7? 1. C BD 1. *2. C = BD (candy bar) B + BC = C; BC + CD = BD 3. (breaking into pieces) 4. BC = BC (why do I need to put this in?) B + BC = BC + CD (renaming, why?) 5. *6. B = CD B CD 7. NOTES: We went from large pieces to smaller pieces. That involves 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 dd= and Sub = must be in the equality sign!

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17 Geo 9 Ch Geo2 2 Proofs in Groups/HW 1. Given : GJ HK Prove : GH JK *WRNING! M Statements G H J K Reasons use cards to recognize reasons 1. GJ HK 2. GJ = HK ( large or small?) GJ = GH + HJ p 3. HK = HJ + JK 4. GH + HJ = HJ + JK a t t e HJ = HJ r n GH = JK GH JK Given 2. H G same proof except with angles.. E F Given GHF HGE FHE EGF Prove GHE HGF WRNING! Statements Reasons Given 2. m GHF = m HGE, m EHF = m FGE ddn Prop of = P Substitution 6. EHG FGH 6.

18 Geo 9 Ch B C 3) D E F recognize this? Given C = DF B = DE Prove BC EF Statements Reasons Given SP 3. B + BC =DE + EF Sub Prop of = Hey, why didn t we have to use definition of congruence? Give a simple explanation. 4. S R P Q T Statements Think about the big idea here. What is the pattern? Reasons Given Given Prove SRT STR *6. SRT STR 6. Def of

19 Geo 9 Ch This is the same diagram. m I doing the same thing? 19 Geo Statements Reasons S P Q R T Given Prove RP TQ PS QS RS TS 6. Statements Reasons S P Z Q R T Given RQ = TP ZQ = ZP Prove RZ = TZ 7. Statements Reasons S P Q R T Given SRT STR 3 4 Prove 1 2

20 Geo 9 Ch Statements Reasons Given m 1+ m 2 = Prove m 1 + m 3 = B C D F E Given BD DE BF DEC Prove FBC CE Statements Reasons

21 Geo 9 Ch Geo E B C D Given: EB DEC Prove: EC DEB Statements Reasons

22 Geo 9 Ch Worksheet Points, Lines and Planes 1. Refer to the diagram: D C B H G E F a) Name 2 planes that intersect in HG. b) re the points, B, C and D collinear? c) re the points, 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 a) The ray opposite to KN is b) nother name for LM is c) LN= d) The coordinate of the midpoint of JM is 3. S T E P 9 4 a) 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.

23 Geo 9 Ch a) n angle adjacent to DB is. b) re, B, and E collinear? c) Can you conclude from the diagram that BE BD? d) What postulate allows you to say m BD + m DBC = m BC? e) m CBE =. f) m BCD =. g) m BD =. 5. Refer to the diagram. OR is a bisector of QOS P Q a) If m 1=2x+15 and m 2=5x 8 then x= O R S b) If m 1=x+7 and m 3=2x then x= 6. Name the definition or postulate that justifies each statement, given the markings on the diagram. R Q T S a) m RSQ + m QST = m RST. b) SQ bisects RT c) Q is the midpoint of RT d) RT = RQ + QT e) re R, Q and T collinear? Use sometimes, always or never. 7. a) djacent angles are congruent. b) Two intersecting lines lie in exactly one plane. c) line and a point not on the line lie in more than one plane.

24 Geo 9 Ch Ch Geometry Worksheet F E Refer to the figure to the right. Given: <1 <2 <3 is a right angle < BF =90 2 C is the midpoint of BD 1 3 B C D Supply a reason for each statement made in the following sequence. G (1) m<1 = m<2 (2) m<3 =90 (3) m<bf = m<3 (4) m<1 + m<2 = m<bf (5) m<1 + m<2 =m <3 (6) m<bf + m<3 =180 (7) m<bf + m<1 + m<2 = 180 (8) m<1 + m<ebd = 180 (9) B + BC = C (10) CD + DG = CG (11) C + CG = G (12) B + BC + CD + DG = G (13) BC = CD (1) D E (2) (x + 39) (3x 3) 5y 2 B 2y + 1 Given the figure above, C = 15, BD bisects BE. Find: x, y, z z C (4x + 5y) (2x 3y) (x + 2y) Given the figure above, find x and y

25 Geo 9 Ch Ch Geometry Review Worksheet (3) (4) B E B E F C D C D Given: B = E Given: m 1 = m 3 C = D m 2 = m 4 Prove: BC = DE Prove: m CD = m DC (6) B C D E Prove: B + BC + CD + DE = E

26 Geo 9 Ch (1) Refer to the figure to the right. Given: FB Supply a reason to justify each statement in the following sequence. (a) 1 BFD (b) 2 and 3 are complementary (c) m 2 + m 3 = 90 (d) 1 is a right angle (e) m 1 = 90 (f) m 2 + m 3 = m 1 D (g) m BFD = m 2 + m 3 (k) m 4 + m 5 = 180 (h) m 3 = m 5 (l) 4 and 5 are supplementary (i) m 2 + m 5 = 90 (m) m 1 + m 2 + m 3 = 180 (j) 2 and 5 are complementary E 5 (n) F + DF = D B 1 F C D (2) Given the figure to the right, FC E, m FD = 155, m 2 = 4 m 3, find the measures of all the numbered angles. G B 1 2 C 3 4 F 6 5 D E (3) Given the figure as marked, find the values of x and y. (3x + 5) (2x + 5) (3x + y) (6y + 15)

27 Geo 9 Ch (4) Find the measure of an angle if 80 less than three times its supplement is 70 more than five times its complement. (5) (6) Given: 1 3 Given: 1 and 7 are supplementary Prove: 2 4 Prove: 6 3 (7) (8) D E F 1 B B C D E Given: C bisects DB Given: BE C D bisects CE 1 4 Prove: 1 3 Prove: C (9) (10) B E D E F 1 3 B 2 4 C 5 6 C D Given: BE bisects C Given: m 1 = m 3 BE bisects D m 2 = m 4 B = E Prove: BC = DE Prove: m 5 = m 6

28 Geo 9 Ch (1) (a) lines form adjacent 's (k) P (b) ESPC (l) defn supplementary (c) defn complementary (m) addition property of = (d) defn (n) SP (e) defn right (f) substitution ( steps c & e ) (g) P (h) VC (i) substitution ( steps c & h ) (j) defn of complementary OR substitution ( steps b & h ) (2) m 1 = 25, m 2 = 72, m 3 = 18, m 4 = 65, m 5 = 25, m 6 = 155 (3) x = 103, y = (4) 30

29 Geo 9 Ch CH DEFINITIONS POSTULTES PROPERTIES Defined terms: DY 1 1. collinear 2. non collinear 3. coplanar 4. segment 5. ray opposite rays 6. distance 7. congruent 8. midpoint 9. bisector segment angle 10. angle vertex obtuse angle right angle acute angle DY 2 straight angle

30 Geo 9 Ch adjacent angles 12. supplementary 13. complementary 14. vertical 15. perpendicular 16. congruent 17. congruent segments 18. SP 19. P 20) dd = 21) Sub = 22) Div = 23) Mult = 24) Reflexive 25) Transitive