GEO 9 CH CH ASSIGNMENT SHEET GEOMETRY Points, Lines, Planes p all,15,16,17,21,25

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1 GEO 9 CH CH ASSIGNMENT SHEET GEOMETRY 9 DAY SECTION NAME PAGE ASSIGNMENT 1 Algebra Review/Assignment #1 Handout 2 Algebra Review/Assignment #2 Handout Points, Lines, Planes p all,15,16,17,21,25 Supplementary Problems CH # Segments Rays Distances p ,5-22all, 32,36,37,39 Supplementary Problems CH # Angles p ,17,29-34,36 Supplementary Problems CH # Postulates & Theorems p all, "If - Then" p , 17,19,22 BRING IN FLASHCARDS TO CLASS SECTION Properties from Algebra p. 40 classroom 11, 12 p. 41 written 6-10 all Properties from Algebra Packet # REVIEW Packet Supplementary Problems CH #1, 2, 3 10 TEST CH

2 Geo 9 Ch

3 Geo 9 Ch Chapter 1 Points, Lines, Planes and Angles In 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, planes B. Definitions - State the meaning of a concept. Definintions are reversible. Definitions contain the least 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 B AB + BC = AC C D. Theorems things we prove to be true Ex. If you have a rectangle, then the diagonals are the same length.

4 Geo 9 Ch Lesson Points, Lines and Planes Points, lines and planes are intuitive ideas that are accepted without definition. These terms are then used in the definitions of other terms. Point - Graph (-4, 0) label Line - Graph (6,0) on the same graph - 2 pts determine a line Plane - Graph ( 1, 5 ) 3 points determine a plane Add point ( 7, 4 ) Create 2 intersecting lines 4 Ways to Determine a plane

5 Geo 9 Ch H G E F E Horizontal plane F Vertical plane A D B C Parallel Lines Perpendicular Lines Skew Lines Collinear Non- Collinear- Sketchpad

6 Geo 9 Ch Segments, Rays and Distance C D A B Segment Ray Opposite rays Distance Congruent Equal Midpoint Angle Straight Angle Vertical Angles

7 Geo 9 Ch Get into groups and find the following : P Ex 1) The ray opposite to EG is 2) The length of MG is. 3) The distance between R and E is G -2 E -1 O 0 M 1 T R 2 3 Y 4 4) 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 would definitely help. 7) Draw two segments, AB and CD for which the intersection of segments AB and CD is the empty 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 the lines, PQ and RS are the same.

8 Geo 9 Ch A. Segment Addition Postulate If B is between A and C, then Draw a picture: AB +BC = AC 1) 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 )

9 Geo 9 Ch B. Angle Addition Postulate If ray OB is between ray OA and ray OC, then the m<aob + m<boc = m< Draw it. Straight Angles in reference to AAP EXAMPLES D A B C 5) K M 3 A 2 1 T L AL bisects < KAT. Find the value of x. a) m<1 = 3x, m<2 = 4x - 15 b) m<2 = x-6, m<3 = 4x

10 6) Geo 9 Ch RECTANGLE TSRP T S O P R a) 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) C D x+10 A x+5 60 B E 8) 1 2; m 1 = x+14; m 2 = x 2-4x 8) Solve for x ) m ABD = 3x; m DBC = x; find m ABD. 9) D A B C

11 Geo 9 Ch ) m FGJ = 3x - 5; m JGH = x + 27; GJ bisects FGH. Find m FGJ. 10) F J G H 11) m ABC = 90 ; m 1 = 2x + 10; m 2 =x+20; m 3=3x 11) C 1 2 B 3 A 12) Has ABC been trisected?

12 Geo 9 Ch 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 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.

13 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.

14 Geo 9 Ch Fill in the correct notation for the lines, segments, rays. Is TW on plane m? Are TSW coplanar? Are 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 A s B Name 3 lines that intersect E? Name 2 planes that intersect at FG? Name 2 planes that don t intersect? Are points RSGC coplanar? R H S G E F D C A B

15 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) Understand the meaning of if-and-only-if. Conditional Statements : hypothesis conclusion If, then. A conditional statement is one that states an assertion, usually called the hypothesis, based on a given condition. It is usually in the form if (given/ hypothesis)., then (prove/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 conclusion ex. 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 to where 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 States If I live in the United States, then I live in North America 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. ex. If a polygon is a quadrilateral, then it has four sides. If a polygon has four sides, then it is a quadrilateral Groups

16 Geo 9 Ch ALL 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.

17 Geo 9 Ch Proof Properties Memorize SOON!!!!! Properties of Equality Make file cards 1. ADDITION PROPERTY If a = b and c = d, then 2. SUBTRACTION PROP If a = b and c = d, then 3. MULTIPLICATION PROP If a = b, and c exists, then 4. DIVISION PROP If a = b, and c 5. SUBSTITUTION 0, then If a = b, then either may replace the other in any equation. 6. REFLEXIVE a = a 7. SYMMETRIC PROP If a = b, then 8. TRANSITIVE PROP 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. TRANSITIVE PROP: If DE FG and FG JK, then If D E, and <E F, then SKETCHPAD *WHICH ONES ARE USED FOR EQUALITY AND CONGRUENCE? DEFINITION OF CONGRUENCE: If AB CD then AB = CD *( WARNING ) If <A <B then m<a = m<b (watch use of = and ) Use this definition to convert congruence to equality and visa versa.

18 Geo 9 Ch In 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) Given 2) 2x + 2 = 5x 3 2) 3) 2x (-2) = 5x 3 + (-2) 3) 4) 2x = 5x 5 4) 5) 2x 5x = 5x 5x 5 5) 6) -3x = -5 6) 7) (- 3 1 )(- 3x) = (- 3 1 ) ( 5) 7) 8) x = 3 5 8)

19 Geo 9 Ch Geo 2.2 Properties from Algebra *Elements of Two-Column Proofs R S P Given: RS = PS; ST = SQ Prove: RT PQ Q T STATEMENTS REASONS 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)

20 Geo 9 Ch Lets try a geometry proof: The first step is ALWAYS to mark your drawing according to the given information. For instance, if segments are given congruent, MARK them congruent with tic marks!! A B C D E F Given: Prove: AB DE, BC EF ***(WARNING!) AC DF Statements Reasons *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?) 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 DF 6. Another, slightly different problem. Given: AB Pr ove : AC CD BD A B C D Statement Reasons 1. AB CD 1. *2. AB = CD BC = BC 3. (isn t this obvious?) 4. AB + BC = BC + CD (here we go again!) AB + BC = AC; BC + CD = BD AC = BD AC BD 7.

21 How am I going to go from #1 to #7? Geo 9 Ch Now, lets try the reverse: Given: AC BD A B C D Pr ove : AB CD (WARNING!) Statements Reasons 1. AC BD 1. *2. AC = BD (candy bar) 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 AB CD 7. 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!

22 Geo 9 Ch Geo2-2 Proofs in Groups/HW 1) Given: GJ HK Prove: GH JK *WARNING! M Statements G H J K Reasons use cards to recognize reasons 1. GJ HK 1. Given 2. GJ = HK ( large or small?) GJ = GH + HJ 3. HK = HJ + JK 4. GH + HJ = HJ + JK HJ = HJ GH = JK GH JK 7. p a t t e r n 2) H G same proof except with angles.. E F Given GHF HGE FHE EGF Prove GHE HGF WARNING! Statements Reasons Given 2. m GHF = m HGE, m EHF = m FGE Addn Prop of = AAP Substitution 6. EHG FGH 6.

23 Geo 9 Ch A B C 3) D E F Given AC DF AB = DE Prove BC EF Statements Reasons Given SAP 3. AB + BC =DE + EF Sub Prop of = 4. S R P Q Given T Think about the big idea here. What is the pattern? Statements Reasons Given Prove SRT STR *6. SRT STR 6. Def of

24 Geo 9 Ch Geo Statements Reasons S This is the same diagram. Am I doing the same thing? P Q R T Given RP PS TQ QS Prove 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 3 4 STR Prove 1 2

25 Geo 9 Ch Statements Reasons Given m 1+m 2 = Prove m 1 m B C D A F E Given ABD ABF DEA DEC Prove FBC CEA Statements Reasons Geo 2.2

26 10. Geo 9 Ch E A B C D Given : AEB DEC Prove: AEC DEB Statements Reasons

27 Geo 9 Ch Worksheet Points, Lines and Planes (1) Refer to the diagram: D C A B H G E F a) 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 a) 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 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.

28 Geo 9 Ch (4) Make a sketch showing the relative position of the four points mentioned in thefollowing statement: 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? B E (6) a) An angle adjacent to ADB is. A 30 D C b) 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 =. P Q (7) Refer to the diagram. Ray OR is a bisector of QOS 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=

29 Geo 9 Ch (8) Name the definition or postulate that justifies each statement, given the markings on the diagram. R Q T a) 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. S EB and EC trisect AD A (10) AB = 7x + 3 AC = 11y 7 CD = 8x + 2y 10 Find AD E B C D (11) In the figure, is < QPS acute, obtuse or right? Justify your anwer. E P 2X+10 X+25 Q 3X R 5X+20 S

30 Geo 9 Ch Ch Geometry Worksheet F E Refer to the figure to the right. Given: <1 <2 <3 is a right angle < ABF =90 2 C is the midpoint of BD 1 3 A 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<abf = m<3 (4) m<1 + m<2 = m<abf (5) m<1 + m<2 =m <3 (6) m<abf + m<3 =180 (7) m<abf + m<1 + m<2 = 180 (8) m<1 + m<ebd = 180 (9) AB + BC = AC (10) CD + DG = CG (11) AC + CG = AG (12) AB + BC + CD + DG = AG (13) BC = CD (1) D (2) E (x + 39) (3x 3) z A 5y 2 B 2y + 1 C (4x + 5y) (x + 2y) (2x 3y) Given the figure above, AC = 15, BD bisects ABE. Find: x, y, z Given the figure above, find x and y

31 Geo 9 Ch Ch Geometry Review Worksheet A A (3) (4) B E B E F C D C Given: AB = AE Given: m 1 = m 3 AC = AD m 2 = m D Prove: BC = DE Prove: m ACD = m ADC (6) A B C D E Prove: AB + BC + CD + DE = AE

32 Geo 9 Ch CH DEFINITIONS POSTULATES PROPERTIES Defined terms: DAY 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 straight angle

33 Geo 9 Ch 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

34 Geo 9 Ch SUPPLEMENTARY HOMEWORK: CH Do your HW in a graph paper notebook 1) The distance from (0,0) to (8,6) is exactly 10. Find other examples of points that are exactly 10 units from (0,0). Using a graph will help. How do you think you can use points to find a distance between 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 a bisector 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. Without looking it up, what do you think this is? Draw a picture 5) Draw a picture of two angles that would be referred to as adjacent. What do you think this means? 6) Several angles have the same vertex at O. Angle AOB is 100 degrees. Angle BOC is 40 degrees. 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 an example. 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<aob + m<boc = m< AOC (b) m<aoc + m<boc + m<aoc = 360 9) Given the following points: A ( -8, 7 ) B ( -4, 1 ) and C (5, 7 ). Graph them and find the length of their sides. Is this a right triangle?

35 Geo 9 Ch For each of the following questions, fill in the blank with always true (A), never true (N), or sometimes true (S). Please write a few sentences explaining your choice. Think of a plane as a piece 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 given plane. 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.

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

GEO 9 CH Chapter 1 Points, Lines, Planes and Angles GEO 9 CH1 2.2 1 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