Unit 1 Workbook - Weebly
Geometry – Unit 1Targets & InfoName:This Unit’s theme – Tools for GeometryAugust - SeptemberUse this sheet as a guide throughout the chapter to see if you are getting the right informationin reaching each target listedBy the end of Unit 1, you shouldknow how to Targetfound in Use inductive reasoning to make aconjecture and prove a conjecturefalseDefine, identify, and name a point,line, plane, collinear points, coplanarpoints, segments, and raysIdentify congruent segments andsolve problems involving propertiesof line segmentsClassify angles and angle pairs andsolve problems using their propertiesChapter 2,Section 1page 82Section 2,pages 11-19Find the midpoint and distancebetween two points when givencoordinatesReview perimeter, circumference,and areaTargetDid IDIAGRAMS &discussed reach the EXAMPLES!on (date) target?Section 3pages 2026Sections 4& 5 pages27-40Section 7pages 5056Section 8pages 5767All material covered on the test willbe based on these targets. So keeptrack of your readiness for the testby updating the “Did I reach thetarget?” column.ESSENTIAL VOCABULARY:inductive reasoning, conjecture, counterexample, Euclid’s undefined terms, collinear, coplanar, line segment,ray, between, intersect, postulate/axiom, theorem, segment addition postulate, midpoint, congruent, angle,angle addition postulate, adjacent angles, vertical angles, linear pair, complementary, supplementary
Suggested Textbook Practice Problems***DAILY QUIZ QUESTIONS WILL BE TAKEN FROM THESE TEXTBOOKPROBLEMS AS WELL AS PRACTICE FROM YOUR UNIT WORKBOOK***Lesson 1: Patterns and Inductive ReasoningPage 86: 6 – 30, 33 – 43, 48 – 50, 53Lesson 2: Points, Lines, and PlanesPage 16: 8 – 46Lesson 3: Postulates and Axioms, Midpoint, DistancePage 24: 8 – 30, 37, 42, 43Page 54: 6 – 51, 58Lesson 4: Angle MeasuresPage 31: 6 – 23, 29 – 39Lesson 5: Angle Pair RelationshipsPage 38: 7 – 40Lesson 6: Perimeter and AreaPage 64: 7 – 502
Lesson 1: Patterns and Inductive ReasoningWhat is Geometry?Who is Euclid?What other types of Geometry exist?Inductive Reasoning:1.Look for a patternEx 1) 1, 2, 4, 16, 64, , ,Ex 2) -5, -2, 4, 13, ,Ex 3),2.Make a conjecture3.Verify the conjecture,,Conjecture: an unproven statement based on observationsIs the conjecture TRUE or FALSE?Ex 1) Every road in Illinois is currently under construction.Ex 2) All math teachers are attractive.Ex 3) The sum of the first n odd integers is n2.To prove a conjecture false, you need ONLY ONE example that shows the conjecture to be false.To prove a conjecture true, however, you must prove that the conjecture is true in ALL CASES.3
Counterexample: example showing a conjecture falseEx 4) For all x , x 2 x . True or False?Ex 5) Every even number greater than 2 can be written as the sum of two prime numbers.True or False?Find a counterexample to show each conjecture false. (TEST QUESTION)a) The product of two numbers is always greater than either number.b) The difference of two integers is less than either integer.c) The sum of two prime numbers is an even number.4
Lesson 1 Practice: Patterns and Reasoning1. Who is considered the Father of Geometry?2. What book did he write?3. How can you prove a conjecture false?4. Sketch the next figure.5. Sketch the next figure.6. Find the next term in the pattern.a. 1, 5, 9, 13,b. 1, 4, 9, 16,c. 1, 3, 9, 27,d. 1, 1, 2, 3, 5, 8, 13,e. O, T, T, F, F, S, S, E,f. 1, 2, 6, 24, 120,g. Aquarius, Pisces, Aries, Taurus,7. What is the next row in the following pattern?11111123451610131410151Complete the conjecture based on the specific cases.8. Conjecture: The sum of any two odd numbers is .1 1 21 3 43 5 87 11 1813 19 32201 305 5065
9. Conjecture: The product of a number (n – 1) and the number (n 1) is always equal to .10. Riley makes a conjecture about slicing pizza. She says that if you use only straight cuts, the number ofpieces will be twice the number of cuts. Is this true? If false, provide a counterexample.Show each conjecture is false by providing a counterexample.11. The square root of a number is alwaysless than the number.12. The sum of two numbers is always greater thanthe larger number.13. All prime numbers are odd.13. If m is a nonzero integer, thenis alwaysgreater than 1.14. The product of two positive numbers isgreater than either number.15. Triangle ABC is a right triangle, so A is aright angle.Make a conjecture for each scenario. Show your work.16. the sum of the first 100 positive odd numbers17. the sum of an even and odd number18. the product of two odd numbers19. Find the perimeter when 100 triangles are put together in the pattern shown. All triangles have side lengthsthat measure 1 cm.A. 100 cmB. 102 cmC. 202 cmD. 300 cm20. Describe a real life situation in which you have used inductive reasoning.6
Lesson 2: Points, Lines, and PlanesDefinition: uses known words to describe a new wordEuclid’s 3 Undefined Terms:1) Point – has no dimension, represented by a small dot and labeled with a capital letter2) Line – extends in one dimension, represented by a straight line with two arrows to show continuing foreverin both directions3) Plane – extends in two dimensions, represented by a shape similar to a slanted tabletop extending foreverin all directionsCollinear Points:Coplanar Points:DAEx 1) a. Name 3 collinear pointsGHb. Name 4 coplanar pointsFc. Name 3 noncollinear pointsBBetween:Consider line AB.ABLine Segment AB is all points between A and B on ABRay AB is the initial point A and all points on AB on the same side of A as point BBe very careful in labeling.ABAB7
If C is between points A and B, then we call CA and CB Opposite Rays.Ex 1) Draw 4 noncollinear points M, N, O, P. Then draw MO, OP, MN, and NMEx 2) Given EH, name two pairs of opposite rays.EFGHAre GF and GE the same ray?Intersect: having one or more points in commonSketch the following:1) line AB and a plane that do not intersect2) two planes that do not intersect and a line that intersects each plane at exactly one point3) a line BC intersecting a plane at one point B4) a line intersecting a plane at infinite points5) Three points that are coplanar but not collinear6) Two lines that intersect in a point and all lie in the same plane7) two planes that intersect at line AB8
Lesson 2 Practice: Points, Lines, and Planes1. Who is considered the Father of Geometry?2. What were his three undefined terms?a.b.c.3. How can you prove a conjecture false?4. How can you prove a conjecture true?Determine if each statement is true or false.TF5. Points A and B are collinear.TF6. Points A, B, and C are collinear.TF7. Points D and E are collinear.TF8. Points J and K are collinear.TF9. Points J, K, and L are collinear.TF10. Points J, K, and L are coplanar.TF11. Points J, K, and M are coplanar.TF12. Points L, M, and N are coplanar.TF13. Points J, K, L, and M are coplanar.DCBAEMKJLNBName a point that is coplanar with the given points.14. A, B, and C15. D, C, and H A16. F, A, and E17. E, F, and G18. A, B, and H19. B, C, and FCDFGEHA20. AB and BC intersect at .21. AD and AE intersect at .22. Plane ABC and plane DCG intersect at .23. Plane EAD and plane BCD intersect at .DBECHG9
Sketch the figure described.24. Draw two points, X and Y. Then sketch XY. Add a point W between X and Y so that WX and WY areopposite rays.25. Two lines that lie in a plane but do not intersect.26. Two lines that intersect and another line that does not intersect either one.27. Two rays that are coplanar but not collinear.28. Two planes that intersect at line MN.29. Three planes that intersect at line AB.30. Points K, L, M, and N are not coplanar. What is the intersection of plane KLM and plane KLN?A. K and LB. M and NC. KLD. KL10
Lesson 3: Postulates and Axioms, Measuring Segments, Distance, MidpointPostulate/Axioms:Ruler postulate: points on a line can be matched on the number line to real numbers so that the distance isequal to the absolute value of the difference between the coordinates (p 20)Between: when 3 points lie on a line we say one is between the othersAB BC ACSegment Addition Postulate: If B is between A and C, thenABCTwo friends leave their homes and walk in a straight line toward the other’s home. When they meet, one haswalked 425 yds and the other has walked 267 yds. How far apart are their homes?11
Finding the distance between two points using coordinates.Find AB.ABADistance Formula:BMidpoint: the point between two endpoints of a segment that divides a segment into two congruent segmentsABMBMidpoint Formula:MA12
Ex 1) Find the midpoint of AB. A(-2, 3) B(5, -2)Ex 2) Find the midpoint of CD. C(5, -8) D(9, -5)Ex 3) Find MN. M(8, -3) N(4, 2)Ex 4) The midpoint of RP is M(2, 4). One endpoint is R(-1, 7). Find the coordinates of the other endpoint P.13
Lesson 4: Angle MeasuresVocabulary Review:Euclid’s Undefined Terms –Midpoint –Collinear –Coplanar –Line Segment –Ray –Congruent –Counterexample –Postulate/Axiom –Find MN and the midpoint of MN.M(-3, 4)N (5, -11)Find RT if RS 2x 3, ST 5x – 12, and RT 4x 15RTSAngle: two rays with the same initial pointACBBName all the angles in the diagram.CADEIf point F is placed between points A and C, have more angles been created?14
Angle Addition Postulate: If point P is in the interior of RST, then m RSP m PST m RST.Ex 1) If m PQR 30o and m RQS 40o, then m PQS PEx 2) Given m PQR 45o, m RQS 20o,m SQT 50o, and m TQV 65oQm PQT Rm SQV m VQP m RQT SVTEx 3)MO(3x 8)No(4x – 9)oNO bisects MNPPBisect: cut in half, make angles congruent ( )15
Ex 4)(2x 5)o(2x – 10)o(x – 20)ox 4 Classifications of Angles:1.Acute –2.Right –3.Obtuse –4.Straight –Adjacent Angles: two angles sharing a common vertex and a common sideMONPName a pair of adjacent angles in the diagram.PQRSVT16
Lesson 4 Practice: Angle Measures1.m QRY 2.m XRZ 3.m XRS 4.m QRS 5.Classify each angle as acute, obtuse, right, or straight.6.a) QRYb) XRZc) XRSd) QRSm LMN m NMO m Why?7. LMN 8.m LMN 9.If m LMN 24 , thenMN bisects LMOm NMO m LMO 10.If m LMO 64 , then11.If m LMN (3x 13) and m NMO (5x – 7) , thenx 12.m LMN m NMO m NMO If m LMN (5x 6) and m LMO 62 , thenx 13.m LMN m LMN m NMO m SXT (4x 1) , m QXS (2x – 2) ,m QXT 125x m QXS 17
m PXR (3x) , m RXT (5x 20) 14.x m RXT 15.Find the distance between (1, 2) and (-5, 4).16.Find the distance between (2, 9) and (-3, -3).17.GH HI Why?18.Determine if each statement is true or false.TFa.All the points shown are coplanar.TFb.P and S are collinear points.TFc.R, S, and T are collinear points.TFd.Q, R, and S are collinear points.TFe.P, R, and S are coplanar points.TFf.Q, R, and S are coplanar points.TFg.RS and PT intersect.TFh.US and PT intersect.TFi.RT and TR are opposite rays.TFj.RS and RT are the same.URPSTQUse the diagram to complete each statement.19. CBJ 20. FJH 21.If m EFD 75, then m JAB 18
22.If m GHF 130, then m JBC 23.Points E and G are collinear, and point F is between points E and G. If EG 49, EF 2x 3, and FG 4x – 2, find x, EF and FG.24.Find the midpoint of ST given endpoints S(-5, 8) and T(-2, 4).25.The midpoint of ST is M(-6, 4). Find the coordinates of point T given endpoint S(8, 2).26.Make a conjecture for the following scenario. Show support for your conjecture.The sum of the first 100 positive even numbers is .Provide a counterexample to show that each conjecture is false.27. 1 and 2 are supplementary, so one of the angles is acute.28.If two angles are adjacent, then they are also congruent.19
Lesson 5: Angle Pair RelationshipsVertical Angles: two angles whose sides form two pairs of opposite raysLinear Pairs: two adjacent angles whose non-common sides form opposite raysa) 1 and 3 are vertical?Ex 1)142b) 2 and 4 are vertical?3c) 3 and 4 are a linear pair?Ex 2) Find x, y, and z.zy40xoMake a conjecture about vertical angles:Make a conjecture about linear pairs:20
Ex 3) Use your conjectures from example 2 to find the values of x and y. Is your conjecture supported?Is your conjecture true or false?(y 20)o(4y – 15)o(3x 5)o(x 15)oComplementary Angles:two angles whose sum Supplementary Angles: two angles whose sum 21
Lesson 6: Perimeter and AreaRectangle:P A Triangle:P A Circle:C A Ex 1) The radius of a circle is 6 cm. Find the circumference and area in exact terms.Ex 2) Find the area of a triangle defined by the points D(1, 3), E(8, 3), and F(4, 7).Ex 3) Find the area and perimeter of the triangle defined by H(-2, 2), J(3, -1), and K(-2, -4).22
Lesson 5 & 6 Practice: Angle Pair Relationships, Perimeter, AreaWrite the Distance FormulaWrite the Midpoint FormulaFind the coordinates of the midpoint of each segment with the given endpoints.1) A(0, 0) B(-8, 6)2) J(-1, 7) K(3, -3)3) C(10, 8) D(-2, 5)Find the coordinates of the other endpoint of a segment with the given endpoint and midpoint M.4) R(2, 6) M(-1, 1)5) T(-8, -1) M(0, 3)6) W(-3, 12) M (2, -1)Use the marks on the diagram to name the congruent segments and congruent angles.7)AD8)9)YWECZXFGB23
QS is the angle bisector of10)PQR. Find the two angle measures not given in the diagram.11)PS22QBD bisects13)o12)P91oRQPR14)RDDB80oQABC. Find the value of x.A(SSx 20)(5x – 22)oo(3x – 85)o(2x 35)oCACBFind the distance between the two points.15) A(3, 12) B(-5, -1)16) C(-6, 9) D(-2, -7)17) Who is considered to be the Father of Geometry?18) What were his three undefined terms? , , and .19) A statement that we accept without proof is called a(n) or a(n) .24
TFS, T, V, and W are coplanar.TFS, R, and X are collinear.PRTFQ, P, and R are collinearTFP, R, Q, and W are coplanar.TFTS and TU are opposite rays.TFPQ and QP are opposite rays.TFT and Q are collinear.TFS, X, and Q are coplanar.TFR is the midpoint of PQ.TFST and SU are opposite rays.TFIf two planes intersect, then they intersect at a line.QXVWSUTUse the figure at the right. Answer with yes or no.20) Are5 and6 a linear pair?21) Are5 and9 a linear pair?22) Are5 and8 a linear pair?23) Are5 and8 vertical angles?24) Are5 and7 vertical angles?25) Are9 and6 vertical angles?56 79814Decide whether the statement is always, sometimes, or never true.26) If m1 40 , then m2 140 27) If m4 130 , then m2 50 28)1 and4 are congruent29) m2 m3 m30) m2 31) m2 90 – m1 m4321325
Use the figure at the right to answer questions #33 – 40.32) If m6 72 , then m7 33) If m8 80 , then m6 934) If m9 110 , then m8 35) If m9 123 , then m7 36) If m7 142 , then m8 37) If m6 13 , them m9 38) If m9 170 , then m6 39) If m8 26 , then m8677 Find the value(s) of the variables(s).41)42)105 (6x 19) x (2x – 11) 44)(3x 20) (5x - 50) (y – 12) (6x – 32) (3y – 8) 45)y 43)46)6x 7x (4x 16) 11y 56 (2x – 20) 2x y 26
A and47)and m B.48)andC andm D.A andB are complementary. The measure ofD are supplementary. The measure ofB are complementary. Find mA and m49)B is three times the measure ofD is eight times the measure ofA. Find mC. Find mACB.50)mmA andA 5x 8B x 4B are supplementary. Find mA and m51)mmA 3x - 7B 11x – 1mmA 6x – 1B 5x – 17B.52)mmA 2xB x 8Given points A(-3, 6), B(4, 8), and C(4, -4). Find the following.53) BC54) midpoint of AB55) AC27
56)FG GH FGHWhy?57)Find the coordinates of endpoint F of segment EF with the midpoint M.E(-2, 8)M(4, -1)F(x, y)Find the perimeter and area of each figure.58)59)60)61) Find the perimeter of the figure with vertices at A(-5, 3), B(7, -2), C(7, -6), D(-5, -6).Find the area of the shaded region. All angles are right angles.62)63)64)28
Review for Unit 1 Test1. Find the area and perimeter (or circumference) of each figure. Give exact answers.a)b)5 cm12 in16 cmsquarearea rectanglearea perimeter perimeter c)d)17 cm8 cm25 cm15 cm8m20 cmtrianglearea circlearea perimeter perimeter 2. Find the perimeter of a square if its area is 100 square meters.3. The length of a rectangle is 4 more than two times its width. If the area of the rectangle is 48 meters, findits length and its width.4. Find the area of a triangle if its vertices are at (-2, 2), (6, 2), and (0, 10)5. Who is considered to be the Father of Geometry?6. What were his 3 undefined terms?7. What do we call a statement that we accept without proof?8. What do we call a statement that we accept because we have proven it?29
TF10.Points A and B are collinear.TF11.Points D and E are collinear.TF12.Points A, B, and C are collinear.TF13.Points D, E, and F are collinear.TF14.Points A, B, and C are coplanar.TF15.Points F, G, and H are coplanar.TF16.Points A, B, D, and E are coplanar.TF17.Points F, G, H, and J are coplanar.TF18.All points shown are coplanar.TF19.AB and FH intersect.TF20.FH and GJ intersect.TF21.AB and AC are the same.TF22.AB and BA are the same.TF23.AB and AC are the same.TF24.AB and BA are the same.TF25.AB and BA are opposite rays.TF26.BC and BA are opposite rays.TF27.AB and BA are the same.TF28.AB and AC are the same.TF29. 1 and 2 form a linear pair.TF30. 4 and 5 form a linear pair.TF31. 1 and 4 are vertical angles.TF32. 2 and 5 are vertical angles.TF33.m 1 m 2 m 3 180TTFF34.35.m 4 m 5 180 1 4DAFBGHJEC2134530
36. If m 1 40 and m 2 90 , thenm 3 m 4 2m 5 134537. CG GF Why?B38. If G is the midpoint of AD, thenACDGWhy?39. If AD bisects BE, thenFEWhy?40. m AGB m BGC m Why?41. If GF bisects AGE, thenWhy?42. One angle is 10 degrees more than 3 times its complement. Find the measure of the angle.43. An angle is 15 degrees less than half its supplement. Find the measure of the angle.H(2x 30) KL(4x 20) x(3x – 8) (2y – 17) J44.Ix m HLJ 45. x y m HLK 31
46.Point B is between A and C. If AB 10 and BC 6, then AC .47. Point B is between A and C. If AB 5 and AC 12, find BC.48.Find the midpoint of AB for A(-2, 8) and B(4, -3).49. If M(-3, 6) is the midpoint of XY, and the coordinates of X are (7, 1), find the coordinates of point Y.50. For A(5, -2) and B(8, 4), find AB.51. Use inductive reasoning to predict the next two terms in the sequence. 2, 6, 7, 21, 22, 66, 67, ,52. Find a counterexample to show that the conjecture is false.a. The sum of two integers is always positive.b. All four sided figures are rectangles.54. Answer each question and provide a reason that supports your answer.a.SSQ QR QWhy?b.RIf Q is the midpoint of SR, then .Why?c.Xm XYW m WYZ m WWhy?d.If YW bisects XYZ, thenYZWhy?32
opposite rays. 25. Two lines that lie in a plane but do not intersect. 26. Two lines that intersect and another line that does not intersect either one. 27. Two rays that are coplanar but not collinear. 28. Two planes that intersect at line MN. 29. Three planes that intersec
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