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Geometry: All-In-One Answers Version B (continued)Name Class DateLesson 1-5Examples.Measuring SegmentsLesson Objectives1Name Class Date1 Using the Segment Addition Postulate If AB 25,NAEP 2005 Strand: MeasurementFind the lengths of segments2x 6find the value of x. Then find AN and NB.Local Standards: NBAN(Vocabulary and Key Concepts.) (2x 6 AB3x that the distance between any two points is the absolute value of the difference of thecorresponding numbers.Simplify the left side.24Subtract 1 from each side.x 8All rights reserved.All rights reserved.correspondence with the real numbers soSubstitute. 1 253x( ) 6 NB x 7 ( 8 ) 7 AN 2x 6 2 8AN and NB 1015BSegment Addition Postulate) 25x 7Postulate 1-5: Ruler PostulateThe points of a line can be put into one-to-onex 7ANUse the Segment Addition Postulate (Postulate 1-6) to write an equation.Topic: Measuring Physical AttributesDivide each side by 3 .10Substitute 8 for x.15, which checks because the sum equals 25.Postulate 1-6: Segment Addition PostulateIf three points A, B, and C are collinear and B2 Finding Lengths M is the midpoint of RT.ABis between A and C, then AB BC AC.8x 365x 9Find RM, MT, and RT.CRMTUse the definition of midpoint to write an equation.RM A coordinate is a point’s distance and direction from zero on a number line.All rights reserved.RAB 5 u aQABabubacoordinate of Acoordinate of BCongruent (艑) segments are segments with the same length.2 cmABADC 2 cmC BAB CD ABCDDmidpoint Pearson Education, Inc., publishing as Pearson Prentice Hall.the length of AB Pearson Education, Inc., publishing as Pearson Prentice Hall.5x 95x segments.B AB MT(RM 5x 9 5(MT 8x 36 81515Substitute. 8xRT RM MT 84RM and MT are each36AddSubtract x) 9 ) 36 midpointDefinition of 8x 3645 3 x15to each side.5xfrom each side.Divide each side by 3 .84Substitute8416815for x.Segment Addition, which is half of168Postulate, the length of RT.Quick Check.1. EG 100. Find the value of x. Then find EF and FG.4x – 20x 15, EF 40; FG 60A midpoint is a point that divides a segment into two congruentA45E2x 30FGC2. Z is the midpoint of XY, and XY 27. Find XZ.BC13.510Geometry Lesson 1-5Daily Notetaking GuideL1Name Class DateLesson 1-6 Pearson Education, Inc., publishing as Pearson Prentice Hall.Examples.1 Naming Angles Name the angle at right in four ways.NAEP 2005 Strand: MeasurementTopic: Measuring Physical Attributes m BOCAB m AOC.0180m AOB m BOC 180T 1 Q TBQ90x5x x x angle9088Substitute 42for m ABC.m 2 46Subtract42Postulatefor m 1 and1882Bfrom each side.Cl2, lDECA O CrightAngle Addition m 2 1. a. Name CED two other ways.Bare the sides of the angle and the endpoint is the vertex of the angle.0,x,42Quick Check.If AOC is a straight angle, thenAn angle ( ) is formed by two rays with the same endpoint. The raysanglem 1 m 2 m ABCBBacuteAOO Cx All rights reserved.40140301502016 017 010Dobtuse90angle,x,straight anglex 5 180180 Pearson Education, Inc., publishing as Pearson Prentice Hall.AOB13050y Pearson Education, Inc., publishing as Pearson Prentice Hall.m 1206010Postulate 1-8: Angle Addition PostulateIf point B is in the interior of AOC, then7017 0A80x20.901AUse the Angle Addition Postulate (Postulate 1-8) to solve.16 0)7 0 10 06 0 11 02050 0 1310 0 110All rights reserved.C8001503014 04)mlCOD 5 u x 2 y u3G2 Using the Angle Addition Postulate Suppose that m 1 42and m ABC 88. Find m 2.b. If OC is paired with x and OD is paired with y,then.Finally, the name can be a point on one side, the vertex, and a pointon the other side of the angle: lAGC or lCGA .a. OA is paired with 0 and OB is paired with 180 .)lGThe name can be the vertex of the angle:Vocabulary and Key Concepts.)Cl3The name can be the number between the sides of the angle:Local Standards:Postulate 1-7: Protractor Postulate))Let OA and OB be opposite rays in a plane.) )OA , OB , and all the rays with endpoint O that can* )be drawn on one side of AB can be paired with thereal numbers from 0 to 180 so that11Geometry Lesson 1-5Name Class DateMeasuring AnglesLesson Objectives1 Find the measures of angles2 Identify special angle pairsDaily Notetaking GuideL1b. Critical Thinking Would it be correct to name any of theangles E? Explain.No, 3 angles have E for a vertex, so youneed more information in the name todistinguish them from one another.G2. If m DEG 145, find m GEF.35DEFAn acute angle has measurement between 0 and 90 .A right angle has a measurement of exactly 90 .Anobtuse anglehas measurement between 90 and 180 .A straight angle has a measurement of exactly 180 .Congruent angles are two angles with the same measure.12L1Geometry Lesson 1-6Daily Notetaking GuideAll-In-One Answers Version BL1L1Daily Notetaking GuideGeometry Lesson 1-6Geometry133

Geometry: All-In-One Answers Version B (continued)Name Class Date2 Constructing the Perpendicular BisectorLesson 1-7Basic ConstructionsLesson ObjectivesUse a compass and a straightedge toconstruct congruent segments andcongruent anglesUse a compass and a straightedge tobisect segments and anglesTopic: Relationships Among Geometric FiguresLocal Standards:sameStep 2 With thecompass setting, put the compasspoint on point B and draw another long arc. Label theintersectpoints where the two arcsas X and Y.is a ruler with no markings on it.A compass is a geometric tool used to draw circles and parts of circles called arcs.Perpendicular linesare two lines that intersect to form right angles.A perpendicular bisector of a segment is a line, segment, or ray that isperpendicular to the segment at its midpoint, thereby bisecting the segmentADCBAll rights reserved.All rights reserved.Construction is using a straightedge and a compass to draw a geometric figure.straightedgeJ* )TTWKM TW Pearson Education, Inc., publishing as Pearson Prentice Hall.MStep 2 Open the compass the length of KM.Step 3 With the same compass setting, put the compass pointon point T. Draw an arc that intersects the ray. Label thepoint of intersection W.* )Geometry Lesson 1-7MBYQuick Check.1. Use a straightedge to draw XY. Then construct RS so that RS 2XY.XRYDaily Notetaking GuideL1L1Name Class DateS2. Draw ST. Construct its perpendicular bisector.S14XAXY AB at the midpoint of AB, so XY is theof AB.perpendicular bisectorLKStep 1 Draw a ray with endpoint T.B* )* )Examples.to KM.XAKN1 Constructing Congruent Segments Construct TW congruentBStep 3 Draw XY . The point of intersection of AB and XY is M,theof AB.midpoint Pearson Education, Inc., publishing as Pearson Prentice Hall.coplanar angles.AYinto two congruent segments.An angle bisector is a ray that divides an angle into two congruentBStep 1 Put the compass point on point A and draw a long arc.greaterBe sure that the opening isthan 12AB.Vocabulary.AAAll rights reserved.2Given: AB.* )* )Construct: XY so that XY AB at the midpoint M of AB.NAEP 2005 Strand: GeometryTDaily Notetaking GuideGeometry Lesson 1-715Name Class Date2 Finding an Endpoint The midpoint of DG is M( 1, 5). One endpoint isLesson 1-8The Coordinate PlaneLesson Objectives1 Find the distance between two pointsin the coordinate plane2 Find the coordinates of the midpointof a segment in the coordinate planeD(1, 4). Find the coordinates of the other endpoint G.Use the Midpoint Formula. Let (x 1, y 1 ) beNAEP 2005 Strand: MeasurementTopic: Measuring Physical Attributesx 1x y 1ya 1 2 2 , 1 2 2 b becoordinates of G.Local Standards:( 1, 5). Solve for(1, 4)x2and the midpointandy2 Pearson Education, Inc., publishing as Pearson Prentice Hall.1Name Class Date, theFind the x-coordinate of G. d (x2 x1 )2(y2 y1 )2All rights reserved.Formula: The Distance FormulaThe distance d between two points A(x 1, y 1 ) and B(x 2, y 2 ) is.All rights reserved.Key Concepts.(x1 x2)y1 y2,22 1 x22d Use the Midpoint Formula. S5 2 1 x2 d Multiply each side by 2 . S10 3 x2Formula: The Midpoint FormulaThe coordinates of the midpoint M of AB with endpoints A(x 1, y 1 ) andB(x 2, y 2 ) are the following:M 1d Simplify. S64 y224 y2 x2The coordinates of G are ( 3, 6) .Quick Check.1. Find the coordinates of the midpoint of XY with endpoints X(2, 5) and Y(6, 13).Examples.1 Finding the Midpoint AB has endpoints (8, 9) and ( 6, 3). Find the coordinates of itsmidpoint M.Use the Midpoint Formula. Let (x 1, y 1 ) be(8, 9)and (x 2, y 2 ) be( 6, 3) .The midpoint has coordinates(x1 x2,y12The x-coordinate isThe y-coordinate is y2) 62) 1Substitute 8 for x1 and 6for x2. Simplify.6) 3Substitute 9 for y1 and 3for y2. Simplify.(1, 3).28 (29 (2 32The coordinates of the midpoint M are164Midpoint Formula.Geometry Lesson 1-8Geometry2Daily Notetaking GuideL1 Pearson Education, Inc., publishing as Pearson Prentice Hall. Pearson Education, Inc., publishing as Pearson Prentice Hall.(4, 4)2. The midpoint of XY has coordinates (4, 6). X has coordinates (2, 3).Find the coordinates of Y.(6, 9)L1Daily Notetaking GuideGeometry Lesson 1-8All-In-One Answers Version B17L1

Geometry: All-In-One Answers Version B (continued)Name Class DateName Class DateLesson 1-92 Finding Area of a Circle Find the area of 䉺B in terms of π.Perimeter, Circumference, and AreaLesson Objectives1Find perimeters of rectangles andsquares, and circumferences of circlesFind areas of rectangles, squares, andcircles2NAEP 2005 Strand: MeasurementIn 䉺B, r Topic: Measuring Physical AttributesA πLocal Standards:A π(A rdOhbSquare with side length s.Perimeter P 4ss2Rectangle with base b andheight h.CCircle with radius r anddiameter d.Perimeter P Circumference C Area A All rights reserved.bhs2b 2hAll rights reserved.Perimeter and Areasyd.BFormula for the area of a circle1.5)22.25πThe area of 䉺B isKey Concepts.Area A 1.5r2Substitute2.25p1.51.5 ydfor r.yd 2.Quick Check.1. a. Find the circumference of a circle with a radius of 18 m in terms of π.36p mpdor C 2prbhArea b. Find the circumference of a circle with a diameter of 18 m to the nearest tenth.pr 2 Pearson Education, Inc., publishing as Pearson Prentice Hall.equalIf two figures are congruent, then their areas are.All rights reserved.Postulate 1-10The area of a region is the sum of the areas of its non-overlapping parts.Examples.1 Finding Circumference 䉺G has a radius of6.5cm. Find thecircumference of 䉺G in terms of π. Then find the circumference to thenearest tenth.C 2 p(6.5C 13πC 13C 2πFormula for circumference of a circle)Substitute6.52. You are designing a rectangular banner for the front ofa museum. The banner will be 4 ft wide and 7 yd high.How much material do you need in square yards?2913 ydfor r.Exact answer40.840704The circumference of 䉺G is186.5 cmGr Pearson Education, Inc., publishing as Pearson Prentice Hall.56.5mPostulate 1-913pUse a calculator., or about40.8cm.Geometry Lesson 1-9Daily Notetaking GuideL1L1Daily Notetaking GuideGeometry Lesson 1-919Name Class DateName Class DateLesson 2-12 Writing the Converse of a Conditional Write the converse of the followingConditional Statementsconditional. Pearson Education, Inc., publishing as Pearson Prentice Hall.Lesson Objectives1 Recognize conditional statements2 Write converses of conditionalstatementsNAEP 2005 Strand: GeometryTopic: Mathematical ReasoningIf x 9, then x 3 12.The converse of a conditional exchanges the hypothesis and the conclusion.Local Standards:ConditionalVocabulary and Key Concepts.If an angle is a straight angle,ConverseIf the measure of an angle isSymbolic FormIf p ,then q .then its measure is 180 .If q ,qSp180 , then it is a straight angle.ConclusionHypothesisConclusionx 9x 3 12x 3 12x 9So the converse is: If x 3 12, then x 9.You read itpSqAll rights reserved.ConditionalAll rights reserved.Conditional Statements and ConversesStatementExampleConverseHypothesisthen p .Quick Check.1. Identify the hypothesis and the conclusion of this conditional statement:If y 3 5, then y 8.Hypothesis:A conditional is an if-then statement.y 3 5hypothesisis the part that follows if in an if-then statement.truth valueThe Pearson Education, Inc., publishing as Pearson Prentice Hall.The conclusion is the part of an if-then statement (conditional) that follows then.of a statement is “true” or “false” according to whether thestatement is true or false, respectively.The converse of the conditional “if p, then q” is the conditional “if q, then p.”Examples.1 Identifying the Hypothesis and the Conclusion Identify the hypothesisand conclusion: If two lines are parallel, then the lines are coplanar.In a conditional statement, the clause after if is the hypothesis and theclause after then is the conclusion. Pearson Education, Inc., publishing as Pearson Prentice Hall.TheConclusion:y 82. Write the converse of the following conditional:If two lines are not parallel and do not intersect, then they are skew.If two lines are skew, then they are not parallel and do not intersect.Hypothesis: Two lines are parallel.Conclusion: The lines are coplanar.20L1Geometry Lesson 2-1Daily Notetaking GuideAll-In-One Answers Version BL1L1Daily Notetaking GuideGeometry Lesson 2-1Geometry215

Geometry: All-In-One Answers Version B (continued)Name Class Date2 Identifying a Good Definition Is the following statement a goodLesson 2-2Biconditionals and DefinitionsLesson Objectives1definition? Explain.An apple is a fruit that contains seeds.NAEP 2005 Strand: GeometryWrite biconditionalsRecognize good definitions2Name Class DateThe statement is true as a description of an apple.Topics: Dimension and Shape; Mathematical ReasoningExchange “An apple” and “a fruit that contains seeds.” The converse reads:Local Standards:A fruit that contains seeds is an apple.Vocabulary and Key Concepts.There are many fruits that contain seeds but are not apples, such as lemonsand peaches. These areBiconditional StatementsA biconditional combines p S q and q S p as p 4 q.Symbolic FormAn angle is a straight angle ifYou read itp if andp4qonly if q .and only if its measure is 180 .All rights reserved.ExampleBiconditionalAll rights reserved.Statementstatement isbiconditionalThe original statementstatementis notcounterexamples, so the converse of the.is nota good definition because thereversible.Quick Check.1. Consider the true conditional statement. Write its converse. If the converseis also true, combine the statements as a biconditional.A biconditional statement is the combination of a conditional statement and its converse.Afalsecontains the words “if and only if.”Conditional: If three points are collinear, then they lie on the same line.Converse:Examples.If three points lie on the same line, then they are collinear.1 Writing a Biconditional Consider the true conditional statement. Write itsTo write the converse, exchange the hypothesis and conclusion.Converse: If x 15 20, then x 5.When you subtract 15 from each side to solve the equation, you get x 5. Becausetrueboth the conditional and its converse areabiconditionalusing the phrase, you can combine them inif and only if.Biconditional: x 5 if and only if x 15 20.22Geometry Lesson 2-2Daily Notetaking GuideL1trueThe converse is.Biconditional:All rights reserved.Conditional: If x 5, then x 15 20. Pearson Education, Inc., publishing as Pearson Prentice Hall. Pearson Education, Inc., publishing as Pearson Prentice Hall.converse. If the converse is also true, combine the statements as a biconditional.Three points are collinear if and only if they lie on the same line.2. Is the following statement a good definition? Explain.A square is a figure with four right angles.It is not a good definition because a rectangle has four right anglesand is not necessarily a square.L1Daily Notetaking GuideGeometry Lesson 2-223Name Class DateName Class DateLesson 2-33 Using the Law of Syllogism Use the Law of Syllogism to draw aDeductive Reasoningconclusion from the following true statements:If a quadrilateral is a square, then it contains four right angles.If a quadrilateral contains four right angles, then it is a rectangle.NAEP 2005 Strand: GeometryTopic: Mathematical ReasoningLocal Standards:The conclusion of the first conditional is the hypothesis of the secondconditional. This means that you can apply theVocabulary and Key Concepts.The Law of Syllogism: Ifthenis true.All rights reserved.conclusionIn symbolic form:If p S q is a true statement and p is true, then q is true.Law of SyllogismAll rights reserved.Law of DetachmentIf a conditional is true and its hypothesis is true, then its Pearson Education, Inc., publishing as Pearson Prentice Hall.Lesson Objectives1 Use the Law of Detachment2 Use the Law of SyllogismpSrpSqandqSrLaw of Syllogism.are true statements,is a true statement.So you can conclude:If a quadrilateral is a square, then it is a rectangle.Quick Check.1. Suppose that a mechanic begins work on a car and finds that the car will notstart. Can the mechanic conclude that the car has a dead battery? Explain.If p S q and q S r are true statements, then p S r is a true statement.No, there could be other things wrong with the car, such as a faulty starter.Deductive reasoning is a process of reasoning logically from given facts to a conclusion.1 Using the Law of Detachment A gardener knows that if it rains, thegarden will be watered. It is raining. What conclusion can he make?The first sentence contains a conditional statement. The hypothesis isit rains.Because the hypothesis is true, the gardener can conclude thatthe garden will be watered.2 Using the Law of Detachment For the given statements, what can youconclude?Given: If A is acute, then m A 90 . A is acute.A conditional and its hypothesis are both given as true.By theLaw of Detachmentconclusion of the conditional, m A 90 , is246Geometry Lesson 2-3Geometry Pearson Education, Inc., publishing as Pearson Prentice Hall. Pearson Education, Inc., publishing as Pearson Prentice Hall.Examples.2. If a baseball player is a pitcher, then that player should not pitch a completegame two days in a row. Vladimir Nuñez is a pitcher. On Monday, he pitchesa complete game. What can you conclude?Answers may vary. Sample: Vladimir Nuñez should not pitch a completegame on Tuesday.3. If possible, state a conclusion using the Law of Syllogism. If it is not possibleto use this law, explain why.a. If a number ends in 0, then it is divisible by 10.If a number is divisible by 10, then it is divisible by 5.If a number ends in 0, then it is divisible by 5.b. If a number ends in 6, then it is divisible by 2.If a number ends in 4, then it is divisible by 2., you can conclude that thetrueNot possible; the conclusion of one statement is not the hypothesis ofthe other statement.Daily Notetaking GuideL1L1Daily Notetaking GuideGeometry Lesson 2-3All-In-One Answers Version B25L1

Geometry: All-In-One Answers Version B (continued)Name Class DateLesson 2-4Examples.Reasoning in AlgebraLesson Objective1Name Class Date1 Justifying Steps in Solving an Equation Justify each step used to solveNAEP 2005 Strand: Algebra and GeometryConnect reasoning in algebra andgeometry5x 12 32 x for x.Topics: Algebraic Representations; MathematicalReasoningGiven: 5x 12 32 xLocal Standards:5x 44 xAddition Property of Equality4x 44Subtraction Property of EqualityKey Concepts.x 11If a b, then a c b c .Multiplication Property If a b, then a c b c .If a b and cb cIf a b, then b a .Transitive PropertyIf a b and b c, then a c .Substitution PropertyIf a b, then b can replace a in any expression.Distributive Propertya(b c) ac .AB A AIf AB CD, then CD AB .If A B, then B A .If AB CD and CD EF , then AB Transitive Property P 艑 SIf P R and R S, thenP

Lesson 1-1 Patterns and Inductive Reasoning reasoning based on patterns you observe. conjecture an example for which the conjecture is incorrect. half 224 12 2 2 Thursday 29 37 Friday Geometry Lesson 1-2 D