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1.1Identify Points, Lines, andPlanesGoalYour Notesp Name and sketch geometric figures.VOCABULARYUndefined term A word without a formal definitionPoint A point has no dimension. It is representedby a dot.Line A line has one dimension. It is represented bya line with two arrowheads.Plane A plane has two dimensions. It isrepresented by a shape that looks like a floor ora wall.Collinear points Points that lie on the same lineCoplanar points Points that lie in the same planeDefined Terms Terms that can be described usingknown wordsLine segment, endpoints Part of a line that consistsof two points, called endpoints, and all the pointson the line between the endpointsRay The ray AB consists of the endpoint A and all@## that lie on the same side of A as B.points on ABAB between A andOpposite rays If point C lies on @## ### and CB### are opposite rays.B, then CAIntersection The intersection of two or moregeometric figures is the set of points that thefigures have in common.Copyright Holt McDougal. All rights reserved.Lesson 1.1 Geometry Notetaking Guide1
Your NotesUNDEFINED TERMSPoint A point has no dimension.It is represented by a dot .Line A line has one dimension.It is represented by a line withtwo arrowheads, but it extendswithout end.Through any two points, there isexactly one line. You can use anytwo points on a line to name it.Plane A plane has two dimensions.It is represented by a shape thatlooks like a floor or wall, but itextends without end.Apoint AABline l, line AB(AB),or line BA(BA)MABCplane M or plane ABCThrough any three points not on the same line, thereis exactly one plane. You can use three points thatare not all on the same line to name a plane.There is a linethrough pointsL and Q that isnot shown in thediagram. Try toimagine whatplane LMQ wouldlook like if it wereshown.Example 1Name points, lines, and planesa. Give two other names for @## LN .Give another name for plane Z.Lb. Name three points that arecollinear. Name four pointsthat are coplanar.M PNbaLN are @### LM and line b . Othera. Other names for @## names for plane Z are plane LMP and LNP .b. Points L, M, and N lie on the same line, so theyare collinear. Points L, M, N, and P lie on the sameplane, so they are coplanar.Checkpoint Use the diagram in Example 1.1. Give two other names for @## MQ. Name a point that isnot coplanar with points L, N, and P.@### QM and line a; point Q2Lesson 1.1 Geometry Notetaking GuideCopyright Holt McDougal. All rights reserved.
Your NotesDEFINED TERMS: SEGMENTS AND RAYSAB ) and pointsLine AB (written as @## A and B are used here to definethe terms below.lineABSegment The line segment AB, or segment}AB, (written as AB ) consists of thesegmentendpoints A and B and all points onendpointendpoint@## ABAB that are between A and B.}}Note that AB can also be named BA .Ray The ray AB (written as ### AB )consists of the endpoint A and allpoints on @## AB that lie on the sameside of A as B .rayendpointAendpointBABAB and ### BA are different rays.Note that ### Example 2In Example 2,### WY and ### WXhave a commonendpoint ,but are notcollinear .So they are notopposite rays.Name segments, rays, and opposite rays}a. Give another name for VX.b. Name all rays with endpoints W.Which of these rays are opposite rays?}}a. Another name for VX is XV .YXWVZ#### , WY#### , WX#### , and WZ#### .b. The rays with endpoint W are WVWV and #### WX ,The opposite rays with endpoint W are #### WY and #### WZ .and #### Checkpoint Use the diagram in Example 2.}2. Give another name for YW.}WY3. Are ### VX and ### XV the same ray? Are ### VW and ### VX thesame ray? Explain.No, the rays do not have the same endpoint;Yes, the rays have a common endpoint, arecollinear, and consist of the same points.Copyright Holt McDougal. All rights reserved.Lesson 1.1 Geometry Notetaking Guide3
Your NotesExample 3Sketch intersections of lines and planesa. Sketch a plane and a line that intersects the plane atmore than one point.Sketching adiagram is a wayto show geometricrelationshipsinformally. A sketchcan be made byhand without usingmeasuring tools.b. Sketch a plane and a line that is in the plane. Sketchanother line that intersects the line and plane at a point.a.b.MMqpExample 4Sketch intersections of planesSketch two planes that intersect in a line.Step 1 Sketch one plane as if youare facing it.1Step 2 Sketch a second plane2that is horizontal .Use dashed lines toshow where one plane is hidden.3mStep 3 Sketch the line of intersection .Checkpoint Complete the following exercises.4. Sketch two different linesthat intersect a plane atdifferent points.MqpMX5. Name the intersection of @## and line a.HomeworkCYpoint MaM6. Name the intersection ofplane C and plane D.DXline a4Lesson 1.1 Geometry Notetaking GuideCopyright Holt McDougal. All rights reserved.
1.2Use Segments and CongruenceGoalYour Notesp Use segment postulates to identify congruentsegments.VOCABULARYPostulate, axiom A rule that is accepted withoutproofTheorem A rule that can be provedCoordinate The real number that corresponds to apointDistance The distance between two points Aand B, written as AB, is the absolute value of thedifference of the coordinates of A and B.Between When three points are collinear, you cansay that one point is between the other two.Congruent segments Line segments that have thesame lengthPOSTULATE 1RULER POSTULATEThe points on a line can bematched one to one with realnumbers. The real number thatcorresponds to a point is thecoordinate of the point.The distance between pointsA and B, written as AB, is theabsolute value of the differenceof the coordinates of A and B.Copyright Holt McDougal. All rights reserved.names of pointsABx1x2coordinates of pointsABx1x2AB 5 x2 2 x1 Lesson 1.2 Geometry Notetaking Guide5
Your NotesExample 1Apply the Ruler Postulate}Measure the length of CD to thenearest tenth of a centimeter.A ruler is a drawingtool that allows youto measure segmentlengths. You canalso use a ruler todraw segments ofgiven lengths.CDSolutionAlign one mark of a metric ruler with C. Then estimatethe coordinate of D. For example, if you align C with 1,D appears to align with 4.7 .Ccm 1D2345678CD 5 4.7 2 1 5 3.7Ruler postulate}The length of CD is about 3.7 centimeters.POSTULATE 2SEGMENT ADDITION POSTULATEIf B is between A and C, thenAB 1 BC 5 AC.If AB 1 BC 5 AC, then B isbetween A and C.Example 2ACABABCBCApply the Segment Addition PostulateStarting pointRoad Trip The locationsSshown lie in a straight line.64 miRest areaFind the distance from theRstarting point to the destination.87 miDestinationSolutionDThe rest area lies between the starting pointand the destination, so you can apply the SegmentAddition Postulate.SD 5 SR 1 RDSegment Addition Postulate5 64 1 87Substitute for SR and RD .5 151Add.The distance from the starting point to the destinationis 151 miles.6Lesson 1.2 Geometry Notetaking GuideCopyright Holt McDougal. All rights reserved.
Your NotesExample 3Find a lengthUse the diagram to find KL.38J15KLUse the Segment Addition Postulate to write an equation.Then solve the equation to find KL.JL 5 JK 1 KLSegment Addition Postulate38 5 15 1 KLSubstitute for JL and JK .23 5 KLSubtract 15 from each side.Example 4Compare segments for congruencePlot F(4, 5), G(21, 5), H(3, 3), and J(3, 22) in a}coordinate plane. Then determine whether FG and}HJ are congruent.yHorizontal segment: Subtract thex-coordinates of the endpoints.F(4, 5)G(21, 5)H(3, 3)FG 5 4 2 (21) 5 51Vertical segment: Subtract they-coordinates of the endpoints.x1J(3, 22)HJ 5 3 2 (22) 5 5}}}}FG and HJ have the same length. So FG HJ.Checkpoint Complete the following exercises.1}1. Find the length of AB to the nearest }inch. Then8draw a segment with the same length.BA71}inches; Check drawingHomework8552. Find QS and PQ.61; 24PQ31R30S3. Consider the points A(22, 21), B(4, 21), C(3, 0), and}}D(3, 5). Are AB and CD congruent? NoCopyright Holt McDougal. All rights reserved.Lesson 1.2 Geometry Notetaking Guide7
1.3Use Midpoint and DistanceFormulasp Find lengths of segments in the coordinate plane.GoalYour NotesVOCABULARYMidpoint The point that divides a segment into twocongruent segmentsSegment bisector A point, ray, line, line segment, orplane that intersects the segment at its midpointExample 1Find segments lengthsFind RS.VR21.7TSWSolution}Point T is the midpoint of RS. So, RT 5 TS 5 21.7.RS 5 RT 1 TSSegment Addition Postulate5 21.7 1 21.7Substitute.5 43.4Add.}The length of RS is 43.4 .Checkpoint Complete the following exercise.1. Find AB.D124ACB14}28Lesson 1.3 Geometry Notetaking GuideCopyright Holt McDougal. All rights reserved.
Your NotesExample 2Use algebra with segment lengths}Point C is the midpoint of BD.}Find the length of BC.3x 2 22x 1 3CBDSolutionStep 1 Write and solve an equation.BC 5 CDWrite equation.3x 2 2 5 2x 1 3Substitute.x22 5 3Subtract 2x fromeach side.x5 5Add 2 to each side.Step 2 Evaluate the expression for BC when x 5 5 .BC 5 3x 2 2 5 3(5) 2 2 5 13}So, the length of BC is 13 units.Checkpoint Complete the following exercise.}}2. Point K is the midpoint of JL. Find the length of KL.8 2 3x2x 1 5KJL16}5THE MIDPOINT FORMULAyThe coordinates of themidpoint of a segmentyare the averages of thex-coordinates and of they-coordinates of the endpoints.11 y2(2y1x1 1 x22,y1 1 y22)A(x1, y1)If A(x1, y1)and B(x2, y2) arepoints in a coordinate plane, then}the midpoint M of AB has coordinates1B(x2, y2)y2x1x1 1 x2x2x2x1 1 x2 y1 1 y2}, } .22Copyright Holt McDougal. All rights reserved.2Lesson 1.3 Geometry Notetaking Guide9
Your NotesUse the Midpoint FormulaExample 3}a. Find Midpoint The endpoints of PR are P(22, 5) andR(4, 3). Find the coordinates of the midpoint M.}b. Find Endpoint The midpoint of AC is M(3, 4). Oneendpoint is A(1, 6). Find the coordinates of endpoint C.Solutiona. Use the Midpoint Formula.M122 142,5yP(22, 5)1322M(?, ?)R(4, 3)1x15 M( 1 , 4 )}The coordinates of the midpoint of PR are M(1, 4) .Multiply each sideof the equation bythe denominator toclear the fraction.b. Let (x, y) be the coordinates ofendpoint C. Use the MidpointFormula to find x and y.Step 1 Find x.y A(1, 6)M (3, 4)Step 2 Find y.C (x, y)11 1x5 321 1x5 66 1y5 426 1y5 8x5 5y5 21xThe coordinates of endpoint C are (5, 2) .Checkpoint Complete the following exercises.}3. The endpoints of CD are C(28, 21) and D(2, 4).Find the coordinates of the midpoint M.13M 23, }22}4. The midpoint of XZ is M(5, 26). One endpoint isX(23, 7). Find the coordinates of endpoint Z.(13, 219)10Lesson 1.3 Geometry Notetaking GuideCopyright Holt McDougal. All rights reserved.
Your NotesyTHE DISTANCE FORMULAB(x2, y2)If A(x1, y1) and B(x2, y2)are points in a coordinateplane, then the distancebetween A and B is y2 2 y1 A(x1, y1) x2 2 x1 C(x2, y1)x}}}AB 5 Ï ( x2 2 x1 )2 1 ( y2 2 y1 )2Example 4Use the Distance FormulaWhat is the approximate length}of RT, with endpoints R(3, 2) andT(24, 3)?yT (24, 3)R(3, 2)1SolutionUse the Distance Formula.1}}}RT 5 Ï ( x2 2 x1 )2 1 ( y2 2 y1 )2}}}5 Ï( 24 2 3)2 1 ( 3 2 2 )2}}5 Ï( 27 )2 1 ( 1 )2Distance FormulaSubstitute.Subtract.}}5 Ï 49 1 1Evaluate powers.}The symbol ømeans "isapproximately equalto."x5 Ï 50Add.ø 7.07Use a calculator.}The length of RT is about 7.07 .Checkpoint Complete the following exercise.Homework}5. What is the approximate length of GH, withendpoints G(5, 21) and H(23, 6)?about 10.63Copyright Holt McDougal. All rights reserved.Lesson 1.3 Geometry Notetaking Guide11
1.4Measure and Classify AnglesGoalYour Notesp Name, measure, and classify angles.VOCABULARYAngle An angle consists of two different rays withthe same endpoint.Sides of an angle In an angle, the rays are calledthe sides of the angle.Vertex of an angle In an angle, the endpoint is thevertex of the angle.OA and ### OB can beMeasure of an angle In AOB, #### matched one to one with real numbers from 0 to180. The measure of AOB is equal to the absolutevalue of the difference between the realOA and ### OB.numbers for #### Acute angle An angle that measures between 08and 908Right angle An angle that measures 908Obtuse angle An angle that measures between 908and 1808Straight angle An angle that measures 1808Congruent angles Angles with the same measureAngle bisector A ray that divides an angle into twoangles that are congruentExample 1You should notname any of theseangles B becauseall three angleshave B as theirvertex .12Name anglesName the three angles in the diagram. ABC , or CBA CBD , or DBC ABD , or DBALesson 1.4 Geometry Notetaking GuideACBDCopyright Holt McDougal. All rights reserved.
80 90 10 070 10 0 90 80 110 170260 0 11060 0 12350 0 150 013A170 1806000 1 20 10150 3014 04Consider ### OB and point A onone side of ### OB. The rays ofthe form ### OA can be matchedone to one with the realnumbers from 0 to 180 .315 0 401 040POSTULATE 3: PROTRACTOR POSTULATE0 10180 170 1 2060Your NotesOBThe measure of AOB is equal to the absolutevalue of the difference between the real numbersOA and ### OB.for ### Measure and classify anglesa. WSRb. TSWc. RSTd. VSTWV0 10180 170 1 2060A protractor isa drawing toolthat allows you tomeasure angles.You can also use aprotractor to drawangles of givenmeasures.80 90 10 070 10 0 90 80 110 170260 0 11060 0 12350 0 150 013170 1806000 1 20 10150 3014 04Use the diagram to find themeasure of the indicated angle.Then classify the angle.315 0 401 040Example 2RSTSR is lined up with the 08 on the outer scale of thea. ### SW passes through 658 on the outerprotractor. ### scale. So, m WSR 5 658 . It is an acute angle.ST is lined up with the 08 on the inner scale of theb. ### SW passes through 1158 on the innerprotractor. ### scale. So, m TSW 5 1158 . It is an obtuse angle.c. m RST 5 1808 . It is a straight angle.d. m VST 5 908 . It is a right angle.Checkpoint Complete the following exercises.1. Name all the angles inthe diagram at the right.JFGH FGH or HGF, FGJ or JGF, JGH or HGJ2. What type of angles do the x-axis and y-axis form ina coordinate plane?right anglesCopyright Holt McDougal. All rights reserved.Lesson 1.4 Geometry Notetaking Guide13
Your NotesA point is in theinterior of an angleif it is betweenpoints that lie oneach side of theangle.interiorPOSTULATE 4: ANGLE ADDITION POSTULATEWords If P is in the interior of RST, then the measure of RST is equal to the sum ofthe measures of RSPand PST .Rm/RSTm/RSP PSm/PSTTSymbols If P is in the interior of RST,then m RST 5 m RSP 1 m PST .Example 3Find angle measuresGiven that m GFJ 5 1558,find m GFH and m HFJ.H(4x 2 1)8(4x 1 4)8SolutionGJFStep 1 Write and solve an equation to find the value of x.m GFJ 5 m GFH 1 m HFJ1558 5 ( 4x 1 4 )8 1 ( 4x 2 1 )8Angle AdditionPostulateSubstitute.155 5 8x 1 3Combine like terms.152 5 8xSubtract 3 fromeach side.19 5 xDivide each sideby 8 .Step 2 Evaluate the given expressions when x 5 19 .m GFH 5 ( 4x 1 4 )8 5 ( 4 p 19 1 4 )8 5 808 .m HFJ 5 ( 4x 2 1 )8 5 ( 4 p 19 2 1 )8 5 758 .So, m GFH 5 808 and m HFJ 5 758 .Checkpoint Complete the following exercise.3. Given that VRS is a right angle,find m VRT and TRS.(x 2 4)8VTm VRT 5 198, m TRS 5 718(3x 1 2)8R14Lesson 1.4 Geometry Notetaking GuideSCopyright Holt McDougal. All rights reserved.
Your NotesExample 4Identify congruent anglesLIdentify all pairs of congruent anglesin the diagram. If m P 5 1208, whatis m N?MPSolutionThere are two pairs of congruent angles:N P N and L MBecause P N , m P 5 m N .So, m N 5 1208 .Example 5Double an angle measureIn the diagram at the right, ### WYbisects XWZ, and m XWY 5 298.Find m XWZ.ZYXWSolutionBy the Angle Addition Postulate,m XWZ 5 m XWY 1 m YWZ .WY bisects XWZ, you knowBecause ### XWY YWZ .So, m XWY 5 m YWZ , and you can writem XWZ 5 m XWY 1 m YWZ5 298 1 298 5 588 .Checkpoint Complete the following exercises.4. Identify all pairs of congruentangles in the diagram. Ifm B 5 1358, what is m D?HomeworkBCAD B D and A C; 13585. In the diagram below, ### KM bisects LKN and m LKM 5 788. Findm LKN.1568Copyright Holt McDougal. All rights reserved.MNLKLesson 1.4 Geometry Notetaking Guide15
1.5Describe Angle PairRelationshipsGoalYour Notesp Use special angle relationships to find anglemeasures.VOCABULARYComplementary angles Two angles whose sum is908Supplementary angles Two angles whose sum is1808Adjacent angles Two angles that share a commonvertex or side, but have no common interior pointsLinear pair Two adjacent angles are a linear pair iftheir noncommon sides are opposite rays.Vertical angles Two angles are vertical angles iftheir sides form two pairs of opposite rays.Example 1In Example 1, BDE and CDEshare a commonvertex. But theyshare commoninteriorpoints, so theyare not adjacentangles.Identify complements and supplementsIn the figure, name a pair ofcomplementary angles, a pairof supplementary angles, anda pair of adjacent angles.ABE5281288388DCSolutionBecause 528 1 388 5 908, ABD and CDBare complementary angles.Because 528 1 1288 5 1808, ABD and EDBare supplementary angles.Because CDB and BDE share a common vertexand side, they are adjacent angles.16Lesson 1.5 Geometry Notetaking GuideCopyright Holt McDougal. All rights reserved.
Your NotesAngles aresometimes namedwith numbers. Anangle measure ina diagram has adegree symbol.An angle namedoes not.Example 2Find measures of complements and supplementsa. Given that 1 is a complement of 2 andm 2 5 578, find m 1.b. Given that 3 is a supplement of 4 andm 4 5 418, find m 3.Solutiona. You can draw a diagram withcomplementary adjacent anglesto illustrate the relationship.12578m 1 5 908 2 m 2 5 908 2 578 5 338b. You can draw a diagram withsupplementary adjacent anglesto illustrate the relationship.34 418m 3 5 1808 2 m 4 5 1808 2 418 5 1398Checkpoint Complete the following exercises.1. In the figure, name a pair ofcomplementary angles, a pairof supplementary angles, anda pair of adjacent angles.D508AFE 1408B408CGcomplementary: DEF and ABC;supplementary; FEG and ABC;adjacent: DEF and FEG2. Given that 1 is a complement of 2 andm 1 5 738, find m 2.1783. Given that 3 is a supplement of 4 andm 4 5 378, find m 3.1438Copyright Holt McDougal. All rights reserved.Lesson 1.5 Geometry Notetaking Guide17
Your NotesIn a diagram, youcan assume thata line that looksstraight is straight.In Example 3, B, C,BD .and D lie on @## So, BCD is astraightangle.Example 3Find angle measuresBasketball The basketball poleforms a pair of supplementaryangles with the ground. Findm BCA and m DCA.ASolution(4x 2 3)8(3x 1 8)8Step 1 Use the fact that 1808is the sum of the measuresof supplementary angles.m BCA 1 m DCA 5 1808( 3x 1 8 )8 1 ( 4x 2 3 )8 5 18087x 1 5 5 1807x 5 175x 5 25BCDWrite equation.Substitute.Combine like terms.Subtract.Divide.Step 2 Evaluate the original expressions when x 5 25 .m BCA 5 ( 3x 1 8 )8 5 ( 3 p 25 1 8 )8 5 838 .m DCA 5 ( 4x 2 3 )8 5 ( 4 p 25 2 3 )8 5 978 .The angle measures are 838 and 978 .Checkpoint Complete the following exercise.4. In Example 3, suppose the angle measures are(5x 1 1)8 and (6x 1 3)8. Find m BCA andm DCA.818 and 99818Lesson 1.5 Geometry Notetaking GuideCopyright Holt McDougal. All rights reserved.
Your NotesIn the diagram,one side of 1 andone side of 4 areopposite rays. Butthe angles are not alinear pair becausethey are notadjacent .Example 4Identify angle pairsIdentify all of the linear pairsand all of the vertical anglesin the figure at the right.213 4 5SolutionTo find vertical angles, look for angles formedby intersecting lines . 1 and 3 are vertical angles.To find linear pairs, look for adjacent angles whosenoncommon sides are opposite rays . 1 and 2 are a linear pair. 2 and 3 are alinear pair.Checkpoint Complete the following exercise.5. Identify all of the linear pairsand all of the vertical anglesin the figure.3 2 14 5 6linear pairs: none; vertical angles: 1 and 4, 2 and 5, 3 and 6Example 5One angle measureshould be fourtimes the otherangle measure. Themeasures shouldalso have a sum of180 for the anglesto form a linear pair.Find angle measures in a linear pairTwo angles form a linear pair. The measure of one angleis 4 times the measure of the other. Find the measureof each angle.4x8x8SolutionLet x8 be the measure of one angle. The measure of theother angle is 4x8 . Then use the fact that the angles ofa linear pair are supplementary to write an equation.x8 1 4x8 5 18085x 5 180x 5 36Write an equation.Combine like terms.Divide each side by 5 .The measures of the angles are 368 and4(368) 5 1448 .Copyright Holt McDougal. All rights reserved.Lesson 1.5 Geometry Notetaking Guide19
Your NotesCheckpoint Complete the following exercise.6. Two angles form a linear pair. The measure of oneangle is 3 times the measure of the other. Find themeaure of each angle.458 and 1358CONCEPT SUMMARY: INTERPRETING A DIAGRAMThere are some things you canconclude from a diagram, andsome you cannot. For example,here are some things that youcan conclude from the diagramat the right.DEABC All points shown are coplanar . Points A, B, and C are collinear , and B is betweenA and C.AC , ### BD, and ### BE intersect at point B. @## DBE and EBC are adjacent angles, and ABCis a straight angle .Homework Point E lies in the interior of DBC.In the diagram above, you cannot}}conclude that AB BC, that DBE EBC, or that ABD isa right angle. This information mustbe indicated, as shown at the right.20Lesson 1.5 Geometry Notetaking GuideDEABCCopyright Holt McDougal. All rights reserved.
1.6Classify PolygonsGoalYour Notesp Classify polygons.VOCABULARYPolygon A polygon is a closed plane figure withthe following properties: (1) It is formed by threeor more line segments called sides. (2) Each sideintersects exactly two sides, one at each endpoint,so that no two sides with a common endpoint arecollinear.Sides The sides of a polygon are the line segmentsthat form the polygon.Vertex A vertex of a polygon is an endpoint of aside of the polygon.Convex A polygon is convex if no line that containsa side of the polygon contains a point in theinterior of the polygon.Concave A concave polygon is a polygon that isnot convex.n-gon An n-gon is a polygon with n sides.Equilateral A polygon is equilateral if all of its sidesare congruent.Equiangular A polygon is equiangular if all of itsangles in the interior are congruent.Regular A polygon is regular if all sides and allangles are congruent.Copyright Holt McDougal. All rights reserved.Lesson 1.6 Geometry Notetaking Guide21
Your NotesIDENTIFYING POLYGONSIn geometry, a figure that lies in a plane is called aplane figure. A polygon is a closed plane figure withthe following properties.1. It is formed by three or more line segmentscalled sides .2. Each side intersects exactly two sides, one ateach endpoint, so that no two sides with a commonendpoint are collinear .Each endpoint of a side is a vertex of thepolygon. The plural of vertex is vertices.BA polygon can be named by listing thevertices in consecutive order. For example,ABCDE and CDEAB are both correctAnames for the polygon at the right.Example 1A plane figure istwo-dimensional.Later, you will studythree-dimensionalspace figures suchas prisms andcylinders.CDEIdentify polygonsTell whether the figure is a polygon and whether it isconvex or concave.b.a.c.Solutiona. Some segments intersect more than two segments, soit is not a polygon .b. The figure is a convex polygon .c. The figure is a concave polygon .Checkpoint Tell whether the figure is a polygon andwhether it is convex or concave.1.2.convex polygon22Lesson 1.6 Geometry Notetaking Guidenot a polygonCopyright Holt McDougal. All rights reserved.
Your NotesExample 2Classify polygonsClassify the polygon by the number of sides.Tell which terms apply to the polygon:equilateral, equiangular, regular, or not regular.SolutionThe polygon has 8 sides. It is equilateral andequiangular, so it is a regular octagon .Example 3Find side lengths(4x 1 3) mmThe head of a bolt is shaped likea regular hexagon. The expressions(5x 2 1) mmshown represent side lengths of thehexagonal bolt. Find the length of a side.SolutionFirst, write and solve an equation to find the valueof x. Use the fact that the sides of a regular hexagonare congruent .4x 1 3 5 5x 2 14 5 xCheck that thisis a reasonableside length for ahexagonal bolt.Write an equation.Simplify.Then evaluate one of the expressions to find a side lengthwhen x 5 4 .4x 1 3 5 4( 4 ) 1 3 5 19The length of a side is 19 millimeters.Checkpoint Complete the following exercises.3. Classify the polygon by the numberof sides. Tell which terms apply tothe polygon: equilateral, equiangular,regular, or not regular.Homeworkquadrilateral; not regular4. The expressions (4x 1 8)8 and (5x 2 5)8 representthe measures of two of the congruent angles inExample 3. Find the measure of an angle.608Copyright Holt McDougal. All rights reserved.Lesson 1.6 Geometry Notetaking Guide23
1.7Find Perimeter, Circumference,and AreaGoalYour Notesp Find dimensions of polygons.FORMULAS FOR PERIMETER P, AREA A, ANDCIRCUMFERENCE CSquareside length sRectanglelength l andwidth wP 5 4sA 5 s2P 5 2l 1 2wA 5 lwsTriangleside lengths a, b,and c, base b,and height h.P5 a1b1cahcCircleradius rC 5 2πrA 5 πr 2wr1bhA5 }Pi (π) is the ratio of acircle's circumference toits diameter.Example 1Find the perimeter and area of a rectangleb2Tennis The in-bounds portion of a singles tenniscourt is shown. Find its perimeter and area.PerimeterAreaP 5 2l 1 2wA 5 lw5 2( 78 ) 1 2( 27 )5 78 ( 27 )5 2105 210678 ft27 ftThe perimeter is 210 ft and the area is 2106 ft2.Checkpoint Complete the following exercise.1. In Example 1, the width of the in-bounds rectangleincreases to 36 feet for doubles play. Find theperimeter and area of the in-bounds rectangle.perimeter: 228 ft, area: 2808 ft224Lesson 1.7 Geometry Notetaking GuideCopyright Holt McDougal. All rights reserved.
Your NotesThe approximations22are3.14 and }7commonly usedas approximationsfor the irrationalnumber π. Unlesstold otherwise, use3.14 for π.Example 2Find the circumference and area of a circleArchery The smallest circle on an Olympic target is12 centimeters in diameter. Find the approximatecircumference and area of the smallest circle.SolutionFirst find the radius. The diameter is 12 centimeters,1( 12 ) 5 6 centimeters.so the radius is }2Then find the circumference and area. Use 3.14 for π.P 5 2πr ø 2( 3.14 )( 6 ) 5 37.68 cmA 5 πr 2 ø 3.14 ( 6 )2 5 113.04 cm2Checkpoint Find the approximate circumference andarea of the circle.2.8mExample 3C ø 50.24 m; A ø 200.96 m2Using a coordinate planeTriangle JKL has vertices J(1, 6), K(6, 6), and L(3, 2).Find the approximate perimeter of triangle JKL.Write down yourcalculations tomake sure you donot make a mistakesubstituting valuesin the DistanceFormula.SolutionFirst draw triangle JKL in a coordinateplane. Then find the side lengths.}Because JK is horizontal, use theRuler Postulate to find JK. Usethe Distance Formula to findJL and LK.yJ(1,6)K(6, 6)L(3, 2)11xJK 5 6 2 1 5 5 units}}}}JL 5 Ï( 3 2 1)2 1 (2 2 6 )2 5 Ï 20 ø 4.5 units}}}}LK 5 Ï( 6 2 3)2 1 ( 6 2 2)2 5 Ï 25 5 5 unitsThen find the perimeter.P 5 JK 1 JL 1 LK ø 5 1 4.5 1 5 5 14.5 units.Copyright Holt McDougal. All rights reserved.Lesson 1.7 Geometry Notetaking Guide25
Your NotesExample 4Solve a multi-step problemLawn care You are using a roller to smooth a lawn. Youcan roll about 125 square yards in one minute. Abouthow many minutes does it take to roll a lawn that is120 feet long and 75 feet wide?SolutionYou can roll the lawn at a rate of 125 square yards perminute. So, the amount of time it takes you to roll thelawn depends on its area .Step 1 Find the area of the rectangular lawn.Area 5 lw 5 120 ( 75 ) 5 9000 ft2The rolling rate is in square yards per minute.Rewrite the area of the lawn in square yards. Thereare 3 feet in 1 yard, and 3 2 5 9 square feetin one square yard.9000 ft2 p1 yd29ft25 1000 yd2Use unit analysis.Step 2 Write a verbal model to represent the situation.Then write and solve an equation based on theverbal model.Let t represent the total time (in minutes) neededto roll the lawn.Rolling rateTotal timeArea of lawn5322(min)(yd )(yd per min)You can roll about1250 yards in 10minutes. Use thisfact to check thatyour solution isreasonable for thelawn area you foundin Step 1.261000 5 125 p t8 5tSubstitute.Divide each sideby 125 .It takes about 8 minutes to roll the lawn.Lesson 1.7 Geometry Notetaking GuideCopyright Holt McDougal. All rights reserved.
Your NotesIn Example 5,you may want tomake and labela sketch of thetriangle. The sketchwill not be exact,but it will help youvisualize the giveninformation.Find unknown lengthExample 5The base of a triangle is 24 feet. Its area is 216 squarefeet. Find the height of the triangle.Solutionh1A5 }bh2Area of a triangle24 ft1(24)(h)216 5 }Substitute.216 5 12hMultiply.218 5 hSolve for h.The height is 18 feet.Checkpoint Complete the following exercises.y3. Find the perimeter of thetriangle shown at the right.B(7, 6)about 17.1 unitsC(7, 3)1A(1, 1)1x4. Suppose a lawn is half as long and half as wide asthe lawn in Example 4. Will it take half the time toroll the lawn? Explain.No, it will take a quarter of the time to roll thelawn because it is a quarter of the original area.Homework5. The area of a triangle is 96 square inches, and itsheight is 8 inches. Find the length of its base.24 inchesCopyright Holt McDougal. All rights reserved.Lesson 1.7 Geometry Notetaking Guide27
Words to ReviewGive an example of the vocabulary word.Point, line, planeCollinear pointsBAplanelinepointA and B are collinearpoints.Coplanar pointsLine segment, endpointsDTDCD and T are coplanarpoints.Ray}CD is a line segmentwith endpoints C and D.Opposite raysYACBX### XY is a ray with initialpoint X.If C is between A andB, then ### CA and ### CB areopposite rays.IntersectionPos
UNDEFINED TERMS Point A point has no dimension. A It is represented by a dot . point A Line A line has one dimension. A B line l, line AB(AB), or line BA(BA) It is represented by a line with
