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1Basics of GeometryPoints, Lines, and PlanesMeasuring and Constructing SegmentsUsing Midpoint and Distance FormulasPerimeter and Area in theCoordinate Plane1.5 Measuring and Constructing Angles1.6 Describing Pairs of Angles1.11.21.31.4Chapter Learning Target:Understand basics of geometry.Chapter Success Criteria: I can name points, lines, and planes.I can measure segments and angles.I can use formulas in the coordinate plane.I can construct segments and angles.SEE the Big IdeaAlamillo Bridge (p.(p 53)Soccer (p. 49)Shed (p(p. 33)Skateboard (p.(p 20)Sulfur Hexafluoride (p. 7)hsnb 2019 geo pe 01op.indd xx2/3/18 2:59 PM

Maintaining Mathematical ProficiencyFinding Absolute ValueExample 1 Simplify 7 1 . 7 1 7 ( 1) Add the opposite of 1. 8 Add. 8Find the absolute value. 7 1 8Simplify the expression.1. 8 12 2. 6 5 3. 4 ( 9) 4. 13 ( 4) 5. 6 ( 2) 6. 5 ( 1) 7. 8 ( 7) 8. 8 13 9. 14 3 Finding the Area of a TriangleExample 2 Find the area of the triangle.5 cm18 cm1Write the formula for area of a triangle.1Substitute 18 for b and 5 for h. —2 (90)1Multiply 18 and 5. 45Multiply —12 and 90.A —2 bh —2 (18)(5)The area of the triangle is 45 square centimeters.Find the area of the triangle.10.11.12.7 yd22 m24 yd16 in.25 in.14 m13. ABSTRACT REASONING Describe the possible values for x and y when x y 0. What does itmean when x y 0? Can x y 0? Explain your reasoning.Dynamic Solutions available at BigIdeasMath.comhs geo pe 01co.indd 111/19/15 8:15 AM

MathematicalPracticesMathematically proficient students carefully specify units of measure.Specifying Units of MeasureCore ConceptCustomary Units of LengthMetric Units of Length1 foot 12 inches1 yard 3 feet1 mile 5280 feet 1760 yards1 centimeter 10 millimeters1 meter 1000 millimeters1 kilometer 1000 meters1in.231 in. 2.54 cmcm123456789Converting Units of MeasureFind the area of the rectangle in square centimeters.Round your answer to the nearest hundredth.2 in.SOLUTION6 in.Use the formula for the area of a rectangle. Convert the units of length from customary unitsto metric units.Area (Length)(Width)Formula for area of a rectangle (6 in.)(2 in.)[ (2.54 cm (6 in.) —1 in.Substitute given length and width.) ] [ (2 in.)( 2.541 in.cm ) ]—Multiply each dimension by the conversion factor. (15.24 cm)(5.08 cm)Multiply. 77.42 cm2Multiply and round to the nearest hundredth.The area of the rectangle is about 77.42 square centimeters.Monitoring ProgressFind the area of the polygon using the specified units. Round your answer to the nearest hundredth.1. triangle (square inches)2 cm2 cm2. parallelogram (square centimeters)2 in.2.5 in.3. The distance between two cities is 120 miles. What is the distance in kilometers? Round your answerto the nearest whole number.2Chapter 1hs geo pe 01co.indd 2Basics of Geometry1/19/15 8:15 AM

1.1Points, Lines, and PlanesEssential QuestionHow can you use dynamic geometry softwareto visualize geometric concepts?Using Dynamic Geometry SoftwareWork with a partner. Use dynamic geometry software to draw several points. Also,draw some lines, line segments, and rays. What is the difference between a line, a linesegment, and a ray?SampleBAGFCEDIntersections of Lines and PlanesUNDERSTANDINGMATHEMATICALTERMSTo be proficient in math,you need to understanddefinitions and previouslyestablished results.An appropriate tool, suchas a software package,can sometimes help.Work with a partner.a. Describe and sketch the ways in which two lines canintersect or not intersect. Give examples of each usingthe lines formed by the walls, floor, and ceiling inyour classroom.b. Describe and sketch the ways in which a lineand a plane can intersect or not intersect.Give examples of each using the walls,floor, and ceiling in your classroom.c. Describe and sketch the ways in whichtwo planes can intersect or not intersect.Give examples of each using the walls,floor, and ceiling in your classroom.QPBAExploring Dynamic Geometry SoftwareWork with a partner. Use dynamic geometry software to explore geometry. Use thesoftware to find a term or concept that is unfamiliar to you. Then use the capabilitiesof the software to determine the meaning of the term or concept.Communicate Your Answer4. How can you use dynamic geometry software to visualize geometric concepts?Section 1.1hs geo pe 0101.indd 3Points, Lines, and Planes31/19/15 8:18 AM

1.1LessonWhat You Will LearnName points, lines, and planes.Name segments and rays.Core VocabulVocabularylarryundefined terms, p. 4point, p. 4line, p. 4plane, p. 4collinear points, p. 4coplanar points, p. 4defined terms, p. 5line segment, or segment, p. 5endpoints, p. 5ray, p. 5opposite rays, p. 5intersection, p. 6Sketch intersections of lines and planes.Solve real-life problems involving lines and planes.Using Undefined TermsIn geometry, the words point, line, and plane are undefined terms. These words donot have formal definitions, but there is agreement about what they mean.Core ConceptUndefined Terms: Point, Line, and PlaneAPoint A point has no dimension. A dot represents a point.point ALineA line has one dimension. It is represented by aline with two arrowheads, but it extends without end.Through any two points, there is exactly one line. Youcan use any two points on a line to name it.Aline , line AB (AB),or line BA (BA)Plane A plane has two dimensions. It is representedby a shape that looks like a floor or a wall, but itextends without end.Through 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.BAMCBplane M, or plane ABCCollinear points are points that lie on the same line. Coplanar points are points thatlie in the same plane.Naming Points, Lines, and Planesa. Give two other names for ⃖ ⃗PQ and plane R.b. Name three points that are collinear. Name fourpoints that are coplanar.SOLUTIONQTVSnPmRa. Other names for ⃖ ⃗PQ are ⃖ ⃗QP and line n. Othernames for plane R are plane SVT and plane PTV.b. Points S, P, and T lie on the same line, so they are collinear. Points S, P, T,and V lie in the same plane, so they are coplanar.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com1. Use the diagram in Example 1. Give two other names for ⃖ ⃗ST . Name a pointthat is not coplanar with points Q, S, and T.4Chapter 1hs geo pe 0101.indd 4Basics of Geometry1/19/15 8:18 AM

Using Defined TermsIn geometry, terms that can be described using known words such as point or line arecalled defined terms.Core ConceptDefined Terms: Segment and RaylineThe definitions below use line AB (written as ⃖ ⃗AB)and points A and B.ASegment The line segment AB, or segment AB,— ) consists of the endpoints A and B(written as ABand all points on ⃖ ⃗AB that are between A and B.— can also be named BA—.Note that ABBsegmentendpointendpointABRay The ray AB (written as ⃗AB ) consists of therayAB that lie on theendpoint A and all points on ⃖ ⃗same side of A as B.endpointANote that ⃗AB and ⃗BA are different rays.BendpointOpposite Rays If point C lies on ⃖ ⃗AB betweenA and B, then ⃗CA and ⃗CB are opposite rays.ABA CBSegments and rays are collinear when they lie on the same line. So, opposite rays arecollinear. Lines, segments, and rays are coplanar when they lie in the same plane.Naming Segments, Rays, and Opposite Rays—COMMON ERRORIn Example 2, ⃗JG and ⃗JFhave a common endpoint,but they are not collinear. So,they are not opposite rays.a. Give another name for GH .Eb. Name all rays with endpoint J. Whichof these rays are opposite rays?SOLUTIONGJFH— —a. Another name for GH is HG .b. The rays with endpoint J are ⃗JE , ⃗JG , ⃗JF , and ⃗JH . The pairs of opposite rayswith endpoint J are ⃗JE and ⃗JF , and ⃗JG and ⃗JH .Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comUse the diagram.MKPLN—2. Give another name for KL . ⃗ the same ray? Explain.3. Are ⃗KP and ⃗PK the same ray? Are ⃗NP and NMSection 1.1hs geo pe 0101.indd 5Points, Lines, and Planes51/19/15 8:18 AM

Sketching IntersectionsTwo or more geometric figures intersect when they have one or more points incommon. The intersection of the figures is the set of points the figures have incommon. Some examples of intersections are shown below.mAqnThe intersection of twodifferent lines is a point.The intersection of twodifferent planes is a line.Sketching Intersections of Lines and Planesa. Sketch a plane and a line that is in the plane.b. Sketch a plane and a line that does not intersect the plane.c. Sketch a plane and a line that intersects the plane at a point.SOLUTIONa.b.c.Sketching Intersections of PlanesSketch two planes that intersect in a line.SOLUTIONStep 1 Draw a vertical plane. Shade the plane.Step 2 Draw a second plane that is horizontal.Shade this plane a different color.Use dashed lines to show where oneplane is hidden.Step 3 Draw the line of intersection.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com4. Sketch two different lines that intersect a planeBat the same point.Use the diagram.5. Name the intersection of ⃖ ⃗PQ and line k.PkM AQ6. Name the intersection of plane A and plane B.7. Name the intersection of line k and plane A.6Chapter 1hs geo pe 0101.indd 6Basics of Geometry1/19/15 8:18 AM

Solving Real-Life ProblemsModeling with MathematicsThe diagram shows a molecule of sulfur hexafluoride, the most potent greenhouse gasin the world. Name two different planes that contain line r.pADqBEGrFCSOLUTIONElectric utilities use sulfur hexafluorideas an insulator. Leaks in electricalequipment contribute to the release ofsulfur hexafluoride into the atmosphere.1. Understand the Problem In the diagram, you are given three lines, p, q, and r,that intersect at point B. You need to name two different planes that contain line r.2. Make a Plan The planes should contain two points on line r and one point noton line r.3. Solve the Problem Points D and F are on line r. Point E does not lie on line r.So, plane DEF contains line r. Another point that does not lie on line r is C. So,plane CDF contains line r.Note that you cannot form a plane through points D, B, and F. By definition,three points that do not lie on the same line form a plane. Points D, B, and F arecollinear, so they do not form a plane.4. Look Back The question asks for two different planes. You need to checkwhether plane DEF and plane CDF are two unique planes or the same planenamed differently. Because point C does not lie on plane DEF, plane DEF andplane CDF are different planes.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comUse the diagram that shows a molecule of phosphorus pentachloride.sGJKHLI8. Name two different planes that contain line s.9. Name three different planes that contain point K.10. Name two different planes that contain ⃗HJ .Section 1.1hs geo pe 0101.indd 7Points, Lines, and Planes71/19/15 8:18 AM

1.1ExercisesDynamic Solutions available at BigIdeasMath.comVocabulary and Core Concept Check1. WRITING Compare collinear points and coplanar points.2. WHICH ONE DOESN’T BELONG? Which term does not belong with the other three?Explain your reasoning.—AB⃖ ⃗FGplane CDE ⃗HIMonitoring Progress and Modeling with MathematicsIn Exercises 11–16, use the diagram. (See Example 2.)In Exercises 3–6, use the diagram.BCtSBADTsEA3. Name four points.CD4. Name two lines.5. Name the plane that contains points A, B, and C.6. Name the plane that contains points A, D, and E.In Exercises 7–10, use the diagram. (See Example 1.)g—11. What is another name for BD ?—12. What is another name for AC ?13. What is another name for ray ⃗AE?14. Name all rays with endpoint E.15. Name two pairs of opposite rays.WQVERSfT16. Name one pair of rays that are not opposite rays.In Exercises 17–24, sketch the figure described.(See Examples 3 and 4.)17. plane P and lineℓ intersecting at one point7.Give two other names for ⃖ ⃗WQ.18. plane K and line m intersecting at all points on line m19. ⃗AB and ⃖ ⃗AC8. Give another name for plane V. ⃗ and NX ⃗20. MN9. Name three points that are collinear. Then name21. plane M and ⃗NB intersecting at Ba fourth point that is not collinear with thesethree points.22. plane M and ⃗NB intersecting at A10. Name a point that is not coplanar with R, S, and T.8Chapter 1hs geo pe 0101.indd 823. plane A and plane B not intersecting24. plane C and plane D intersecting at ⃖ ⃗XYBasics of Geometry1/19/15 8:18 AM

ERROR ANALYSIS In Exercises 25 and 26, describeand correct the error in naming opposite rays inthe diagram.In Exercises 35–38, name the geometric term modeledby the object.35.CABXYDE25.26. 36. ⃗AD and ⃗AC are opposite rays. 37.38.— and YE— are opposite rays.YCIn Exercises 27–34, use the diagram.BIACDFIn Exercises 39–44, use the diagram to name all thepoints that are not coplanar with the given points.G39. N, K, and LKL40. P, Q, and NEJHN27. Name a point that is collinear with points E and H.42. R, K, and N28. Name a point that is collinear with points B and I.43. P, S, and K29. Name a point that is not collinear with points E44. Q, K, and Land H.30. Name a point that is not collinear with points B and I.M41. P, Q, and RRQSP45. CRITICAL THINKING Given two points on a line and31. Name a point that is coplanar with points D, A, and B.a third point not on the line, is it possible to drawa plane that includes the line and the third point?Explain your reasoning.32. Name a point that is coplanar with points C, G, and F.46. CRITICAL THINKING Is it possible for one point to be33. Name the intersection of plane AEH and plane FBE.in two different planes? Explain your reasoning.34. Name the intersection of plane BGF and plane HDG.Section 1.1hs geo pe 0101.indd 9Points, Lines, and Planes91/19/15 8:18 AM

47. REASONING Explain why a four-legged chair may53. x 5 or x 2rock from side to side even if the floor is level. Woulda three-legged chair on the same level floor rock fromside to side? Why or why not?54. x 055. MODELING WITH MATHEMATICS Use the diagram.J48. THOUGHT PROVOKING You are designing the livingKroom of an apartment. Counting the floor, walls, andceiling, you want the design to contain at least eightdifferent planes. Draw a diagram of your design.Label each plane in your design.LPNQM49. LOOKING FOR STRUCTURE Two coplanar intersectinglines will always intersect at one point. What is thegreatest number of intersection points that exist if youdraw four coplanar lines? Explain.50. HOW DO YOU SEE IT? You and your friend walka. Name two points that are collinear with P.in opposite directions, forming opposite rays. Youwere originally on the corner of Apple Avenue andCherry Court.b. Name two planes that contain J.c. Name all the points that are in more thanone plane.NApple AveECRITICAL THINKING In Exercises 56–63, complete the.statement with always, sometimes, or never. Explainyour reasoning.Cherry Ct.RoseSRd.W56. A line has endpoints.Daisy Dr.57. A line and a point intersect.58. A plane and a point intersect.59. Two planes intersect in a line.a. Name two possibilities of the road and directionyou and your friend may have traveled.60. Two points determine a line.b. Your friend claims he went north on CherryCourt, and you went east on Apple Avenue.Make an argument as to why you know thiscould not have happened.61. Any three points determine a plane.62. Any three points not on the same linedetermine a plane.MATHEMATICAL CONNECTIONS In Exercises 51–54,63. Two lines that are not parallel intersect.graph the inequality on a number line. Tell whether thegraph is a segment, a ray or rays, a point, or a line.64. ABSTRACT REASONING Is it possible for three planes51. x 3to never intersect? intersect in one line? intersect inone point? Sketch the possible situations.52. 7 x 4Maintaining Mathematical ProficiencyReviewing what you learned in previous grades and lessonsFind the absolute value. (Skills Review Handbook)65. 6 2 66. 3 9 67. 8 2 68. 7 11 Solve the equation. (Skills Review Handbook)69. 18 x 4310Chapter 1hs geo pe 0101.indd 1070. 36 x 2071. x 15 772. x 23 19Basics of Geometry1/19/15 8:18 AM

1.2 LessonWhat You Will LearnUse the Ruler Postulate.Copy segments and compare segments for congruence.Core VocabulVocabularylarrypostulate, p. 12axiom, p. 12coordinate, p. 12distance, p. 12construction, p. 13congruent segments, p. 13between, p. 14Use the Segment Addition Postulate.Using the Ruler PostulateIn geometry, a rule that is accepted without proof is called a postulate or an axiom.A rule that can be proved is called a theorem, as you will see later. Postulate 1.1 showshow to find the distance between two points on a line.Postulatenames of pointsPostulate 1.1 Ruler PostulateThe points on a line can be matched one toone with the real numbers. The real numberthat corresponds to a point is the coordinateof the point.ABx1x2coordinates of pointsThe distance between points A and B, writtenas AB, is the absolute value of the differenceof the coordinates of A and B.ABABx2x1AB x2 x1 Using the Ruler Postulate—Measure the length of ST to the nearest tenth of a centimeter.STSOLUTIONAlign one mark of a metric ruler with S. Then estimate the coordinate of T. Forexample, when you align S with 2, T appears to align with 5.4.Scm1T234ST 5.4 2 3.456Ruler Postulate— is about 3.4 centimeters.So, the length of STMonitoring ProgressHelp in English and Spanish at BigIdeasMath.comUse a ruler to measure the length of the segment to the nearest —18 inch.1.3.12Chapter 1hs geo pe 0102.indd 12MUNV2.4.PWQXBasics of Geometry1/19/15 8:19 AM

Constructing and Comparing Congruent SegmentsA construction is a geometric drawing that uses a limited set of tools, usually acompass and straightedge.Copying a SegmentUse a compass and straightedge to construct a line segment—.that has the same length as ABABSOLUTIONStep 1Step 2ABStep 3ACBCDraw a segment Use a straightedge—.to draw a segment longer than ABLabel point C on the new segment.Measure length Set your—.compass at the length of ABABCDCopy length Place the compass atC. Mark point D on the new segment.— has the same length as AB—.So, CDCore ConceptCongruent SegmentsREADINGIn the diagram, thered tick marks indicate—AB —CD . When thereis more than one pair ofcongruent segments, usemultiple tick marks.Line segments that have the same length are called congruent segments. You— is equal to the length of CD—,” or you can say “AB— iscan say “the length of AB—congruent to CD .” The symbol means “is congruent to.”ABLengths are equal.AB CDSegments are congruent.— CD—ABCD“is equal to”“is congruent to”Comparing Segments for CongruencePlot J( 3, 4), K(2, 4), L(1, 3), and M(1, 2) in a coordinate plane. Then determine— and LM— are congruent.whether JKSOLUTIONyJ( 3, 4)K(2, 4)L(1, 3)2 4 2JK 2 ( 3) 5Ruler PostulateTo find the length of a vertical segment, find the absolute value of the difference of they-coordinates of the endpoints.2 2Plot the points, as shown. To find the length of a horizontal segment, find the absolutevalue of the difference of the x-coordinates of the endpoints.4 xM(1, 2)LM 2 3 5Ruler Postulate—— have the same length. So, JK— LM—.JK and LMMonitoring ProgressHelp in English and Spanish at BigIdeasMath.com5. Plot A( 2, 4), B(3, 4), C(0, 2), and D(0, 2) in a coordinate plane. Then— and CD— are congruent.determine whether ABSection 1.2hs geo pe 0102.indd 13Measuring and Constructing Segments131/19/15 8:19 AM