Study Guide and Intervention - Mrs. Kim's Geometry

1 Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, 16 Glencoe GeometryStudy Guide and InterventionPoints, Lines, and PlanesNAME _____ DATE _____ PERIOD _____1-1 Name Points, Lines, and PlanesIn Geometry , a pointis a location, a linecontainspoints, and a planeis a flat surface that contains points and lines. If points are on the sameline, they are collinear. If points on are the same plane, they are the figure to name each of the a line containing point AThe line can be named as . Also, any two of the three points on the line can be used to name ,AC ,or BC b. a plane containing point DThe plane can be named as plane Nor can be named using three noncollinear points in the plane, such as plane ABD,plane ACD,and so to the a line that contains point is another name for line m? a point not on AC . the intersection ofAC and DB . a point not on line or line and label a plane Qfor each is in plane intersects AB at Xis collinear with points Aand Yis not collinear with points Tand contains points Xand Y.

2 PmABECD NABCDE xampleExercises05-56 Geo-01-873958 4/3/06 6:29 PM Page 6 Chapter 17 Glencoe GeometryCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Guide and Intervention (continued)Points, Lines, and PlanesNAME _____ DATE _____ PERIOD _____1-1 Points, Lines, and Planes in SpaceSpaceis a boundless, three-dimensional set ofall points. It contains lines and How many planes appear in the figure?There are three planes: plane N, plane O, and plane Are points A,B, and Dcoplanar?Yes. They are contained in plane to the a line that is not contained in plane a plane that contains point three collinear to the many planes are shown in the figure? points B, E, G,and Hcoplanar? a point coplanar with D,C,and and label a figure for each MandNintersect in HJ . ris in plane N, line sis in plane M, and lines rand sintersect at point tcontains point Hand line tdoes not lie in plane Morplane 1-1 ExampleExercises05-56 Geo-01-873958 4/3/06 6:29 PM Page 7 Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, 114 Glencoe GeometryStudy Guide and InterventionLinear Measure and PrecisionNAME _____ DATE _____ PERIOD _____1-2 Measure Line SegmentsA part of a line between two endpoints is called a linesegment.

3 The lengths of M N and R S are written as MNand RS. When you measure asegment, the precision of the measurement is half of the smallest unit on the the length of M N .The long marks are centimeters, and theshorter marks are millimeters. The length of M N is centimeters. The measurement is accurate to within millimeter, so M N isbetween centimeters and the length of R S .The long marks are inches and the short marks are quarter inches. The length of R S is about 1 34 inches. The measurement is accurate to within one half of a quarter inch,or 18 inch, so R S is between 1 58 inches and 1 78 inches the length of each line segment or the precision for each 12 1 Example 2 Exercises05-56 Geo-01-873958 4/3/06 6:29 PM Page 14 Chapter 115 Glencoe GeometryCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Guide and Intervention (continued)Linear Measure and PrecisionNAME _____ DATE _____ PERIOD _____1-2 Calculate MeasuresOn PQ , to say that point Mis between points Pand Qmeans P, Q, and Mare collinear and PM MQ AC ,AB BC 3 cm.

4 We can say that the segments arecongruent, or A B B C . Slashes on the figure indicate whichsegments are EFby adding EDand DF EFTherefore,E F is centimeters cmFDFind xand between Aand BC ACx 2x 2x 53x 2x 5x 5AC 2x 5 2(5) 5 15Ax2xCB2x 5 Find the measurement of each segment. Assume that the art is not drawn to T C Z X Find xand RSif Sis between Rand 5x,ST 3x, and RT 2x,ST 5x 4, and RT 6x,ST 12, and RT 4x,R S S T , and RT the figures to determine whether each pair of segments is B and C D Y and Y Z XYZ3x 55x 19x2 ADCB11 cm11 cm5 cm5 cmWY6 cmXXZY3 4 2 4 cmSLesson 1-2 Example 1 Example 2 Exercises05-56 Geo-01-873958 4/3/06 6:29 PM Page 15 Chapter 121 Glencoe GeometryCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Guide and InterventionDistance and MidpointsNAME _____ DATE _____ PERIOD _____1-3 Distance Between Two PointsDistance on a Number LineDistance in the Coordinate PlaneAB |b a|or |a b|xyOC(1, 1)A( 2, 1)B(1, 3)Pythagorean Theorem:a2 b2 c2 Distance Formula:d (x2 x1)2 (y2 y1)2 ABabFind |( 4) 2| | 6| 6 5 4 3 2 10123 ABFind the distance between A( 2, 1) and B(1, 3).

5 Pythagorean Theorem(AB)2 (AC)2 (BC)2(AB)2 (3)2 (4)2(AB)2 25AB 25 5 Distance Formulad (x2 x1)2 (y2 y1)2 AB (1 ( 2))2 (3 ( 1)) 2 AB (3)2 (4)2 25 5 Use the number line to find each the Pythagorean Theorem to find the distance between each pair of (0, 0),B(6, 8) ( 2, 3),S(3, 15) (1, 2),N(9, 13) ( 12, 2),F( 9, 6)Use the Distance Formula to find the distance between each pair of (0, 0),B(15, 20) ( 12, 0),P( 8, 3) (11, 12),D(6, 2) ( 2, 10),F( 4, 3) 10 8 6 4 202468 ABCDEFGL esson 1-3 Example 1 Example 2 Exercises05-56 Geo-01-873958 4/3/06 6:29 PM Page 21 Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, 122 Glencoe GeometryStudy Guide and Intervention (continued)Distance and MidpointsNAME _____ DATE _____ PERIOD _____1-3 Midpoint of a SegmentMidpoint on aIf the coordinates of the endpoints of a segment are aand b, Number Linethen the coordinate of the midpoint of the segment is a 2b .Midpoint on aIf a segment has endpoints with coordinates (x1, y1) and (x2, y2),Coordinate Planethen the coordinates of the midpoint of the segment are x1 2x2 , y1 2y2.

6 Find the coordinate of the midpoint of P Q .The coordinates of Pand Qare 3 and Mis the midpoint of P Q , then the coordinate of Mis 32 1 22 or the midpoint of P Q for P( 2, 4) and Q(4, 1). Find thecoordinates of x1 2x2 , y1 2y2 22 4 , 4 21 or (1, )Use the number line to find the coordinate of the midpoint of each E G F G B G D E Find the coordinates of the midpoint of a segment having the given (0, 0),B(12, 8) ( 12, 8),S(6, 12) (11, 2),N( 9, 13) ( 2, 6),F( 9, 3) (10, 22),T(9, 10) ( 11, 2),N( 19, 6) 10 8 6 4 202468 ABCDEFG 3 2 1012 PQExample 1 Example 2 Exercises05-56 Geo-01-873958 4/3/06 6:29 PM Page 22 Chapter 129 Glencoe GeometryCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Guide and InterventionAngle MeasureNAME _____ DATE _____ PERIOD _____1-4 Measure AnglesIf two noncollinear rayshave a common endpoint, they form an angle. The rays are the sidesof the common endpoint is the vertex. The angle at the right can be named as A, BAC, CAB, or right angleis an angle whose measure is 90.

7 An acute anglehas measure less than 90. An obtuse anglehas measure greater than 90 but less than Name all angles that have Ras angles are 1, 2, and 3. Forother angles, use three letters to namethem: SRQ, PRT, and Name the sides of ,RP SRTPQ123 Measure each angle andclassify it as right,acute,or ABDU sing a protractor,m ABD 90, so ABDis an acute DBCU sing a protractor,m DBC 115 90, so DBCis an EBCU sing a protractor,m EBC 90. EBCis a right to the the vertex of the sides of another name for each angle in the figure and classify it as right,acute,or MPR5. RPN6. NPSPNMRSCBA1234 DLesson 1-4 Example 1 Example 2 Exercises05-56 Geo-01-873958 4/3/06 6:29 PM Page 29 Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, 130 Glencoe GeometryStudy Guide and Intervention (continued)Angle MeasureNAME _____ DATE _____ PERIOD _____1-4 Congruent AnglesAngles that have the same measure are congruent angles. A ray that divides an angle into two congruent angles is called an angle bisector.

8 In the figure,PN is the angle bisector of MPR. Point Nlies in the interior of MPRand MPN to the figure above. If m MPN 2x 14 and m NPR x 34, find xand find m PN bisects MPR, MPN NPR,or m MPN m 14 x 34m NPR (2x 14) (x 34)2x 14 x x 34 x 54 54x 14 34 108x 14 14 34 14x 20QS bisects PQT, and QP and QR are opposite m PQT 60 and m PQS 4x 14, find the value of m PQS 3x 13 and m SQT 6x 2, find m and BC are opposite rays,BF bisects CBE, and BD bisects m EBF 6x 4 and m CBF 7x 2, find m m 1 4x 10 and m 2 5x, find m m 2 6y 2 and m 1 8y 14, find m DBFa right angle? Geo-01-873958 4/3/06 6:29 PM Page 30 Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, 136 Glencoe GeometryStudy Guide and InterventionAngle Relationships NAME _____ DATE _____ PERIOD _____1-5 Pairs of AnglesAdjacent anglesare angles in the same plane that have a commonvertex and a common side, but no common interior anglesare twononadjacent angles formed by two intersecting lines.

9 A pair of adjacent angles whosenoncommon sides are opposite rays is called a linear each pair of angles as adjacent angles,vertical angles,and/or as a linear SRTand TRUhave a commonvertex and a common side, but nocommon interior points. They areadjacent 6 and 5 are adjacent angles whosenoncommon sides are opposite angles form a linear 1 and 3 are nonadjacent angles formedby two intersecting lines. They are verticalangles. 2 and 4 are also vertical Aand Bare two angles whose measureshave a sum of 90. They are complementary. Fand Gare two angles whose measureshave a sum of 180. They are 60 60 120 NRPSM1432 Identify each pair of angles as adjacent,vertical, and/or as a linear 1 and 22. 1 and 63. 1 and 54. 3 and 2 For Exercises 5 7, refer to the figure at the two obtuse vertical two acute adjacent an angle supplementary to the measures of two complementary angles if the difference in their measures is Geo-01-873958 4/3/06 6:29 PM Page 36 Chapter 137 Glencoe GeometryCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Guide and Intervention (continued)Angle Relationships NAME _____ DATE _____ PERIOD _____1-5 Perpendicular LinesLines, rays, and segments that form four right angles are perpendicular.

10 The right angle symbol indicates that the lines are perpendicular. In the figure at the right,AC is perpendicular to BD ,or AC BD .Find xso that D Z P Z .If D Z P Z , then m DZP DZQ m QZP m DZPSum of parts whole(9x 5) (3x 1) 90 Substitution12x 6 84 Subtract 6 from each 7 Divide each side by xand yso that NR MQ . m EBF 3x 10,m DBE x, and BD BF .Find m EBF 7y 3 and m FBC 3y 3, find yso that EB BC . x,m PQS,and m y,m RPT,and m (4y 5) (2y 5) QRPS3x (8x 2) BCADEFMNRSQPx 5x (9y 18) ZDPQ(9x 5) (3x 1) BCDAL esson 1-5 ExampleExercises05-56 Geo-01-873958 4/3/06 6:29 PM Page 37 Chapter 143 Glencoe GeometryCopyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Guide and InterventionTwo-Dimensional FiguresNAME _____ DATE _____ PERIOD _____1-6 PolygonsA polygonis a closed figure formed by a finite number of coplanar linesegments. The sides that have a common endpoint must be noncollinear and each sideintersects exactly two other sides at their endpoints.