Segment And Angle Bisectors - Miami Senior High

Page 1 of 91.5Segment and Angle BisectorsWhat you should learnGOAL 1Bisect a segment.GOAL 1BISECTING A SEGMENTGOAL 2Bisect an angle, asapplied in Exs. 50–55.The midpoint of a segment is the point that divides, or bisects, the segmentinto two congruent segments. In this book, matching red congruence marksidentify congruent segments in diagrams.Why you should learn itA segment bisector is a segment, ray, line, or plane that intersects a segment atits midpoint.RECMAMBABDFE To solve real-life problems,such as finding the anglemeasures of a kite inExample 4.AL LI ÆM is the midpoint of AB ifÆM is on AB and AM MB.ÆCD is a bisector of AB .You can use a compass and a straightedge (a ruler without marks) toÆconstruct a segment bisector and midpoint of AB. A construction is ageometric drawing that uses a limited set of tools, usually a compass and astraightedge.A C T IACTIVITYVITYConstructionSegment Bisector and MidpointÆUse the following steps to construct a bisector of AB and find the midpointÆM of AB.AB1 Place the compasspoint at A. Use acompass settinggreater than halfÆthe length of AB.Draw an arc.34Chapter 1 Basics of GeometryAB2 Keep the samecompass setting.Place the compasspoint at B. Drawan arc. It shouldintersect the otherarc in two places.AMB3 Use a straightedgeto draw a segmentthrough the pointsof intersection.This segmentÆbisects AB at M,the midpoint ofÆAB.

Page 2 of 9If you know the coordinates of the endpoints of a segment, you can calculatethe coordinates of the midpoint. You simply take the mean, or average, of thex-coordinates and of the y-coordinates. This method is summarized as theMidpoint Formula.THE MIDPOINT FORMULAyIf A(x1, y1) and B(x2, y2) are pointsin a coordinate plane, then theÆmidpoint of AB has coordinates B (x2, y2)y2 y1 y22 x1 x2 y1 y2 , .THE MIDPFORMULA2 OINT 2y1x 1 x22Æy 3 12 M 2,1Use the Midpoint Formula as follows. 3 1 , 2 2º2 5 3 (º2)M , 22UsingAlgebraxA( 2, 3)SOLUTIONEXAMPLE 2x2Finding the Coordinates of the Midpoint of a SegmentFind the coordinates of the midpoint of ABwith endpoints A(º2, 3) and B(5, º2).xy A (x1, y1)x1EXAMPLE 1x 1 x2 y 1 y2, 221 xB(5, 2)Finding the Coordinates of an Endpoint of a SegmentÆThe midpoint of RP is M(2, 4). One endpoint is R(º1, 7). Find the coordinates ofthe other endpoint.SOLUTIONSTUDENT HELPStudy TipSketching the points in acoordinate plane helpsyou check your work.You should sketch adrawing of a problemeven if the directionsdon’t ask for a sketch.yLet (x, y) be the coordinates of P.Use the Midpoint Formula to writeequations involving x and y. R( 1, 7)M (2, 4) º1 x 227 y 42º1 x 47 y 8x 5y 1 1 2 x , 7 2 y P (x, y)xSo, the other endpoint of the segment is P(5, 1).1.5 Segment and Angle Bisectors35

Page 3 of 9GOAL 2 BISECTING AN ANGLEAn angle bisector is a ray that dividesan angle into two adjacent angles thatare congruent. In the diagram at theÆ right, the ray CD bisects ABC becauseit divides the angle into two congruentangles, ACD and BCD.ADCBIn this book, matching congruence arcsidentify congruent angles in diagrams.m ACD m BCDACTIVITYConstructionAngle BisectorUse the following steps to construct an angle bisector of C.BBBDDCCA1 Place the compassCA2 Place the compass3 Label the intersec-tion D. Use astraightedge todraw a ray throughC and D. This isthe angle bisector.point at A. Draw anarc. Then place thecompass point at B.Using the samecompass setting, drawanother arc.point at C. Draw anarc that intersectsboth sides of theangle. Label theintersections A and B.AACTIVITYAfter you have constructed an angle bisector, you should check that it divides theoriginal angle into two congruent angles. One way to do this is to use a protractorto check that the angles have the same measure.Another way is to fold the piece of paper along the angle bisector. When you holdthe paper up to a light, you should be able to see that the sides of the two anglesline up, which implies that the angles are congruent.BCAÆ Fold on CD .36DChapter 1 Basics of GeometryABDCThe sides of angles BCD and ACD line up.

Page 4 of 9EXAMPLE 3Dividing an Angle Measure in HalfÆ The ray FH bisects the angle EFG.Given that m EFG 120 , what are themeasures of EFH and HFG?EH120 FSOLUTIONGAn angle bisector divides an angle into two congruent angles, each of which hashalf the measure of the original angle. So,120 2m EFH m HFG 60 .EXAMPLE 4FOCUS ONPEOPLEDoubling an Angle MeasureKKITE DESIGN In the kite, two angles are bisected.Æ 45 EKI is bisected by KT .IÆ ITE is bisected by TK .Find the measures of the two angles.ESOLUTIONREFELAL IJOSÉ SAÍNZ,You are given the measure of one of the two congruentangles that make up the larger angle. You can find themeasure of the larger angle by doubling the measure ofthe smaller angle.a San Diego kitedesigner, uses colorfulpatterns in his kites. Thestruts of his kites oftenbisect the angles theysupport.27 Tm EKI 2m TKI 2(45 ) 90 m ITE 2m KTI 2(27 ) 54 EXAMPLE 5Finding the Measure of an AngleÆ xyUsingAlgebraIn the diagram, RQ bisects PRS. Themeasures of the two congruent anglesare (x 40) and (3x º 20) . Solve for x.P(x 40) qRSOLUTIONm PRQ m QRS(x 40) (3x º 20) x 60 3x (3x 20) SCongruent angles have equal measures.Substitute given measures.Add 20 to each side.60 2xSubtract x from each side.30 xDivide each side by 2.So, x 30. You can check by substituting to see that each of the congruentangles has a measure of 70 .1.5 Segment and Angle Bisectors37

Page 5 of 9GUIDED PRACTICE Concept Check Vocabulary Check1. What kind of geometric figure is an angle bisector?2. How do you indicate congruent segments in a diagram? How do you indicatecongruent angles in a diagram?3. What is the simplified form of the Midpoint Formula if one of the endpointsof a segment is (0, 0) and the other is (x, y)?Skill Check Find the coordinates of the midpoint of a segment with the givenendpoints.4. A(5, 4), B(º3, 2)5. A(º1, º9), B(11, º5)6. A(6, º4), B(1, 8)Find the coordinates of the other endpoint of a segment with the givenendpoint and midpoint M.7. C(3, 0)8. D(5, 2)M(3, 4)M(7, 6)9. E(º4, 2)M(º3, º2)Æ 10. Suppose m JKL is 90 . If the ray KM bisects JKL, what are the measuresof JKM and LKM?Æ QS is the angle bisector of PQR. Find the two angle measures not givenin the diagram.11.P12.SPS13.PS40 52 64 qqRqRRPRACTICE AND APPLICATIONSSTUDENT HELPExtra Practiceto help you masterskills is on p. 804.CONSTRUCTION Use a ruler to measure and redraw the line segmenton a piece of paper. Then use construction tools to construct a segmentbisector.14.AB15. C16.DSTUDENT HELPHOMEWORK HELPExample 1:Example 2:Example 3:Example 4:Example 5:Exs. 17–24Exs. 25–30Exs. 37–42Exs. 37–42Exs. 44–49FFINDING THE MIDPOINT Find the coordinates of the midpoint of a segmentwith the given endpoints.17. A(0, 0)B(º8, 6)21. S(0, º8)T(º6, 14)38EChapter 1 Basics of Geometry18. J(º1, 7)K(3, º3)22. E(4, 4)F(4, º18)19. C(10, 8)D(º2, 5)23. V(º1.5, 8)W(0.25, º1)20. P(º12, º9)Q(2, 10)24. G(º5.5, º6.1)H(º0.5, 9.1)

Page 6 of 9xy USING ALGEBRA Find the coordinates of the other endpoint of asegment with the given endpoint and midpoint M.25. R(2, 6)26. T(º8, º1)M(º1, 1)M(0, 3)28. Q(º5, 9)29. A(6, 7)M(º8, º2)27. W(3, º12)M(2, º1)30. D(º3.5, º6)M(10, º7)M(1.5, 4.5)RECOGNIZING CONGRUENCE Use the marks on the diagram to name thecongruent segments and congruent angles.31. A32.33.DZCEWBFXYGCONSTRUCTION Use a protractor to measure and redraw the angle ona piece of paper. Then use construction tools to find the angle bisector.34.35.36.Æ ANALYZING ANGLE BISECTORS QS is the angle bisector of PQR. Find thetwo angle measures not given in the diagram.37.P38.22 q39.SPSS91 qR40.qRq41.S80 PR42.PP45 PR75 RINTSTUDENT HELPNEER TSOFTWARE HELPVisit our Web sitewww.mcdougallittell.comto see instructions forseveral softwareapplications.43.124 SqTECHNOLOGY Use geometrysoftware to draw a triangle.Construct the angle bisector ofone angle. Then find the midpointof the opposite side of the triangle.Change your triangle and observewhat happens.qRSBDACDoes the angle bisector always passthrough the midpoint of the oppositeside? Does it ever pass through the midpoint?1.5 Segment and Angle Bisectors39

Page 7 of 9INTSTUDENT HELPNEER THOMEWORK HELPVisit our Web sitewww.mcdougallittell.comfor help with Ex. 44–49.Æ xy USING ALGEBRA BD bisects ABC. Find the value of x.44.A(x 15) D45.46.47.(2x 7) AD49.D(4x 9) A(15x 18) CBC 12 x 20 D(3x 85) (23x 14) AC(6x 11) 48.BDBACB(10x 51) (5x 22) C(2x 35) (4x 45) BADCBSTRIKE ZONE In Exercises 50 and 51, use the information below. Foreach player, find the coordinate of T, a point on the top of the strike zone.In baseball, the “strike zone” is the region a baseball needs to pass through inorder for an umpire to declare it a strike if it is not hit. The top of the strike zoneis a horizontal plane passing through the midpoint between the top of the hitter’sshoulders and the top of the uniform pants when the player is in a batting stance. Source: Major League Baseball50.51.6360TT4542242200AIR HOCKEY When an air hockey puck is hit into the sideboards, itbounces off so that 1 and 2 are congruent. Find m 1, m 2, m 3,and m 4.52.53.106 123440Chapter 1 Basics of Geometry54.130 123460 3124

Page 8 of 9PAPER AIRPLANES The diagram55.Arepresents an unfolded piece of paper usedto make a paper airplane. The segmentsrepresent where the paper was folded tomake the airplane.BLCUsing the diagram, name as many pairs ofcongruent segments and as many congruentangles as you can.NKDJEIF56.MGHWriting Explain, in your own words, how you would divide a line segmentinto four congruent segments using a compass and straightedge. Then explainhow you could do it using the Midpoint Formula.57. MIDPOINT FORMULA REVISITED Another version of the Midpoint Formula,for A(x1, y1) and B(x2, y2 ), is 1212M x1 (x2 º x1 ), y1 ( y2 º y1) .Redo Exercises 17–24 using this version of the Midpoint Formula. Do youget the same answers as before? Use algebra to explain why the formulaabove is equivalent to the one in the lesson.TestPreparation58. MULTI-STEP PROBLEM Sketch a triangle with three sides of different lengths.a. Using construction tools, find the midpoints of all three sides and the anglebisectors of all three angles of your triangle.b. Determine whether or not the angle bisectors pass through the midpoints.c. ChallengeWriting Write a brief paragraph explaining your results. Determine ifyour results would be different if you used a different kind of triangle.INFINITE SERIES A football team practices running back and forth on thefield in a special way. First they run from one end of the 100 yd field to theother. Then they turn around and run half the previous distance. Then theyturn around again and run half the previous distance, and so on.59. Suppose the athletes continue therunning drill with smaller andsmaller distances. What is thecoordinate of the point thatthey approach?001005010060. What is the total distance that theathletes cover?075100EXTRA CHALLENGEwww.mcdougallittell.com062.51001.5 Segment and Angle Bisectors41