Segment and Angle Bisectors - Miami Senior High

1 34 Chapter 1 Basics of GeometryBisect a an Angle , asapplied in Exs. 50 55. To solve real-lifeproblems,such as finding the anglemeasures of a kite in Example you should learn itGOAL2 GOAL1 What you should and Angle BisectorsBISECTING ASEGMENTThe of a Segment is the point that divides, or the segmentinto two congruent segments. In this book, matching red congruence marksidentify congruent segments in is a Segment , ray, line, or plane that intersects a Segment atits the midpoint of AB if CD is a bisector of AB .Mis on AB and AM= can use aand a (a ruler without marks) toa Segment bisector and midpoint of AB.

2 A is ageometric drawing that uses a limited set of tools, usually a compass and bisectorbisects,midpointGOAL1 AMBAMBCDS egment Bisector and MidpointUse the following steps to construct a bisector of AB and find the midpointMof AB .ConstructionACTIVITYP lace the compasspoint at A. Use acompass settinggreater than halfthe length of AB .Draw an the samecompass the compasspoint at B. Drawan arc. It shouldintersect the otherarc in two a straightedgeto draw a segmentthrough the pointsof segmentbisects AB at M,the midpoint ofAB . and Angle Bisectors35If you know the coordinates of the endpoints of a Segment , you can calculate the coordinates of the midpoint.

3 You simply take the mean, or average, of the x-coordinates and of the y-coordinates. This method is summarized as theTHE MIDPOINT FORMULAF inding the Coordinates of the Midpoint of a SegmentFind the coordinates of the midpoint of AB with endpoints A( 2, 3) and B(5, 2).SOLUTIONUse the Midpoint Formula as , = 32 , 12 Finding the Coordinates of an Endpoint of a SegmentThe midpoint of RP is M(2, 4). One endpoint is R( 1, 7). Find the coordinates ofthe other (x, y) be the coordinates of P. Use the Midpoint Formula to write equations involving xand y. 12+x = 2= 4 1 + x = 47 + y= 8x =5y = 1 So, the other endpoint of the Segment is P(5, 1).

4 7+y 2 EXAMPLE 23 + ( 2) 2 2 + 5 2 EXAMPLE 1 Midpoint A(x1, y1) andB(x2, y2) are points in a coordinate plane, then the midpoint of AB has coordinates , .y1+ y2 2x1+x2 2 THE MIDPOINT FORMULA UsingAlgebraxyxyxyB(x2, y2)y2y1 y22y1x1x1 x22x2A(x1, y1) , x1 x22y1 y22xyR( 1, 7)M(2, 4) , 1 x 27 y2P(x, y)xy11B(5, 2)A( 2, 3)M , 3212 STUDENTHELPS tudy TipSketching the points in acoordinate plane helpsyou check your should sketch adrawing of a problemeven if the directionsdon t ask for a 1 Basics of GeometryBISECTING ANANGLEAn is a ray that divides an Angle into two adjacent angles thatare congruent.

5 In the diagram at theright, the ray CD bisects ABCbecauseit divides the Angle into two congruentangles, ACD and this book, matching congruence arcsidentify congruent angles in 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 bisectorGOAL2 ADBCA ngle BisectorUse the following steps to construct an Angle bisector of the compasspoint at C.

6 Draw anarc that intersectsboth sides of theangle. Label theintersections Aand B. Place the compasspoint at A. Draw anarc. Then place thecompass point at the samecompass setting, drawanother the intersec-tion D. Use astraightedge to draw a ray through Cand D. This is the Angle on CD .The sides of angles BCDand ACDline ACD=m and Angle Bisectors37 Dividing an Angle Measure in HalfThe ray FH bisects the Angle that m EFG = 120 , what are the measures of EFH and HFG?SOLUTIONAn Angle bisector divides an Angle into two congruent angles, each of which hashalf the measure of the original Angle . So,m EFH= m HFG= 1220 = 60.

7 Doubling an Angle MeasureKITEDESIGNIn the kite, two angles are bisected. EKIis bisected by KT . ITE is bisected by TK .Find the measures of the two are given the measure of one of the two congruent angles that make up the larger Angle . You can find the measure of the larger Angle by doubling the measure of the smaller EKI= 2m TKI= 2(45 ) = 90 m ITE = 2m KTI= 2(27 ) = 54 Finding the Measure of an AngleIn the diagram, RQ bisects PRS. The measures of the two congruent angles are (x+ 40) and (3x 20) . Solve for PRQ= m QRSC ongruent angles have equal measures.(x+ 40) = (3x 20) Substitute given + 60 = 3xAdd 20 to each = 2xSubtract xfrom each = xDivide each side by 2.

8 So, x = 30. You can check by substituting to see that each of the congruentangles has a measure of 70 .RSqP(3x 20) (x 40) EXAMPLE 5 EXAMPLE 4 FGHE120 EXAMPLE 3 JOS SA NZ,a San Diego kitedesigner, uses colorfulpatterns in his kites. Thestruts of his kites oftenbisect the angles ONPEOPLEU singAlgebraxyxyKTE45 27 I38 Chapter 1 Basics of kind of geometric figure is an Angle bisector? do you indicate congruent segments in a diagram? How do you indicatecongruent angles in a diagram? is the simplified form of the Midpoint Formula if one of the endpointsof a Segment is (0, 0) and the other is (x, y)?Find the coordinates of the midpoint of a Segment with the (5, 4), B( 3, 2) ( 1, 9), B(11, 5) (6, 4), B(1, 8)Find the coordinates of the other endpoint of a Segment with the givenendpoint and midpoint (3, 0) (5, 2) ( 4, 2) M(3, 4)M(7, 6)M( 3, 2) m JKLis 90.

9 If the ray KM bisects JKL, what are the measuresof JKMand LKM?QS is the Angle bisector of PQR. Find the two Angle measures not givenin the a ruler to measure and redraw the line segmenton a piece of paper. Then use construction tools to construct a THEMIDPOINTFind the coordinates of the midpoint of a segmentwith the given (0, 0) ( 1, 7) (10, 8) ( 12, 9)B( 8, 6)K(3, 3)D( 2, 5)Q(2, 10) (0, 8) (4, 4) ( , 8) ( , )T( 6, 14)F(4, 18)W( , 1)H( , )EFCDABPRACTICEANDAPPLICATIONSRq52 PSRq64 PSRqP40 SGUIDEDPRACTICEV ocabulary Check Concept Check Skill Check STUDENTHELPHOMEWORK HELPE xample 1:Exs.

10 17 24 Example 2:Exs. 25 30 Example 3:Exs. 37 42 Example 4:Exs. 37 42 Example 5:Exs. 44 49 Extra Practiceto help you masterskills is on p. and Angle Bisectors39 USINGALGEBRAFind the coordinates of the other endpoint of asegment with the given endpoint and midpoint (2, 6) ( 8, 1) (3, 12)M( 1, 1)M(0, 3)M(2, 1) ( 5, 9) (6, 7) ( , 6)M( 8, 2)M(10, 7)M( , )RECOGNIZINGCONGRUENCEUse the marks on the diagram to name thecongruent segments and congruent a protractor to measure and redraw the Angle ona piece of paper. Then use construction tools to find the Angle is the Angle bisector of PQR. Find thetwo Angle measures not given in the geometry software to draw a triangle.