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S \ 6L"  _P    X*  \ 6`  _    Z* H \ 0޽h ? 3380___PPT10.I!",0^N0 meD  (  D @  D# po,$D  0N   0 D   0N   0 D   0n D 0"`> 0n2 D 0"` 00n2 D 0"` 0 D 0|k  A EXAMPLE 1 2 Z D C *Aj0311784@ F l& D l& D 0ԃD(l & ;5.1(2( D 0ȇ TPerpendiculars and Bisectors 2 D(F   D  T  p` D#  p``" D 0v:  0 D 0"` P <GOAL 2`2 D 0p` D 0l 91 2  D 0p _+USING PROPERTIES OF PERPENDIUCLAR BISECTORS, 2,H D  fl33o?"6@ NNN?Nb ` ,$D0 @THEOREMS (2  D  f䘄 o?"6@ NNN?N\ o@,$@0 N Perpendicular Bisector Theorem Converse of the Perpendicular Bisector Theorem"O 2N D  f o?"6@ NNN?N0 @ ,$D 0 =VOCABULARY perpendicular bisector equidistant from two points& 23 2>H D 0!޽h ? 33E=___PPT10.]R+q$D' = @B D<' = @BA?%,( < +O%,( < +D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*D %(D' =-s6Bwipe(left)*<3<*D D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*D "%(D' =-s6Bwipe(left)*<3<*D "D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*D">%(D' =-s6Bwipe(left)*<3<*D">D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =-g6B fade*<3<*DD'' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =-g6B fade*<3<*DD7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*D %(D' =-s6Bwipe(left)*<3<*D D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*D O%(D' =-s6Bwipe(left)*<3<*D OD' =%(D@' =%(D' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*DD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*DD' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*DD' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*D+p+0+D0 ++0+D0 +t" 0^N0 B:0K ( (    P 6&    CExtra Example 1N & 3 r. o? "6@ NNN?N  n<In the diagram shown, is the perpendicular bisector of= 2=f (  s *A ?? p  f )  s *A ??   I8 @`p = @`p@ @`p ; @`p@   p 3 `@pB + B # lDo?"0@NNN?N @ 0B ,  # lDo?"0@NNN?N 0@ PB - # lDo?"0@NNN?N@ 0 PB . # lDo?"0@NNN?N@  0B /   fDo?"0@NNN?N 0 0B 0   `Do?"0@NNN?N@ @ pB 1  # lDo?"0@NNN?N  `B 2  # lDo?"0@NNN?N  ` 4   f. o?"6@ NNN?N 7T(2 5   f. o?"6@ NNN?N` 7P(2 6   f. o?"6@ NNN?N` 7D(2 7   fL#. o?"6@ NNN?N@ 7C(2 8   fd'. o?"6@ NNN?N7 7Q(2 9   fP*. o?"6@ NNN?Np 57(2 :   ft/. o?"6@ NNN?Np 57(2 <  # l o?"6@ NNN?NpN8    A    ?  # l/. o?"6@ NNN?N   La. What segment lengths in the diagram are equal? b. Explain why T is onLM0 h @  c $A ??  m B  fT8. o?"6@ NNN?NP ,$ 0 eQC = QD, PC = PD, and TC = TD, 23333k C # l=. o?"6@ NNN?N P ,$ 0 ]CT = DT, so T is equidistant from C and D. By the Converse of the Perpendicular Bisector Theorem, T is on the perpendicular bisector of the segment. 233333333333333:3333233H  0޽h ? 33___PPT10._`+UVD'  = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*B %(D' =-s6Bwipe(left)*<3<*B D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*C %(D' =-s6Bwipe(left)*<3<*C +p+0+B 0 ++0+C 0 + 0^N0 ]U& (  ~  s *4' `   D  C xP. o?"6@ NNN?N  ^Explain why FA = FB..(2 8 pP    R   f o?"6@ NNN?N 0 pB   `Do?"0@NNN?N@ @ P   # l o?"6@ NNN?N@  0B  # lDo?"0@NNN?NP P `B  # lDo?"0@NNN?N` ` `   fdX. o?"6@ NNN?N  7A(2   fX\. o?"6@ NNN?N 0 0  7F(2   fP`. o?"6@ NNN?N @  7G(2   fd. o?"6@ NNN?NP p 7B(2l    `,$D 0  # l@h. o?"6@ NNN?N   cSince is the perpendicular bisector of , FA = FB by the Perpendicular Bisector Theorem.,d 25(h  c $A '??  'h  c $A (?? (H  0޽h ? 33IA___PPT10!.av+D'  = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*+)0^N0 c[0  (  0F   c0  T  p` 0#  p``" 0 0v:  0  0 0v."` P <GOAL 2`2  0 0p`  0 0Tz. 92 2   0 6Tp W#USING PROPERTIES OF ANGLE BISECTORS$ 2$ @  0#  |,$D 0N   0 0   0N   0 0   0n 0 0"`> 0n2 0 0"` 00n2 0 0"` 0 0 0k  A EXAMPLE 2 2 Z 0 C *Aj0311784@ F l& 0 l& 0 0D(l & ;5.1(2( 0 0. TPerpendiculars and Bisectors 2 D(H 0  f.33o?"6@ NNN?N  ,$D0 @THEOREMS (2  0  f8. o?"6@ NNN?N d,$@0 |> Angle Bisector Theorem Converse of the Angle Bisector Theorem"? 2> 0  fL. o?"6@ NNN?N0 @ ,$D 0 EVOCABULARY distance from a point to a line equidistant from two lines& 2; 2FH 0 0!޽h ? 33___PPT10.]R+:pD' .= @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0 %(D' =-s6Bwipe(left)*<3<*0 D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0 +%(D' =-s6Bwipe(left)*<3<*0 +D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0+F%(D' =-s6Bwipe(left)*<3<*0+FD' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0D'' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-s6Bwipe(left)*<3<*0D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0?%(D' =-s6Bwipe(left)*<3<*0?D)' =%(D' =%(Dy' =4@BB5BB%()))D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*0D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*0D' =-g6B fade*<3<*0+p+0+00 ++0+00 +}` 0^N0 I9A9&;8(    66    CExtra Example 2`  c $A  ??P @  P*8  0 & 0B  3 rDo?"0@NNN?N pp  3 r o?"6@ NNN?Npf   0e0e    B{CDEF Ao 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||{@ "0e@     @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab\rB  3 rDo?"0@NNN?N@ pB  3 rDo?"0@NNN?Np   0e0e    B~C\DEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||~\@ "0e@     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abkG  3 rE o?"6@ NNN?NFf    0e0e    BJC3DEF A@  Ao 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||$0<E,J3@  s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abX !  0e0e    B7CXDEF A@  Ao 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||%.'765N7X@  s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `aby "  fh o?"6@ NNN?N0 7P(2 #  f? o?"6@ NNN?N @ 7S(2 $  fD o?"6@ NNN?N 7R(2 %  fG o?"6@ NNN?N 7Q(2l 0` )0`,$D0 '  f$L o?"6@ NNN?N` nStatements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5.r(2Z 2 0   ? (  fV o?"6@ NNN?N 0`P c/Note: The given statements will not be listed.0 20 * c $A  ?? a 8  $D 0 + c $A ?? p  8 $D 0 , c $A S??f ,t. 8 S$D 0 - c $A ??d P 8 $D 0 . c $A ?? 8@w 8 $D 0 / c $A ?? x  8 $D 0 0 c $A ?? @P 8 $D  0 1 c $A  ??( `  8  $D  0 2 c $A #??@(@88 #$D  0 3 c $A &??`x 8 &$D  0 @  4# p o,$D 0N   0 5   0N   0 6   0n 7 0"`> 0n2 8 0"` 00n2 9 0"` 0 : 0Xak  A EXAMPLE 3 2 Z ; C *Aj0311784@ H  0޽h ? 33&&___PPT10&._`+Dp&' = @B D+&' = @BA?%,( < +O%,( < +D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-g6B fade*<3<*)D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-s6Bwipe(left)*<3<**D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =-s6Bwipe(left)*<3<*+D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-s6Bwipe(left)*<3<*,D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =-s6Bwipe(left)*<3<*-D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =-s6Bwipe(left)*<3<*.D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =-s6Bwipe(left)*<3<*/D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-s6Bwipe(left)*<3<*0D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-s6Bwipe(left)*<3<*1D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =-s6Bwipe(left)*<3<*2D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =-s6Bwipe(left)*<3<*3D)' =%(D' =%(Dy' =4@BB5BB%()))D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*4D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*4D' =-g6B fade*<3<*4+  0^N0 |m (  x  c $.   . 8   g@R ]  f o?"6@ NNN?N B ^  `Do?"0@NNN?N`` ` # l o?"6@ NNN?N`B a # lDo?"0@NNN?N@@0B b # lDo?"0@NNN?N0 c  f. o?"6@ NNN?N p  7A(2 d  f. o?"6@ NNN?Np 7B(2 e  fķ. o?"6@ NNN?N0P 7C(2 f  f. o?"6@ NNN?Np 7D(28 (  l( g h # l. o?"6@ NNN?N(  SIn the figure, is the perpendicular bisector of What kind of triangle is T 2Th i c $A )??9 ! )h j c $A *??( p  *h k c $A +?? *0  +K m  fd. o?"6@ NNN?N ,$ 0 C isosceles (2 33H  0޽h ? 33___PPT10f.``+]^D'  = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*m%(D' =-s6Bwipe(left)*<3<*m+8+0+m0 +  0 d(  dX d C \    d S n \ 0    H d 0޽h ? 3380___PPT10.J@P$0 ((  ^  S \     c $t \ 0    H  0޽h ? 3380___PPT10.J0W-0 ((  ^  S \     c $y \ 0    H  0޽h ? 3380___PPT10.J xXO Qޛv.l"k#$"[a8U1llXMM$օd.$:Dp$A'}XkZo77}ok7i LU:V BfTR+-4@PA49(M 3b!x܃c.˲zmj׾0{= F@K>t+sg.(!+OhywѬT?y';bA?MAp;:gm.fSfcH.+G?q)Fsg5oZԦ7C됇^sE?{mb6D#|c^L䟉fN5{o7n`)FuxYq"cIcG!f[/.VیTQz2w0PWi*±RV(5KWK}^^>048k_2ٚ'gQ=ao$L@X*fPgS 'SzduO&HC38j' u nGb-EyA0f4&Q )Ќ5Ƅ("kq͠$`hG91Um,`m4iws~ZZƞ^U-& LZ0Zm [=d `l5t QG:EYn$tjIٰSӎ!Πb…_Eި~S+oFga-nݝuocGM$}J=_9|(SeLwl/} w&?Ǥ~{?rֺIǾ';mrs@By 5Gg¢ PIX̭f$J?4oꌚJ?$-S?nZ*}oU*ߗ +J br8_}/1R xXMhA~36MMӪE (zDzhZlcIj (yG "Ƀ$Q/DЊHO{;u nK6Λ7{y7yP}ljmy^ ʛ- PŒaH4H`\Q55Vr?fm7Z)g A_N;VM# B gn:$p"VꜣQ;N"C 8q74/B2 m,?9Vhz0^E堜'G7ɧ +f^/M-D?qnڒFsg7́;թr-u#u@@<αz0^)X sL6eduVL5څʂ{ƭ1, ʨO786| -OHCa:7O@g,כKeSV#G&=Z`X8JԻc \ғO %ӆV#z!Y=Ut }k]k67yF7n@y7y-9??R7DшovGDN;Ɨ:PEQ-OpG HVp]t͚FGk#!ق6 _r ƲF2!q W"u3oZT#-ø.ۈ?$Ń*I;1&3}{~9{MmM,@PhU~V=RCPia J_}HHXA2v-ja,;[ZeK=UE{ד13|k2sފXyy9!m:A;)VNY 6LYy3gzV7/L{OSJ0p"5(Tʾj$QK,'Q%F}9GKkմ7sԚ6Z|5޽v-}"<E̤%qS< 7ZeèNoXҕUğ~R~Qc8¨k3ni.q~gGcc|aƃ/2b|0ˌG_њn x`ߩwB޸M̵!tGc|j,30`XsgݕU&6s/ePG{ASnAH!L#h~Zy}NطheuB;AHgO QT$AnЧ6X硉y7ovO' c>qz5/]U? 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