C92Q'u?Rqj%2suGsԣI5՝9<}>_w \uq y+oN7(P,+fO+dy|dugu;7Z:\.Ug|(I(l嬳 >ue^+R*u--!٧/W~R|P^
z0
yB뙎Zdz&Kԓy <] 2!ׄwR7r)s-CzS;gdqZN=Қ7I,3 }ڠM֝s--7#;b ىˌ bًڈmK bb:%jHEˬhHጙN[|/x)F]afK8?Ɉ4hv"ɹW8Im6aC1OMǢq'~v /|˽B~u+ZiǙ (Eī@Pq wF 'S8Wf&pza6MGAQ Loj`Ϛ6mfMGGW@f ` ߋؕ@> g#HJ=r 5i_Ha- K}Rh.EB?8~S8lts^{Cӹ|pX6e ]#7N47~3AگT8͡wstQ&om7UT1Ħ9jv2%ON^ڸ߉D^}?.ƸʢhTڏ:qq֣ q+Jn-j '%̖ou3R?܀mUR*'kR~*?a8
Eb*&LIꔏ8]rh[]${mȴ1vUw&Q RW Ï%LUMJ`! rH]ij @. ` `G : Q x}SMlQv
zhI=v.$*۲V` `p1^k´q2kO 3L+pXa%\á\Dp bJHiq X
5T'"?[ۂMKbaKX߶Gu=_ISB~i7H; آ 8 O 0 ]H?'`!U m_?Ab}0ۊՓ|
# xMN0;
)- 06]R"1
AV + < ! ξ~m6
ʦU9c51^eEl֚|:DGk&vQf "eEJLWЉfe
rVR? :x"~vy]4C[y &hkezlQn
8f,
'h7\r y- \8}"<2}oґD-Rn0EQ~n`T[-Nj>fcZ%Y`!9 7\,&YOҕ ` e% : xuRkQؤ?APXcb/!IqMlz!=zG k^'87f $ x1r2D(.ıE9Ky~D1dl"KC'?niTgQuQW$'ϥN ?%~GrVDF´_L~#ڏNgLFXG]=5twY nѼ& q b o{E>3$f:C ƨ'!GFj_*9WFHŽ(Y/
0dQm;;=QjCΘ rF*07ՀQ=f8;զi<
wqU -m3 ^ -bLWZ8> b*-;ʤ)5c^Π 2 Pc|.C1>_څ3ē6 ũM'*4F߄d2^Jp%ټL߉(to`! z$[o֛ xt xuRMkA~fv7&ݤ R~`/x\V]JҔ$%,
^AЃ//^'d3<0,cp@.3!gBLW_X^"t`a3 ؤH2%E22p=>MuˢK̟s[W*J$|S* |g5 d g~(
$<_\{ k|3m_&Y01٭q^VKŷ21+E5 Sb:j6Y^/ [M`n/: 7$ރ=Z+ZIz$;;N$v>*zOUĿ.m
EۥU,4H7$d[&A nlX0H'20NZ#FuS*k8d:vϓʠV?7yRC9
[agsN"Zk<]ʞ!.e *T?J[i ) (
h u E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n v E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n w E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n x E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n y E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n z E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n { E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n | E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n
} E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n ~ E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n ) E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n % E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n & E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n ' E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n * E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n M E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n 4 E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n : E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n ? E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n B E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n G E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n H E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n K E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n N E q u a t i o n E q u a t i o n . D S M T 4 0 * M a t h T y p e 5 . 0 E q u a t i o n d / 0 D A r i a l a e0Ԗ 0 l l | -a0 D V e r d a n a e0Ԗ 0 l l | -a0 "
B (. 2
@ n ? " d d @ @@ `` r j 8 0 T n H I ,
m 9 %
2 # 5 < = ~ + 7 | 7 ] 1
i
2 K J s K + & Y 5 _ ` 2 $ (^% ~ 2 $ _"i6. ~ 2 $ HI٩+ee 2 $ &=J 2 $ Hp̾ee) 8 2 $ T; vNJ!ѭ 2 $ k}TFɲ)K,<
P 2 $ _0Ύe ( Z 2 $ rH]ij 2 $ V5XC
F
2 $ $В1])@^= S 2 $ lIDG]S ;E 7 l 2 $ 0ދU.ˀn"5 " 2 $ r! ג5 J < (% 2 $ *5ᒊ{rV!u } d' 2 $ V f;LA ) 2 $ !G7̵ $0 ( a, 2 $ $S9wf[Q 1 . 2 $ f G > 0 2 $ _fܬ,qY@ 4 2 2 $ 8>[L;Ж(? 5 ,5 2 $ t, K^A3 E a8 2 $ _y3~u*r w : 2 $ I3L'E f = 2 $ m_?Ab}0ۊՓ ] ? 2 $ 7\,&YOҕ A @ 2 $ z$[o֛ $ !C c $ 0e 0e
A @ A o 8c 8c
? 1 d 0u 0@T y2 N P ' p < ' p A) B C D| E | | s
"
0e @ @ A B C D EE F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab N E 5% N E 5% N F
5%
!"? N @ A B C D EF F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab 33 33 f 3f ̙ @ 33 , ʚ;AU8 ʚ; g 4 E d E d -a0 p p p @ G 4 B d B d ䷍ -a0 < 4 d d d d 8 ))0l | < 4 ! d ! d 8 ))0l | 8 0 _ _ _ P P T 1 0
?
% O
=
g +
3 $
Checkpoint
Extra Example 3 -
Checkpoint u
/ p 4
P P
0` 33 ` S f 3 f ` 33 g ` f ` www 3 PP ` ZX db mo ` \ ғ 3 y` Ӣ ` 3f 3f f ` 3f 3 FK f ` hk] www f ܹ ` ff >>\ `Y{ ff ` R>& - {p _/ ̴ > ? " d d @ , | ? " d d @ " @ ` n ? " d d @ @@ `` P R @ ` ` p > > ^N0 j b (
6 " `}
T Click to edit Master title style !
! $
0 h " `
R Click to edit Master text styles
Second level
Third level
Fourth level
Fifth level !
S
0 \ " ^ `
X *
0 " ^
Z *
0 " ^ `
Z * H
0 h ? 33 8 0 _ _ _ P P T 1 0 . ]R D e f a u l t D e s i g n 0 p \ * (
\
\
0
P
X *
\
0
Z * d
\
c $ ?
\
0 d;
0
R Click to edit Master text styles
Second level
Third level
Fourth level
Fifth level !
S
\
6
_ P
X *
\
6
_
Z * H
\ 0 h ? 33 8 0 _ _ _ P P T 1 0 . I! - 0 ^N0 D @ N (
D
mD
N B o ? "6 @ N N N ? N x , $ D 0
Z Be able to identify the parts of isosceles triangles:
legs and base
vertex and base angles $ 6 2 % [ @
D # @j? , $ D 0 N 0
D 0 N 0
D 0 n
D
0 " ` > 0 n 2
D
0 " ` 0 0 n 2
D
0 " ` 0
D
0 J k
A EXAMPLE 1
2
Z
D
C * A j 0 3 1 1 7 8 4 @
F
D
D
6 P D( l &
; 4.6 ( 2 (
D
6 dS
c + Isosceles, Equilateral, and Right Triangles , 2 , D(
D
f X o ? "6 @ N N N ? N `
> ( 2
F
D T p `
D # p ` ` "
D
0 v: 0
D
0 [ " ` P
< GOAL 2 ` 2
D
0 p `
D
0 `
9 1 2
D
0 c p
[ ' USING PROPERTIES OF ISOSCELES TRIANGLES ( 2 (
D
# l g o ? "6 @ ` N N N ? N 0
, $ D
0
8
Base Angles Theorem
Converse of the Base Angles Theorem 8 9 2 7 33 33 a
D
# l m 33 33 o ? "6 @ ` N N N ? N
7 , $ @
0
S ISOSCELES TRIANGLE THEOREMS ( 2
D
# l o ? "6 @ N N N ? N 0
, $ D
0 H
D 0 ! h ? 33 n f _ _ _ P P T 1 0 F . ]R + p D '
=
@B D ' =
@B A ? %, ( < + O %, (
< + D ' = % ( D ' = % ( D3 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * mD 6 % ( D ' = -o 6 B d i s s o l v e *< 3 < * mD 6 D ' = % ( D ' = % ( D3 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * mD 6 D % ( D ' = -o 6 B d i s s o l v e *< 3 < * mD 6 D D ' = % ( D ' = % ( D3 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * mD D [ % ( D ' = -o 6 B d i s s o l v e *< 3 < * mD D [ D[ ' = % ( D ' = % ( D@ ' =A @B B B B 0B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * D % ( D ' = -o 6 B w i p e ( u p ) *< 3 < * D Dc ' =4 @B B B B % ( ) ) ) D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * D % ( D ' = -o 6 B w i p e ( u p ) *< 3 < * D D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * D % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * D D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * D 9 % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * D 9 D ' = % ( Dz ' = % ( D" ' =4 @B B " B B % ( ) ) ) D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * D % ( D ' X = + 4 . B ( - # p p t _ w / 2 ) 0 B ( # p p t _ x ) *Y 3 > B
p p t _ x < * D % ( D ' = + 4 . B 0 0 B - 1 . 0 *[ 3 > B x s h e a r < * D % ( X ) ) ? ) ? D ' = 0l 9 B B B B B B *< 3 < * D % ( X ) ) ? ) ? D2 ' = + 4 ) B- ( # p p t _ h / 3 + # p p t _ w * 0 . 1 ) *Y 3 > B
p p t _ x < * D % ( X ) ) ? ) ? +8 +0 + D 0 + :
0 ^N0 {@ s@ 0 , S
(
? (
P
6 $'
C Extra Example 1 7
&
f o ? "6 @ N N N ? N @
c 1 Use the diagram to prove the Base Angles Theorem. 2 2 2 R 8 P 0
1
P 0 R
'
3 r o ?
"6 @ N N N ? N
p B
(
3 r D o ?
"0 @ N N N ? N ` 0 B
)
B
3 r D o ?
"0 @ N N N ? N ` ` B
*
3 r D o ?
"0 @ N N N ? N p
,
# l \ o ?
"6 @ N N N ? N 0
7 W ( 2
-
# l ( o ?
"6 @ N N N ? N 0
7 Z ( 2
.
# l o ?
"6 @ N N N ? N
7 Y ( 2
/
# l o ?
"6 @ N N N ? N P p
7 X ( 2
2
c $ A ? ? hP P 8 $ @
0 z l 0 p
9
0p , $ D 0 @ p
5
p B
3
f D 33 o ? "0 @ N N N ? N p B
4
f D 33 o ? "0 @ N N N ? N p N p
6
0 p B
7
f D 33 o ? "0 @ N N N ? N p B
8
f D 33 o ? "0 @ N N N ? N p
;
c $ A ? ? ` p 8 $ D
0
<
c $ A ? ? `Ppp 8 $ @
0 l
@
, $ D 0 B
=
f D o ? "0 @ N N N ? N B
>
f D o ? "0 @ N N N ? N B
?
f D o ? "0 @ N N N ? N
A
c $ A ? ? P
8 $ D
0
B
c $ A ? ? P
8 $ @
0
C
c Z 0e 0e
B8 C D E F A @ o 8c 8c
? 1 d 0u 0@T y2 N P ' p < ' p A) B C D| E | | ( 0 0 8 H 4 l 0 @ s
"
0e @ @ A B C D EE F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab N E 5% N E 5% N F
5%
!"? N @ A B C D EF F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab
D
c Z 0e 0e
B8 C D E F A @ o 8c 8c
? 1 d 0u 0@T y2 N P ' p < ' p A) B C D| E | | ( 0 0 8 H 4 l 0 @ s
"
0e @ ` @ A B C D EE F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab N E 5% N E 5% N F
5%
!"? N @ A B C D EF F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab H l P
H
P , $ D 0
F
c Z 0e 0e
B8 C D E F A @ 33 o 8c 8c
? 1 d 0u 0@T y2 N P ' p < ' p A) B C D| E | | ( @ 0 ` 8 4 0 @ s
"
0e @ @ A B C D EE F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab N E 5% N E 5% N F
5%
!"? N @ A B C D EF F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab P H
G
B
s ` 0e 0e
B8 C D E F @ 33 o 8c 8c
? 1 d 0u 0@T y2 N P ' p < ' p A) B C D| E | | ( @ 0 ` 8 4 0 @ s
"
0e @ @ A B C D E F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab N 5% N 5% N
5%
!"? N @ A B C D E F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab P
I
f o ? "6 @ ` N N N ? N `
, $ D
0
l
If a triangle is equilateral, then it is equiangular.
If a triangle is equiangular, then it is equilateral. " m 2 l j
J
f $ 33 33 o ? "6 @ ` N N N ? N
;K7 , $ @
0
b * COROLLARIES TO ISOSCELES TRIANGLE THEOREMS + ( 2 +
K
# l o ? "6 @ N N N ? N `
, $ @
0 @
L
# pU o , $ D 0 N 0
M
0 N 0
N
0 n
O
0 " ` > 0 n 2
P
0 " ` 0 0 n 2
Q
0 " ` 0
R
0 k
A EXAMPLE 2
2
Z
S
C * A j 0 3 1 1 7 8 4 @
H
0 h ? 33 OD GD _ _ _ P P T 1 0 'D . _` + z DSC ' x =
@B DC ' =
@B A ? %, ( < + O %, (
< + D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * 2
% ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * 2
D+ ' =4 @B B
B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * 9
% ( D ' = -g 6 B f a d e *< 3 < * 9
D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * ;
% ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * ;
D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * <
% ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * <
D+ ' =4 @B B
B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * @
% ( D ' = -g 6 B f a d e *< 3 < * @
D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * A
% ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * A
D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * B
% ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * B
D+ ' =4 @B B
B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * H
% ( D ' = -g 6 B f a d e *< 3 < * H
D! ' = % ( D! ' = % ( D@ ' =A @B B B B 0B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * J
% ( D ' = -o 6 B w i p e ( u p ) *< 3 < * J
Dc ' =4 @B B B B % ( ) ) ) D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * K
% ( D ' = -o 6 B w i p e ( u p ) *< 3 < * K
D6 ' =A @B B
B B 0B % ( D ' = -g 6 B f a d e *< 3 < * P D ' = 1 : B h i d d e n *o 3 >+ B# s t y l e . v i s i b i l i t y < * P % ( D6 ' =A @B B
B B 0B % ( D ' = -g 6 B f a d e *< 3 < * &
D ' = 1 : B h i d d e n *o 3 >+ B# s t y l e . v i s i b i l i t y < * &
% ( D) ' =4 @B B
B B % ( D ' = -g 6 B f a d e *< 3 < * 1
D ' = 1 : B h i d d e n *o 3 >+ B# s t y l e . v i s i b i l i t y < * 1
% ( D) ' =4 @B B
B B % ( D ' = -g 6 B f a d e *< 3 < * 2
D ' = 1 : B h i d d e n *o 3 >+ B# s t y l e . v i s i b i l i t y < * 2
% ( D) ' =4 @B B
B B % ( D ' = -g 6 B f a d e *< 3 < * 9
D ' = 1 : B h i d d e n *o 3 >+ B# s t y l e . v i s i b i l i t y < * 9
% ( D) ' =4 @B B
B B % ( D ' = -g 6 B f a d e *< 3 < * ;
D ' = 1 : B h i d d e n *o 3 >+ B# s t y l e . v i s i b i l i t y < * ;
% ( D) ' =4 @B B
B B % ( D ' = -g 6 B f a d e *< 3 < * <
D ' = 1 : B h i d d e n *o 3 >+ B# s t y l e . v i s i b i l i t y < * <
% ( D) ' =4 @B B
B B % ( D ' = -g 6 B f a d e *< 3 < * @
D ' = 1 : B h i d d e n *o 3 >+ B# s t y l e . v i s i b i l i t y < * @
% ( D) ' =4 @B B
B B % ( D ' = -g 6 B f a d e *< 3 < * A
D ' = 1 : B h i d d e n *o 3 >+ B# s t y l e . v i s i b i l i t y < * A
% ( D) ' =4 @B B
B B % ( D ' = -g 6 B f a d e *< 3 < * B
D ' = 1 : B h i d d e n *o 3 >+ B# s t y l e . v i s i b i l i t y < * B
% ( D) ' =4 @B B
B B % ( D ' = -g 6 B f a d e *< 3 < * C
D ' = 1 : B h i d d e n *o 3 >+ B# s t y l e . v i s i b i l i t y < * C
% ( D) ' =4 @B B
B B % ( D ' = -g 6 B f a d e *< 3 < * D
D ' = 1 : B h i d d e n *o 3 >+ B# s t y l e . v i s i b i l i t y < * D
% ( D) ' =4 @B B
B B % ( D ' = -g 6 B f a d e *< 3 < * H
D ' = 1 : B h i d d e n *o 3 >+ B# s t y l e . v i s i b i l i t y < * H
% ( D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * I
7 % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * I
7 D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * I
7 m % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * I
7 m D) ' = % ( D ' = % ( Dy ' =4 @B B 5 B B % ( ) ) ) D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * L
% ( D ' = + 4 8 ?R C B B C B # p p t _ w B *Y 3 > B
p p t _ w < * L
D ' = + 4 8 ?R C B B C B # p p t _ h B *Y 3 > B
p p t _ h < * L
D ' = -g 6 B f a d e *< 3 < * L
+ +0 + P 0 + +0 + &
0 + +0 + J
0 + S>
0 ^N0 # # ! 4 # (
6 0
C Extra Example 2 c
f 2 o ? "6 @ N N N ? N
1 a. Find the value of x.
b. Find the value of y. B 2 2
8
$
` Y @ P
P N @ p
P R
# l G> o ? "6 @ N N N ? N @ p B
B
f D o ? "0 @ N N N ? N p !
f l; o ? "6 @ N N N ? N x ]
E 50 & ( 2 " B
# l D o ? "0 @ N N N ? N P B
# l D o ? "0 @ N N N ? N
0e 0e
BK C D E F A o 8c 8c
? 1 d 0u 0@T y2 N P ' p < ' p A) B C D| E | | K @ " 0e @ @ A B C D EE F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab N E 5% N E 5% N F
5%
!"? N @ A B C D EF F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab e
"
f @ o ? "6 @ N N N ? N
F x ( ( 2 " "
f XE o ? "6 @ N N N ? N
@
F y ( ( 2 "
f I o ? "6 @ N N N ? N
P p
7 A ( 2
!
f G o ? "6 @ N N N ? N
7 D ( 2
"
f P o ? "6 @ N N N ? N
7 C ( 2
#
f T o ? "6 @ N N N ? N @
7 B ( 2 m l 0 &
4 0& , $ @
0 {
%
# l Y o ? "6 @ N N N ? N 0 &
Q a. Looking only at triangle BCD, we know that by the Base Angles Theorem. , R 2 2 n
&
s * A ? ? \
'
s * A
? ? 9
8 $ D
0 V
*
f _ o ? "6 @ N N N ? N ` , $ 0
N x 0 ( 2 33 & 33 < l `
, ` , $ D
0 P
)
f c o ? "6 @ N N N ? N `
t B b. First, use the Linear Pair Postulate to find the measure of C 2 C h
+
c $ A ? ?
-
c $ A
? ? P
8
$ @
0 T
.
f Hi o ? "6 @ N N N ? N
, $ 0
L 115 , 2 & l
1
, $ @
0
/
f m o ? "6 @ N N N ? N
C We also see that 2 h
0
c $ A ? ?
2
c $ A ? ? 8 $ D
0 V
3
f r o ? "6 @ N N N ? N , $ 0
N y 0 ( 2 33 & 33 H
0 h ? 33 \ T _ _ _ P P T 1 0 4 . _` + | D` ' =
@B D ' =
@B A ? %, ( < + O %, (
< + D' ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * 4 % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * 4 D8 ' =A @B B
B B 0B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * * % ( D ' = -g 6 B f a d e *< 3 < * * D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * ' % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * ' D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * , % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * , D' ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * - % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * - D8 ' =A @B B
B B 0B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * . % ( D ' = -g 6 B f a d e *< 3 < * . D' ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * 1 % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * 1 D8 ' =A @B B
B B 0B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * 3 % ( D ' = -g 6 B f a d e *< 3 < * 3 D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * 2 % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * 2 + +0 + * 0 + +0 + . 0 + +0 + 3 0 + G
0 ^N0 ? ? r 0 .? (
x
c $ ؉
G: 8 0
`
o ` @ B
]
# l D o ? "0 @ N N N ? N 0
0 B
^
# l D o ? "0 @ N N N ? N
0 B
` B
# l D o ? "0 @ N N N ? N P
0 B
a
# l D o ? "0 @ N N N ? N 0 P
0 B
b
# l D o ? "0 @ N N N ? N 0 B
c B
# l D o ? "0 @ N N N ? N 0 0 "
e
f \ o ? "6 @ N N N ? N g E e
F x ( ( 2 " "
f
f o ? "6 @ N N N ? N > r
F y ( ( 2 " !
g
f o ? "6 @ N N N ? N @ @ `
E 50 & ( 2 "
h
s ` 0e 0e
B` C D E F A @ o 8c 8c
? 1 d 0u 0@T y2 N P ' p < ' p A) B C D| E | |
7 # G ; W S [ ~ ` @ s
"
0e @ @ A B C D EE F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab N E 5% N E 5% N F
5%
!"? N @ A B C D EF F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab
0
j B
f 0e 0e
B` C D E F @ o 8c 8c
? 1 d 0u 0@T y2 N P ' p < ' p A) B C D| E | |
7 # G ; W S [ ~ ` @ s
"
0e @ @ A B C D E F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab N 5% N 5% N
5%
!"? N @ A B C D E F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab
0
k B
f 0e 0e
B` C D E F @ o 8c 8c
? 1 d 0u 0@T y2 N P ' p < ' p A) B C D| E | |
7 # G ; W S [ ~ ` @ s
"
0e @ @ A B C D E F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab N 5% N 5% N
5%
!"? N @ A B C D E F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab >
l
0e 0e
B C D E F A o 8c 8c
? 1 d 0u 0@T y2 N P ' p < ' p A) B C D| E | | @ " 0e @ @ A B C D EE F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab N E 5% N E 5% N F
5%
!"? N @ A B C D EF F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab 2
` P
m
0e 0e
B` CE D E F A o 8c 8c
? 1 d 0u 0@T y2 N P ' p < ' p A) B C D| E | | ` E @ " 0e @ @ A B C D EE F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab N E 5% N E 5% N F
5%
!"? N @ A B C D EF F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab `
n B
0e 0e
B` CE D E F o 8c 8c
? 1 d 0u 0@T y2 N P ' p < ' p A) B C D| E | | ` E @ " 0e @ @ A B C D E F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab N 5% N 5% N
5%
!"? N @ A B C D E F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab c
p
f t o ? "6 @ N N N ? N
1 a. Find the value of x.
b. Find the value of y. B 2 2 Z
q
f ̣ o ? "6 @ N N N ? N P`
p , $
0
R
a. x = 40 , 2 Z
r
f o ? "6 @ N N N ? N `
, $
0
R
b. y = 80 , 2 H
0 h ? 33 _ _ _ P P T 1 0 . `` + ] D '
=
@B D ' =
@B A ? %, ( < + O %, (
< + D ' = % ( D ' = % ( DD ' =A @B B B B 0B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * q % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * q D ' = % ( D ' = % ( DD ' =A @B B B B 0B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * r % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * r +p +0 + q 0 + +0 + r 0 + 0 ^N0 0
(
0 F `
0 P0 T p `
0 # g ` ` "
0
0 v: 0
0
0 " ` P
< GOAL 2 ` 2
0
0 p `
0
0
9 2 2
0
6 g
W # USING PROPERTIES OF RIGHT TRIANGLES $ 2 $ @
z0 # a
j` , $ D 0 N 0
{0 0 N 0
|0 0 n
}0
0 " ` > 0 n 2
~0
0 " ` 0 0 n 2
0
0 " ` 0
0
0 k
A EXAMPLE 3
2
Z
0
C * A j 0 3 1 1 7 8 4 @
F
0
0
0 t D( l &
; 4.6 ( 2 (
0
0
c + Isosceles, Equilateral, and Right Triangles , 2 , D( V
0
f 33 33 o ? "6 @ ` N N N ? N ;K
, $ @
0
N HYPOTENUSE-LEG THEOREM ( 2
0
f o ? "6 @ N N N ? N `
, $ D
0
0
f o ? "6 @ N N N ? N ` `f , $
0
Y The fifth way to prove two triangles are congruent is applicable only to right triangles: , Z 2 A H
0 0 ! h ? 33 _ _ _ P P T 1 0 b . ]R +
w D ' =
@B D ' =
@B A ? %, ( < + O %, (
< + D$ ' = % ( D ' = % ( Dt ' =A @B B B B 0B % ( ) ) ) D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * 0 % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * 0 D[ ' = % ( D ' = % ( D@ ' =A @B B B B 0B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * 0 % ( D ' = -o 6 B w i p e ( u p ) *< 3 < * 0 Dc ' =4 @B B B B % ( ) ) ) D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * 0 % ( D ' = -o 6 B w i p e ( u p ) *< 3 < * 0 D) ' = % ( D ' = % ( Dy ' =4 @B B 5 B B % ( ) ) ) D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * z0 % ( D ' = + 4 8 ?R C B B C B # p p t _ w B *Y 3 > B
p p t _ w < * z0 D ' = + 4 8 ?R C B B C B # p p t _ h B *Y 3 > B
p p t _ h < * z0 D ' = -g 6 B f a d e *< 3 < * z0 +p +0 + 0 0 + +0 + 0 0 + A
0 ^N0 % % = @ % (
<
<
s 5 0e 0e # " 0e
8
< p N
<
<
C x o ? "6 @ N N N ? N B
< B
C x Z D o ? "0 @ N N N ? N
<
3 r o ? "6 @ N N N ? N
<
3 r o ? "6 @ N N N ? N @ B
<
C x D o ? "0 @ N N N ? N x x B
<
C x D o ? "0 @ N N N ? N x x
<
f o ? "6 @ N N N ? N @
7 A ( 2
<
f o ? "6 @ N N N ? N @
7 D ( 2
<
f o ? "6 @ N N N ? N
7 C ( 2
<
f o ? "6 @ N N N ? N
7 B ( 2 `
<
c $ A M ? ? P M F 3
< 3
<
# l 8 o ? "6 @ N N N ? N 3
t Statements Reasons
1. 1. Given
2. 2.
3. 3.
4. 4.
5. 5. r ( 2 ` 2
6
n
<
s * A 4 ? ? x 4
<
c $ A : ? ? @ 8 : $ D
0
=
c $ A ? ? ? `
P 8 ? $ D
0
=
c $ A B ? ? G
@@/ 8 B $ D
0 ?
=
f + o ? "6 @ N N N ? N >
^ , $
0
7 Given 2
=
c $ A G ? ? @ 8 G $ @
0 `
=
c $ A H ? ? pl H
=
c $ A K ? ? x
8 K $ D
0 l 6 f
= 6f , $ D 0
=
0e 0e
BT CH D E F 33 o 8c 8c
? 1 d 0u 0@T y2 N P ' p < ' p A) B C D| E | | T H @ " 0e @ @ A B C D EE F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab N E 5% N E 5% N F
5%
!"? N @ A B C D EF F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab 6 K ~
=
0e 0e
BT CH D E F 33 o 8c 8c
? 1 d 0u 0@T y2 N P ' p < ' p A) B C D| E | | T H @ " 0e @ @ A B C D E F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab N 5% N 5% N
5%
!"? N @ A B C D E F G H I J K5% L M N O P Q R S T U W YZ[ \ ]^_ ` ab W f
=
c $ A N ? ? )
8 N $ D
0 >
=
f p2 o ? "6 @ N N N ? N
, $
0
6 HL ( 2 H
< 0 h ? 33 _ _ _ P P T 1 0 g . av + ) D '
=
@B D ' =
@B A ? %, ( < + O %, (
< + D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * < % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * < D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * = % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * = D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * = % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * = D ' = % ( D ' = % ( DD ' =A @B B B B 0B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * = % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * = D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * = % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * = D+ ' =4 @B B
B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * = % ( D ' = -g 6 B f a d e *< 3 < * = D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * = % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * = D ' = % ( D ' = % ( D7 ' =4 @B B B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < *
= % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < *
= D ' = % ( D ' = % ( DD ' =A @B B B B 0B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * = % ( D ' = -s 6 B w i p e ( l e f t ) *< 3 < * = +p +0 + = 0 + +0 + = 0 + m(
0 ^N0 | t (
~
s * tH `
`
c $ A ) ? ? ) F ` ` 0
@ N ` P
P @ p N P
P
b
# l o ? "6 @ N N N ? N P
# l o ? "6 @ N N N ? N P B
# l D o ? "0 @ N N N ? N ` ` ` B
f D o ? "0 @ N N N ? N ( ( B
f D o ? "0 @ N N N ? N X X
f M o ? "6 @ N N N ? N `
7 A ( 2
f hO o ? "6 @ N N N ? N ` 0
7 B ( 2
f S o ? "6 @ N N N ? N
` 0
7 C ( 2 F ` ` 0
@ N ` P
P @ p N P
P
b
# l o ? "6 @ N N N ? N P
# l o ? "6 @ N N N ? N P B
# l D o ? "0 @ N N N ? N ` ` ` B
f D o ? "0 @ N N N ? N ( ( B
f D o ? "0 @ N N N ? N X X
f Y o ? "6 @ N N N ? N `
7 D ( 2
f ] o ? "6 @ N N N ? N ` 0
7 E ( 2
f a o ? "6 @ N N N ? N
` 0
7 F ( 2
f de o ? "6 @ N N N ? N $ , $ 0
v Statements Reasons
1. 1. Given
2. 2. Given
3. 3.
4. 4. SAS r ( 2 b 2
=
c $ A ? ?
p 8 $ @ 0
c $ A % ? ?
i 8 % $ @ 0
c $ A & ? ?
8 & $ @ 0
c $ A ' ? ? p@
8 ' $ @ 0 $
S ~ A * ? o ? "6 @ N N N ? N @ 8 * $ D 0 H
0 h ? 33 y _ _ _ P P T 1 0 Y . av + ӯ D '
=
@B D ' =
@B A ? %, ( < + O %, (
< + D
' = % ( D
' = % ( D8 ' =A @B B
B B 0B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * % ( D ' = -g 6 B f a d e *< 3 < * D+ ' =4 @B B
B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * % ( D ' = -g 6 B f a d e *< 3 < * D+ ' =4 @B B
B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * % ( D ' = -g 6 B f a d e *< 3 < * D+ ' =4 @B B
B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * % ( D ' = -g 6 B f a d e *< 3 < * D+ ' =4 @B B
B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * % ( D ' = -g 6 B f a d e *< 3 < * D+ ' =4 @B B
B B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * % ( D ' = -g 6 B f a d e *< 3 < * +8 +0 + 0 +
0 ^N0 p ' (
N d~ o ? "0 @ N N N ? N 0D
Q QUESTION: ,
( 2 $ $
N o ? "0 @ N N N ? N @T
O ANSWER: , ( 2 $ $ _
f ( o ? "6 @ N N N ? N |
Q Can you use the HL Congruence Theorem to prove two isosceles triangles congruent? R ( 2 R l
f 4 o ? "6 @ N N N ? N pv
, $ 0
d * Yes, but only if they are right triangles. + ( 2 + H
0 3 h ? 33 _ _ _ P P T 1 0 x . { + ]% D ' =
@B D ' =
@B A ? %, ( < + O %, (
< + D ' = % ( D ' = % ( DV ' =A @B B 5 B B 0B % ( D ' = 1 : B v i s i b l e *o 3 >+ B# s t y l e . v i s i b i l i t y < * % ( D ' = + 4 8 ?R C B B C B # p p t _ w B *Y 3 > B
p p t _ w < * D ' = + 4 8 ?R C B B C B # p p t _ h B *Y 3 > B
p p t _ h < * D ' = -g 6 B f a d e *< 3 < * +8 +0 + 0 + 0 d (
d X
d C \
d S %
\
0
H
d 0 h ? 33 8 0 _ _ _ P P T 1 0 . J@P 0 x (
x X
x C \
x S *
\
0
H
x 0 h ? 33 8 0 _ _ _ P P T 1 0 . J $ 0 ( (
^
S \
c $
\
0
H
0 h ? 33 8 0 _ _ _ P P T 1 0 . J0W - 0 ( (
^
S \
c $ 4
\
0
H
0 h ? 33 8 0 _ _ _ P P T 1 0 . J xXMlE~3?)lR" KzHTi{I7-IHH8pRzȁCB2HTBPUB
E 8c.Ƽ7u&uig|oΛ}?{>~J%0`)_ "( l)| +: 9`B &S0Sj
R3x
ϼƷRɳn |v.rEj:
c_b`+z,`;q &p zfY2A.rbyI@' $0)eY-X좵_> {T1M?yyr*ʹ/(E?]+B_u=P Nc~I!
W>m[G!, =\ZJ_