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2
1 ; ,
== . .
2 ; xy ,
, x = y= .
.
3 ; , , , .. xy yz .
, . , , .
4 ; ; . , .. .
. .
( ) .
5 : . , , . , .
. , , B . , .
6 ; , .
.. xy xy' yx' xy' .
7 .
xy xy' . yx'.
8 .
xy xy' xy. x = . ' , 1 = x 2 = ( ) . 1 = 2, ' x'y'.
9 . , . + = 2 2 + 2 = 2 + = 1 = 1 .
.
10 ; ; ; ; ; ; ; , . . 1, , 2. , . , . , . , . , . . To . . . , , . . , .
3
1 ; . , , , , ,
, . , , , , , , , , . + + , 2.
2 ; , , : , . , : , (.2) , (.3). = , , (.4).
3 ; , , , (.5), , (.6). , , (.7).
4 ;
, . . . .
, . A . B .
, . , , . . , .
5 1 . (1 ): , .
6 :
. . ( =). . = , A1 = A2 (), , B = . = , 1 = 2. 1 + 2 = 180 1 = 2= 90, .
7 .
=
= = = = = .
8 . . =, 1 = 2 = 90 (), , = .
9 , . (,) . = . = (= ), = (= ) = . , = .
10 2 . (2 ) : , . ''') = '', B= B' = '. = ''. AB '', .. > ''. , = ''. ''' = '', = '' B= B', () , = '. ' = , = , < . '', =''. , , '' = '', = '' B = B', () .
11 3 . (3 ): , .
''' = '', = '', = '' (.17). A= A. . x, Bx = B' , = ''. ''' , = '', = '' B= B'. = '' = A.
= '' '' = , ,
A1=1 (1). , = '' '' = ,
A2= 2 (2). A , (1) (2) A= . = A, A= A, .
12 , , . = . =, =, () , A1 = A2, 1 = 2. .
13 . , , = . , , .
14 . .
15 , , , , . (,) , = . : = (= ), = (= ) = , () . , = , = . = .
16 . : (), (), ().
17 . x'x, x'x . x'x, . x'x. x'x y xy = AM x , = . , x'x, x'x , . = 1 = 2, ,
x'x. x'x . , , = 1 = 2, 1 = 2. 1= 90, 2 = 90, 1 + 2= 180 ,, , , .
18 ; , , 1 () 2 () : , , . , , .
19 , . ''' A = A'=90, = '' B = B'. = ''. '' , .. > ''. , = ''. ''' = '', = '' = ',
, = ' = 90, . ( ). ''. = '', ''' , = '', = '' = ' ().
20 , . ''' A = A' = 90, = '' = ''. B = B'. '' ', = '' = ''. '' ''. , , = '' = ''. = '' = ''. ''' = '', = '' = '' ( ''), , B = B'. ().
21 . : = ( ) = ( ) 1=2=90 ( ) ( ) = 1=2 .
22 .
(,), , . (= = ), , 1 = 2. 1 = 2 = .
23 . : , :
. .
24 . . ' (,) , . , = = 90, = (= ) = ( = ). , = . . . = = 90, = = , , AK = AB2 = 2 = .
25 . . xy .
MAOx MBOy. A = B = 90, = , = . . .
MAOx MBOy = . , A = B = 90, = = , .
26 ; .
26 ; , () . .. : , . , . , ( ) .
27 ; ; 180 ; , ' , ' .
, '. , ', , . . A , , 180 , .
28 , ; , : (.). (.). (.).
29 ; ; ;
, ' , ' . '. , ', , . . , , , , .
30 , ; (.). x'x
x'x x'x (.). (.). (=) (.). (.).
31 . ; . (.47) , , , = . Ax A Ax = A. : = , = 1 = 2, = . A A A >. A> B. : ) . ) 180.
32 . ; > . , =. B1 = 1 = .
(i) , . , , > . (ii) . , , , . , , . > (.57). > . , , B , B > = 1, . , , >. . > . = , = , . R, = R < R.
> R. , , OM > OA > R. , x'x .
= R. , x'x (,R), > = R. , x'x . . , x'x (,R) . : . .
< R . . x , = R. , > = R. x, , , , , , . x' , '., x'x . x'x, . .
: >R, . =R, .
37 . (,) , , , . = (= ) = (= ), , , , . , , . : .
38 (,) ; (, ) , . , , .
39 , . A = = 90, = (= ), , = . : , : (i) , (ii) .
40 ; . : (i) (, ) (, R), < R (.) (ii) (, R) (, ) , > R + (.). (i) , (, ) (, R), = R - (.).
(ii) , , = R + (.). . , , R - (.). .
41 . (KR) (,) , . = = R, . = = . ,
. (, R) (, ) , R = , . , R = , = = . .
42 ;
, , .
4
1 ; ; 1 2, , : i) (.),ii) (.),iii) (.). 1 2 , : 1 2 . 1 2 , 1//2.
2 :
, . = . 1, 2 ,
, . 1//2.
3 : , . 1//2 . .. = . , x xAB . x//2 . 2, . = .
4 : , . i) ,ii) .
19 :
5 : 1 2 , , 1// 2//, 1//2. 1 2 , , . 1//2.
6 1 2 , . ; 1 . 2, //2 2, . 2. : , .
7 : 2 , . ; 1 ,2 + 2. 1 2 , + 2 . 1 2 , , . , , A1 > A1= 2- + > 2, . 1,2 . : , .
8 ; ', . ' . , . A = //, , . '.
20 :
9 , , , . xAy x'y' x//x' y//y', , . x' By' x y . . : . . , . : , , , .
10 1 2 . (i) . (ii) . i) x, By A . = A/2 = /2. A = ( ). = . x By . x//By.
(ii) z A, z x ( ). x//By, Az By.
11 ; . .
12 , . , , , . x y , , . , =. =, . =, . , (O,OA) .
21 :
13 ;
, . , , .
14 , . . I + = 2 + 2 < + < 2 , = . , = . A. , . , . 15 ; . . a . , , , , 16 : , . x y , . x y I, : x + y = /2 + /2 = 2 - ( + )/2 < 2. , . A, .
22 :
17 2 .
, .. , xy//B. = (1) = (2), xy . + A + = 2 (3). (1), (2) (3) : A + + = 2 17 : 2 :
i) .
ii) , , . iii) . iv) 60.
i) A + B + = 2 + = 2, A + B + = + = A + B. ii) - iv) .
18
xy = x''y'= xO'x' OyO'y'. ' A = B =1 1 = 2(). , = . 19 . ii) . i) , ( ) + =2, ' + = 2, =', = . ii) , ( ) + = 2, + = 2, = .
23 :
20 - 2-4 . , . 2 . 1+2+3+ ... + = 4 : 1 + 2 + 3 + ... + = (2 - 4) . . 12... , .. 1 . -2 , , 12 1 1, . -2 2(-2)=(2-4) , : 1 + 2 + 3 + ... + = (2 - 4) . 21 4 .
24 :
5
O -
1 ; ; ; . , . , , . . 2 . . , // //. 3 : (i) . (ii) . (iii) . i), ii) , : 1 = 1 = ( ). . 2 = 2 = ( ). , , = = . A = = = + . iii) , . : = 1 = 1 = ( ). A1 = 1 = ( ). , , , = = .
25 :
4 ; . . 5 ; ; , . , . 6 ; : . (ii) . (iii) . (iv) . , , . (i) = = . , , = , = . 1 = 1 = 2 = 2 = , // //, . (ii) // = . , = , 1 = 1 = . , (i), . (iii) = = = = + + + = 4 2 + 2 = 4 + = 2. , + = 2, // + = 2, // , . (iv) = = . , . , (i), // // , .
26 :
7 ; : , .
8 ; : . , ( ), . 9 . . . . (A = = 90, , = ), = 10 . , : (i) . (ii) . (iii) . (iv) . (i) . (ii) = . ( = , = , ), A = . A + = 2, A = = 1. , . (iii) , 4. (iv) , .
27 :
11 ; : . . 12 . (i) . (ii) . . , A.
A. . 13 . , : (i) . (ii) . (iii) . (iv) . (i) (ii) .
(iii) . , . , ,
. , = . . (iv) A. ( ), . 14 ; : .
28 :
15 ; . , : (i) . (ii) . (iii) . (iv) , , . 16 . , . , : (i) . (ii) . (iii) . (iv) . (v) . (vi) . 17 : . , , . // = B/2 . EZ = . , . = // , = // , = . , : (i) // // (ii) // 2 = = /2 . 18 .
29 :
: , . . . . Z , , // . , , , . . 19 ; : () , . 1, 2, 3 1 , , , (.22). 2 1, 2, 3 , , , = . // . , = (1) = (2). // . , = (3) . (1), (2) (3) = . 20 1 2 1 2, . 1 2. 1 2 = , 1 2. 1 2, 1 2 , = = /2 . , 1 2, + = = , = = /2 . (// = , // = ), // 1, 2. , .
30 :
21 , , , , , , . . . , =// /2 (1) =// /2 (2) (1) (2) = // , . : . 22 . x , , . , , . , = = . 23 ; 2/3 . . . i) = . , E =// /2 (1). =// /2 (2). (1) (2) // // , (3). , = . , , ( ) . ii) = = 2 , = 2 = 2. (1) (3) = 2 = 2 = 2. (2) (3) = 2. , , . = + = 2 + = 3. = 1/3, = 2/3. = 2/3 = 2/3. : . .
31 :
24 ; : , , , . , , , . , : = = , = = = = . , , . . , . , , . , , , . , , , ( , ) . , , . . : , , 25 ; : , , , , , . .. , . , , . 26 .
32 :
: . ( = 90) . = /2. . , // .
, . , , = , = /2 . : , . . = /2 . = , 1 = (1) = , 2 = (2). (1) (2) 1 + 1 = + , = + . + + = 2 , 2 = 2 = 1. 27 30 60 : 30, . ( = 90) = 30 . = /2. = 30, = 90 - 30 = 60. 2 = = 60, . = = /2 . , = /2 , = 30. , = 2= = ( = 2). , = 60. = 90 - 60 = 30. 28 . ; ; : .
.
.
.
33 :
29 ; : . , , :
i) // , E =
(1).
( // ) , . . : //, E = /2 (1). //, = /2 (2). i) // , ( ). ii) (1) (2)
K + KZ =
+
E =
30 . : . , , // , = 2 (3). , , // , . (1) (3) :
- = =
-
=
( > ).
31 ; : 32
34 :
, : (i) . (ii) . (i) (// =). . ( = = 90, = = = ), = . + = 180 + = 180 ( ), = . (ii) ( = , = ), = . 33 . , . (i) . (ii) . 34 : i) , ii) . i) (//) . , A1 = 1 = ( ). ii) , . //. , , . 35 ; , , . 36 ; ; ; : . (,) . . I I I , . . . . .
35 :
z
z
y
y
2
1 , , . ,
, ,
2 , , i)
ii) .
xx, yy, zz
i) xx, yy .
yy, zz
, zz, xx .
.
ii) y, Ay, z, Az, . .
, , .
, , . . .
, , .
3 , , = . = .
= + = + =
36 :
4 , . , = 2.
= 2
= 2
.
+ = 2 + 2
= 2( + )
= 2 5 , , .
i) = 2
ii) =
2
.
, , , ..
, , = , = =
, . = = , = = , = =
= = AB
2=
2
i) = = 2
=
2
2
(1)
2
=
22 2 2
(2)
(1), (2) = 2
ii) M = = 2
=
2
2
(3)
2
=
2
2 2 2
(4)
(3), (4) = 2
37 :
1
2
A O
2
1
O
1
34
2
y
x
6
,
, .
2
()
.
, = , =
1
= 2
= 2
= + 2
= + 2
=
2
2
2
=
2
=
2
2
2
7
,
.
2
()
.
, = , =
1
= 2
= 2
= 2
= 2
=
2
2
2
=
2
=
2
=
2
2
2
8
, , .
x, Oy , ,
xOy = AO
2
()
, . ,
38 :
O
21
= , =, = .
x 1
= 2
= 2
Oy 3
= 4
= 2
xOy = 2
+ + 3
= 2
+ +
2
=
2
2
AO
2
=
2
=
2
=
2
2
xOy = AO
2
9
1 2 .
;
10 (, R) (, ) R > . , , , . = = . . = = R = = R = = + = R + = + = + R = 11 20 . . + 20 = 90 2 = 70 = 35 90 35 = 55 .
39 :
O
O
A
M
12
, , , , ,
, , 1, 2, 3, 4. .
1 2 3 4
2
3
4
+ + + = 360 + 2 + 3 + 4 = 360 10 = 360 = 36 = 36 , = 2. 36 = 72 , = 3. 36 = 108 , = 4. 36 = 144 . 13 , . , , ,
, , , .
= x, = y
, , x, y.
x
2 2 2 2 2
(1)
x
2 2
(2)
(1), (2) = .
, . 14
(, R) , , , , 0
150 , 0
45 0
105 . .
.
.
0 0 0 0 0
360 150 45 105 60 .
0
06030
2 2
0 0 0
150 30 180 , .
21A
1
1
M
3
1 , = = . = .
1 2
. = .
=
2 , , = = . .
,
1 1
= =
( ) . = . =
=
3 .
, .
. = .
,
() . = =
21
1
12
2
2
1
4
=
= .
,
1 2
(-) . = .
5 ''' = ', = ' = '. .
''' . = '', = '' ( ) ='', ''. , ''' (), = '. , ' '' = ', = ' =', () .
6 . , , = , = , = , .
1 2
() . = .
1 1 (1)
2 2
(2)
(1) + (2)
7 . , , . ,
.
A
B
21 1
2
''
'
= . = .
=
=
8 . ,
. .
. = .
= =
2 2
9
= , = A A . , : i) = = ii) = =
i) ( ) . = .
( ) . = . =
( ) . = . =
ii) ( ) . = . = = .
10 . .
, .
( ) = . =
( ) = . =
( ) = .
B
11
11
A
' ''
'
11 , , , , .
( ) . = . =
= =
. = .
12
= .
.
, .
( ) . = . =
( ) . = .
13 . , . i) =
ii) BAM B A M iii) iv) = = .
. = . A , , = .. .
1 1
i) ( ) . = . = =
ii) ( ) . = . 1 1
A
iii) ( ) . = . iv) iii) = =
(i) = , = .
E
14 , , . ,
, .
, , .
=
,
,
.
15 ( = ). .
= . :
i)
ii) .
i) =
.
ii) .
. ,
= , = .
, . = . ,
:
= , B = AE
, .
16 . , = , =. , , : (i) , , (ii) = .
K
B
A
B
A
(i) ( = = 90) = ( ) 1 = 2 (). , . = = 90, = 1 = 2. (ii) =. = . = .
17 , , , . = . , = , , . ( = = 90) = , 1= 2 . , = .
18 : i) ii)
i)
= , ,
. = . , =
ii) ,
. = .
, =
19 .
, , ,
.
. = . =
B '
A'
'
A
1 2
112 2''
'
''
20 , , , .
. . = .
= . =
21 ( = ) . : i) ii) .
, i) . = .
, 1 2
. =
ii) . = .
.
22
,
.
. = .
2 2
1 1
( ) . = . =
( ) . =
'
''
21
2
1
23
,
.
. = . = =
. = . .
( ) . = .
24
( 1 ) .
, . . . = .
, 1 2 .
= = (1)
. = .
, = 1 2
= (2)
(1) + (2) = . .
25 (, R), , . , : i) ii) MA = M =
i) =
. = .
ii) i) MK = M (1)
= = = (2)
(1) + (2) =
(1) (2) =
11' '
'
'
M
26 , , .
= + + = + +
+ = + (1)
= =
(1) = = A A
. = . (2)
. 1
(3)
1
(3)
2
2
(2)
. = . = .
27 , .
. = (
) ( , ).
.
(, ), 1
A , 2
A
, .
28 . , , , , , .
. .
= . . = = . = .
4 321
'' '
29 , . , : i) = ii) , , .
i) = =
=
ii) 1 2
O O 3 4
O O
,
1 2
O O +3 4
O O =2 2
O O + 3 3
O O
= 2 2
O + 23
O = 2 (2
O +3
O ) = 2. 90 = 180 .
30 , . ''' = '', = '' > ' . B > ''. > ', Ax , x = '. x , = ''. ''' (). , = ''. , , = . , ii) + i) . > 1. 1 , 1 > . > . (ii) : + +
21 2 1
1 12 2
A
32 . :
(i) , (ii) , (iii) ,
. . . ( = , = , 1= 2 ). = (1) 1 = . 1 = = (2), , 1 = 2= . (1) (2) =. , = .
33
1 1 B .
0
1B 90 .
1 1 B (1)
1 2 B . (2)
(1) + (2) 1 1 2 2
0
12 180
0
1 90
34
= , = . ;
= 1 1
(1)
(2)
(2) (1) 2 2
. =
,
.
.
.
1
M
122
1
O
21 1
2
35 , < .
. 1
1
,
>
36
( = 90), . < .
= (1) .
. < (2) (1), (2) <
37 . . = = .
( ) . = .
1 1 B (1)
. 2 2
B (2)
(1) + (2) B , . .
38 ( = ) , , .
, , .
:
B , B , 1 2 1 2
. =
( ) . = .
=
2 2 B
1 2
1 2
39
2
, .
2
2
;
2
<
.
1
< .
2
+
, 2
2
40 < .
i) ii)2 2
iii) 2
i) =
= .
() . = .
= < 1
(1)
< , . 2
(2)
(1), (2) 1 2
ii) :
2.
2 2
iii) ii) 2.
(1)
2.
(2)
2.
(3)
(1) + (2) + (3) 2 2. 2. 2.
2.
53 :
O
41 (,R) . i) . .
< < + < < + < < ii) = 2
ii) + + + ;
i) . : + >
. : OB + O >
. : + >
. : + >
2 ( + + + ) > + + +
+ + + > 2
54 :
123
3 2
1
B
A
ii) , + > , + =
+ (1)
+ (2)
(1) + (2) + + + +
, , + + + +
. 44 ( < ) = = . . : i)
ii) .
i) ( ) . = .
1 1 E B
. 2 2
E B
3 3
E B
. ii) , M , , . , , , .
45 , ,
. i) E < EB ii) E < B
i) , < ,
< (1) ii) B BE, B < ,
< (2) (1) (2) < < < 46 . ,
= , < , <
55 :
21
47 ,
, , ,
. ( ) ( ) i) = ii) iii) H OM , , i)
, 48 , .
49
(, ), 1 ,
2 ,
. 1 ,
2 , , = 90
.
.
. , , . 50 (,R) . , . .
56 :
3
1 2
= = = + + = + + = 2 51 , , . (, ) = , , = . (--) . = . = = = 52 (,R) , .
.
.
2 3
,
1 2
1 2
=3
= 3 3
53 . , , . : (i) . (ii) .
57 :
(i) () . (1). ,
, (1) , (). (ii) . = , () = , (). = .
54 (, ) (, 2)
i) = 2
ii) K =
iii) = 2
iv) = 3
v) = 4
2 1R R 2
2 1R R 2 3
i) < 2 1
R R (, ) (, 2)
ii) = 2 1
R R
iii) 2 1
R R < < 2 1
R R
iv) = 2 1
R R
v) > 2 1
R R .
55 (,R) , < 2R. (, 2R). : i) (, 2R) (, )
ii) (, R)
iii) (, R).
= 1
R = R
= 2
R = 2R
= 3
R < 2R
i)
2 1
R R < < 2 1
R R
2R R < O < 2R + R
R < < 3R
58 :
M
AB
ii) (,R) , . .
iii) = 2R = R ,
(, ), , . (,R) . 56
1 1O ,R 2 2O ,R 1 2 1 2 2O O R R 2R . i) N .
ii) (1
O ) , (2
O )
, , , , . N
, , (1
O ) (2
O ) .
i) 1 2 1 2
O O R R
.
1 2 2
R R 2R 1 2
R R
ii) 1 2
O O 1
O A + AB + B2
O
1
O M2
MN NO 1
O A2
AB BO
1 2 MN AB
AB A1
O + 1
O2
O + 2
O B
AB 1
O M +1
O2
O + 2
O
57 .
. = 90 , .
,
, , . . 58 , , (, R), < >
59 :
1122
1
1
1
2
2
2
12
'
'
'
''
, , . : AB < BE OK > O . AB < BE
< ) 1 1
( ))
2 >
2 (
1 1 , )
> ( ). 59 , , , , , = = . , , , , . () . = . = .
1 1 1
2 2 2
() . = . = .
. .
60
= 2
=
2
. = 1.
. = = .
.
1
= 2
=
( - ) . = . =
2 = . .
= 1, = 1. 61 , , . , = , = = , . ( ) . = . . = . = = = ( ) . = . . = = . ( ) . = ..
60 :
1
1x y
O
A
2
1
x y
2
O
4
1 = 120 = 60 + xx ||
+ = + = 120 + 60+ = 180 xx ||
2
x y . x y , .
|| x 1 1
2 1
1 2
. .
3
x y . y , x . = .
2
( )
1
( )
1 2
. =
x
x
61 :
1
21
x
M
1 2
1
1
1
x
1
2
4 ( = ) .
(, ) , .
= () 1
=
1
5 ( = ) . x = .
.
x
x 1
(1) = = .
2
(2)
(1) (2) 1 2
6 . // . = + =
2
1 1
1 2
1
. = 7 < x. //x . = . =
1 1
//x 1 1
1 2
1 2
1 1
62 :
21
1
I
21 1
I
34
x
12
1
M
K
8 , . = +
// 1 2 I B
1 2 B B
1 1 I B .
= (1) = (2)
(1) + (2) + = + = + 9 // //. .
// 1 1 I B
1 2 B B
1 2 I B .
= (1) = (2) (1) + (2) + = + + + = + + . = 10 , Bx. . (, ) , (, ) x , //. // .
= () 1 2 Z B
1 2 B B
1 1 Z B
, //
// 11 x y. , x, y
, = = . .
63 :
1
1
1
x y
42 3
2
2
1
1
1
112
//x y
2 1
( )
= 1 1
2 1
.
12
< ,
. .
= .
// 1
2 1
2
.
.
(1) // ..
(2) (1) (2) = 13 < .
,
.
i) .
ii) + =
iii) A ) = = 2
) = = 2
i) // 1 1
1 2
1 2
1 1
. =
64 :
112
2
x
55
ii) BE + = + + = = + =
iii) ,
.
(--) . = .
= (1)
i) 1 2
2
.
= (2)
(1) (2) =
(ii) 2 BE = AB + A
2
= = 2
-
= 2
2
=
2
14
= x ,
= 110 =70 ,
= 1802 = 180 140 = 40
15 60 = 60 =1802 =180 120 = 60 = 60 2 = 180 = 120
= 60 , , .
65 :
1
2 1
108
1 2
16
B .
= 144
.
= + B 144 = 3 B + B 144 = 4 B B = 36
= 180 -
= 180 144 = 36
B = . = 17
= = x = 108 . .
= 2
A
1
1
= + 2
A = + = 2
= 1
2
0
1 1 180
0
1 2 2 180
0
1 180 4
2
A + 1
+ 108 = 180
+ 180 4 + 108 = 180 3 = 108 = 36 18
0 90 , , //. 20
, .
: = + 20
+ = 90
+ 20 + = 90
2 = 70
=35 = 55
// = = 55 = 1
=
2
= 45
19 900 . . .
( 2 4) 90 = 900 2 4 = 10 2 = 14 = 7
66 :
20 A . , x//. ) = = = .
A = + = 2, A/2 = A1 = = . x//, . (ii) x// A1 = ( ) A2 = ( ). = ( A1 = A2), = . 21 . (i) 90 + A/2 . (ii) A/2. (iii) 90 - A/2.
I.
. (i) :
+ 1 + 1 = 180 = 180 - 1 - 1
= 180 - 2 - 2 = 90 + 90 - 2 - 2
= 90 + A/2( A2 + 2 + 2 = 90). (1)
(ii) .
: = 90 = 90 + (2) ( ).
(1) (2) = A/2. (3)
(iii) I = 90, + = 90 = 90 - . (4)
(3) (4) = 90 - A/2.
67 :
1
1 2
2
1
E
22
0
90
2
. =
0
902
0
902
0 2 2 180
0 180 2
0 0 180 180 2
=
23
.
i)
ii) 0
90
2
,
0
902
i) 1
A = 2
A = A
2
1
1
= + A
2 (1)
2
2
= +A
2 (2)
(2) (1) 2
1
=
ii) H (1) 1
= + 0
90
2
2
=
090
2
+
2
=
090
2
2
24
.
=
2
.
= - 1
= A
2
1
= 90 .
68 :
1
1
1
12
1
2
1
= A
2 ( 90 )
= A
2 90 + =
= A
2
2
2
2
+ =
2
25
, ,
2
= 180 1
1
= 180 A
2
2
= 0 360
2
360 ,
=
2
=
2
26
. 2
0 180
0 2 180 (1)
. : 0
1 90
(1) + 2 (0
190 ) = 180
+ 180 21
= 180 = 21
27
( A = 90 )
. .
. :
1
E = 2
E = 90 - 1
B = 90 B
2 (1)
. :
1
Z = 90 - 2
B = 90 B
2 (2)
(1) (2) 1
E = 1
Z . .
69 :
1
2
1
1
1
1
2
1
28 ( = ) . , , = . .
0
2 90
= 1
= 1 .
= 12
1
2
2
2
(1)
= . + = 180
+ 2 = 180 0 2 180
0 90
2
(2)
(1) + (2) 0
2 90
29
.
,
2
. 1
B = 90 1
1
= + 1
= +
2
1
B = 90 ( +
2
) =
=
2
+
2
+
2
2
=
2
2
=
2
30 = .
, ,
= . , , .
, .
1
+ + 2
= 180
1
+ 2
= 90 = 90
B
70 :
1
12
1
1
1 1
B = 21
1
=
2
(1)
= 2
+ 2
+1
= 180
22
= 180 1
2
= 90 1
2
(2)
1
= B
(2) 2
= 90
2
(3)
(1) + (3) 1
+ 2
= 90
31 ( = ) . ,
.
i) B = ii) >
i) 1
=
. : 1
= 90 -
. : = 90 -
: =
1
=
ii) .
> , 2
> 1
.
, , 2
= 90 , 1
.
, 1
.
, 1
.
32 ,
= = .
i) AZ = AH ii) AZ AH
i)
, .
= =
,
,
71 :
11
M
1 1
.
, . , 1
1
.
ii) . = . 1
=
. , , 1
.
33
A = 60 .
B= .
.
= A + 1
B= 60 +
2
.
= + 1
= +
2
.
60 +
2
= +
2
60 =
2
+
2
120 = , A = 60. 34 ,
, = = 1
3. .
//
= = 60 = 60 ,
= = 2
3
= 1
3, .
, .,
. 35
A = = 90 = .
= . .
72 :
23
32
1
1
M
1
= 3
,
. = . ,
1
= .
1
= 2
+ 2
= 90 3
+ 90 3
= 180 3
3
=
36 < . :
i) To =
.
ii) =
.
iii)
= .
i)
90 < 90 ( . , )
. < <
ii) > . =
( ) . = .
= = . = <
< =
iii) i) <
2
= (1)
ii) >
2
= (2)
(1) (2)
= .
73 :
1
12
E
Z
O
E
O
5
1
. . = .
1
= 2
|| 1
= 2
1
= 1
. = , / = = . 2 . , = , .
, . 3 , . i) ii) , , .
i) = || = || ii) , . , , . , , .
74 :
1 2
1
1
1
1 1
1
4 .
. ,
=
/, =
, , = ,
, 1
= 2
. 1 2
1 1
1
= 2
.
5 ( = ) . // ( ) // ( ). + =
/, = , , = ,
1
= .
1
= ||
1
= .
6 . // ( ). //
+ =180 .
. 1
.
1
.
. = .
1
= 1
,
1
=1
, ,
= , = . , , //.
75 :
7 . = = . , . , / = || = ||
/ // (1) / = // = //
/ // (2) (1), (2) . 8 . , = = . i) AH = AZ ii) , .
/
= // (1) /
= // (2) (1) (2) i) AH = AZ ii) , // 9 , , , , , = = . i) ii) , , .
/ = /
, . = /
, . i) , ( ), /
ii) , , .
76 :
12 3
4
2
1
Z
1
B
10 = , = .
Z = 90.
, 1
=
2
=
1
=2
3
=4
.
11 . = = . , . .
1
+ + 2
= 180
= + 1
.
. 1
=
= 2 1
1
=
2
(1)
2
=
2
(2)
= , + = 180 . (3)
(1), (2), (3) 1
+ + 2
=
2
+ +
2
= + = 180 .
12 ( = ) .
= .
.
,
1
= .
. = = .
, =//
/, .
77 :
Z
O
1
1
A
13
.
.
, . 14 = 2 . ,
, .
, = 2 = , , . 15 , , .
A ,
0
0
1 1
180 902 2 2 2
, , = 90
0 90 .
. 16 , , . / , . : = ,
= . = , / . : . ,
= . =
78 :
O
1
1
32
K
11
12
17 . , = = = . . : , / , , . . 18 . , , , , , = = = . . , .
. 1
K = 1
=
, .
1
+ 3
= 90
1
K + 3
= 90 .
2
K = 90
. , , , . 19 , . , . , .
1 1
2 1
= = , .
/. , = . , / .
79 :
2
1
1
20 , = . i) = ii) AZ i) ,
= 1
= 1
ii) 1
+2
= 90
1
+ 2
= 90
, . = 90, . AZ 21 , . , .
.
= (1) =// /
// (2) =// /
// (3) (2) (3) /. (1) . 22 , , . .
==
, , =
K N ( ). = .
80 :
1
1
1
32
1
1
11
M
23
B = 45. ( ). . ,
1
= 1
= B = 45 ( )
= = , , , .
1
= 45 , =
= . 24
. , = .
1
= 2
+ 3
2
= 1
1
= 1
+ 3
1
,
1
= 1
+ 1
.
1
+ 1
= 1
+ 3
, 1
= 3
,
. 25 : i) ( ). ii) , , , (
).
i) MK, , + = ( ) = , , = .
81 :
, .
NM ( ) = . , , , = . ii) MK, ,
. = (1) i) .,
+ = (2) ( ) (1) + (2) + + = 26 =
= 90 ,
= 2
.
=2
,
= = 2
27
( = 90), ; .
, =2
= = .
.
28 , . , , // .
82 :
M
E
21
29 . , , , . , ,
= //AB
2 (1)
= //AB
2 (2)
(1) (2) = // . 30 . , = .
, = B
2
(1)
, = B
2
(2)
(1) (2) = 31
( A = 90) B = 30 . , , =
A = 90 B = 30 = B
2
, = B
2
= 32
, .
=
, =
.
= 2
3
= 2
3
= (1)
= 1
3
= 1
3
= (2)
1 2
(3) (1), (2), (3) . = .
= =
83 :
33
( A = 90). .
, . . , , . , . 34
( A = 90) = 30 , . = .. .
= //2
= //
/ (1)
= 30
= 2
= (2)
(1) (2) . 35
( A = 90) . i) , ,
= = 90.
ii) , = 4
.
i)
= 2
(1)
= 2
(2)
(1) (2) . = . ( )
= = 90
ii) = 2
= 2
= 4
84 :
1
E
36 .
, = 4
.
. ,
,
,
= 2
=
2
=
4
37
( A = 90)
. .
= - 1
(1) .
= 2
= ,
= (2)
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