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 = .

B, B > B1 1 , 1> . B > > , B > . . B > . >, =

(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)