" OM Z~ _ [6 i 9 0 O ~ 996] Z~ _ ~ M u Ok Owi < Q } " K i x H Q } " H O> Z ~ K ioz Z~ _ H K O^. OM " H H J< H ~ OQ K Ok O^ H k x } H x K. Q } Ja z JQ O ~ O_ OK H <. J _ XH x+ rqok H H O_ Q } OK. Z~ _ O ^ ^ Q \ OK H ^. z < i O H + L " ih i Oz H ~ ^ Q WK. P rqoz L C_ D rq K i N is a Numer A Portrait of Paul Erdos J x Q f _ k. I know numers are eautiful. If they aren t eautiful, nothing is. Jx Z~ _ K.. ik. i ~ : OM ~Y JH ~ } OY zok @. JH ~ } OY H < Q ^ ~. _ O_ H ~ } H ey H H iok @. : ~ OH Q Q OY f ~ O " @.6 ~ x.7 ~, ~,,.8 OY H L* ~ O q.9 OY P_ H @. ik x rq '' OY Y " OK K q. OY k i H ~ H L Q _ K q. > e < OY, ~+ < J@ Q OY OH M OY H sh ~ OY " ^ Q q Q i OK @ L y _. _ + Q P zok @ _ _. 0,000 O ~ OY " ^ u ~ OkOK _ k. OM ~ O^ @ ~ ^ XH Y HO ^ Q " k. OM i Q Ja $k K O^ @ 000 O ~ H >. L yok K JO H ^ u O^ J~ ah JO H ^ u J _. D ^ u 0,,,,,,6,7,8,9 J OY L yok ^ ~. nxx P^ ~ 0 H ey ^ O OM ^ u (Numerical System) Jx _ JO^ ~ ) ''Decima J ^ Decem J \ < ^ O_ z k. Decem JQ '' k Jx J~. Q } ''q* H ~ } Jx, OM ''Q } H ~ } Jx JO^ ~.

" OM 7 ~ Q u [ OY, N {,,, g}, ~ OY W {0,,, g}, ~ OH Z {g,,, 0,, g} i JH ~ } OY Q g\ ^ q H i H L yok @ Q i < ~ H x <. H Ok " x X C H C. Z) Jx ~ OH JH ~ } OY Q. a) Jx [ OY ~ OH Q. ) Jx ~ OH [ OY Q. _ ) Jx ~ OY [ OY Q. W) Jx [ OY ~ OY Q. Z ) Jx JH ~ } OY ~ OY Q.. i ~ : OM ~Y JH ~ } OY zok @ JH ~ } OY p JH ~ } OY Q ~ O " Q q OY. WH _ p i q ~ OH i q! 0, g\ x Q Jx Q i OK ^ ~. q p WH _ q > 0 J ~ Q OM Q Jx Q i OK ^. JH ~ } OY [ OY, ~ OY, ~ OH, ^ < H a i } H a LO_. WK @ XH eh Z@ JH ~ } OY XH Oz Q K K ^ K _ K. i OM Q i OK. JH ~ } OY _ OM ~Y zok K i WK @ XH eh OM ~Y _ z @ Q xok K. JH ~ } OY OM ~Y Yz Q zok @ H ~, " ^ @ } _ W _ JH ~ } OY OM ~Y Q i OK. q~ } (i) OM ~Y 7 Q i OK @. 0 i ^ 7 LO_. OY OM N W Z Q 6 N W Z Q / N W Z Q 0 N W Z Q 9 N W Z Q 8 N W Z Q.7 N W Z Q J ~ " Q Q, JH ~ } OY H ~ Q q_ n. ^ Q } G

J^ (ii) 7 i ^ 7 LO_. Q } G (iii) - i 0 ^ - LO_. u [ OY JH ~ } OY JQ. nx q~ _ x[ Q <?. JH ~ } OY H < Q ^ ~.. (a) OH (i) O$ ^ ~ (Closure property) ~O_ JH ~ } OY " Z C_ XH JH ~ } OY JQ. nxx JH ~ } OY OH H O$ ^ ~ JO^ ~. Q O$ ^ ~ H ey O_. (ii) a i c d J q U" < ~O_ JH ~ } OY, a + c _ JH ~ } OY JQ d (i) + 6 XH JH ~ } OY. q~ }: 9 9 9 (ii) + + 6 + XH JH ~ } OY. ^ ~ (Commutative property) ~O_ JH ~ } OY OH ^ ~ H ey O_. a i c d J q U" < ~O_ JH ~ } OY a q~ }:, J q ~O_ JH ~ } OY LHS + + + RHS + 0 + 9 0 OH ^ ~ x[o. LHS RHS 0 + (iii) K ~ ^ ~ (Associative property) JH ~ } OY OH K ~ ^ ~ H ey O_. a, c d i e f J q U" < _ JH ~ } OY, a + c c + d d 9 0 a + c e a c + d f j + + d j e f

" OM q~ }:, i J q _ JH ~ } OY. LHS + + j + + + + j RHS + + j + + j + + 6 6 j + + j + 7 + 7 + 6 6 + 9 7 6 6 + 9 6 6 6 6 LHS RHS OH K ~ ^ ~ x[ Q. j JQ. (iv) OH (Additive identity) : i U^ < XH JH ~ } OY " J^ OY JQ. a U^ XH JH ~ } OY J~ a JH ~ } OY OH JQ. q~ }: (i) (ii) + 0 0 + 7 7 7 7 0 7-0 j + - + -7 j + 0 a 0+ (v) OH q (Additive inverse) : -a jxh } H H a H OH q a JH ~ } OY J~ -a j J JH ~ } OY OQ LO_. H a a 0 q~ }: + - j (i) H OH q - JQ. (ii) - H OH q JQ. (iii) 0 H OH q 0 JQ. OY [ OY ~ OY ~ OH JH ~ } OY O$ ^ ~ i a OH ^ ~ XH H JH ~ } OY 0 q 0 WH _ q! 0 Jx " K. K ~ ^ ~ i Q } G

J^ Q } G.. () f " ( H ) (Sutraction) (i) O$ ^ ~ (Closure Property) U" ~O_ JH ~ } OY ^ ^ Z C_ JH ~ } OY JQ. H f " Q O$ ^ ~ H ey O_. a i c d J q U" < ~O_ JH ~ } OY J~ a (ii) q~ }: (i) 7 (ii) 7 7 - XH JH ~ } OY JQ. - c _ XH JH ~ } OY JQ. d - - XH JH ~ } OY JQ. ^ ~ (Commutative Property) JH ~ } OY f " ^ ~ H eylo_ ^. a i c d J q U" < ~O_ JH ~ } OY J~ a q~ }: 9 i J q ~O_ JH ~ } OY, LHS! 9 9 - RHS - 9 9 0-8 8 0 - - - - JQ. LHS! RHS f " ^ ~ x[ H ^. (iii) K ~ ^ ~ (Associative property) JH ~ } OY f " K ~ ^ ~ H eylo_ ^. a, dc, fe J q U" x _ JH ~ } OY J~ q~ }:, i J q _ JH ~ } OY. - -! - - j j JQ. LHS a c! d c d a - - JQ. - c e a c -! d f j - - d j - - j RHS - - j - - j - - 6 j - j 6 - - 6 - - LHS! RHS f " K ~ ^ ~ x[ H ^. ~O_ JH ~ } OY " " \ ^ ~ x[ Q. e f 6

" OM OY O$ ^ ~ f " ^ ~ K ~ ^ ~ [ OY ~ OY ~ OH JH ~ } OY H ^.. (c) Q }H ~ (Multiplication) (i) O$ ^ ~ (Closure property) ~O_ JH ~ } OY Z C_ XH JH ~ } OY JQ. Q }H ~ Q' O$ ^ ~ H ey O_. a, c d J q U" ~O_ JH ~ } OY J~ a c d ac d XH JH ~ } OY JQ Q } G q~ }: (i) (ii) 7 7 XH JH ~ } OY JQ. 9 0 7 XH JH ~ } OY JQ. (ii) ^ ~ (Commutative property) : JH ~ } OY Q }H ~ XH ^ ~ H ey O_ a i c d J q U" ~O_ JH ~ } OY J~ a q~ }: i 8 - J q JH ~ } OY -8 j -8 LHS -8 j RHS -8 j - LHS RHS Q }H ~ ^ ~ x[ Q. j JQ - c c a d d JQ (iii) K ~ ^ ~ (Associative property) JH ~ } OY Q }H ~ K ~ ^ ~ H ey O_. a, c, e d f J q U" _ JH ~ } OY J~ a c e a c e j j JQ d f d f 7

Q } G J^ q~ }:, - j i J q _ JH ~ } OY - j - LHS - j - RHS - 8 j - LHS RHS Q }H ~ K ~ ^ ~ x[ Q. jj JQ. (iv) Q }H ~ (Multiplicative identity) U^ XH JH ~ } OY i H Q }H ~ J^ JH ~ } OY JQ. H JH ~ } OY H Q }H ~ JQ. (v) a J k U^ XH JH ~ } OY J~, a a a JQ. q~ }: 0 K Q }H ~ (i) (ii) 7 7-8 j - 8 u JH ~ } OY 0 Q } Oz 0 K. a U^ JH ~ } OY J~, a q~ }: (i) - 0 0 (ii) 7-0 0 j (vi) Q }H ~ q H a 0 0 0 a JQ. a! 0, a J u JH ~ } OY, c d J JH ~ } OY OQ LO_ c J~ a H Q }H ~ q c d JQ. d a J k JH ~ } OY J~, a Q }H ~ q H. q~ }: (i) H JQ. (ii) - j H Q }H q - ~ OH ~ OY H Q }H ~ JQ <? j JQ. i) 0 ^. ii), J JH ~ } OY J^ OY Q H ey O_.? H 0. JQ <? 8

" OM OY [ OY ~ OY ~ OH JH ~ } OY O$ ^ ~ Q }H ~ ^ ~ i K ~ ^ ~.. (d) Q ~ (Division) (i) O$ ^ ~ (Closure property) H x JH ~ } OY Q ~ O$ ^ ~ H ey O_. (ii) q~ }: a i c d J q ~O_ JH ~ } OY, c 0 d ' Z C_ XH JH ~ } OY JQ. (i) (ii) a c d!, J~ ' XH JH ~ } OY JQ. ' 8 XH JH ~ } OY JQ. ^ ~ (Commutative property) JH ~ } OY Q ~ ^ ~ H ey LO_ ^. a i c d J q U" ~O_ JH ~ } OY J~ a q~ }: i 8 J ~O_ JH ~ } OY LHS '! 8 8 ' JQ. 8 RHS 8 LHS! RHS Q ~ ^ ~ x[ H ^. ' c d c d ' a i! JQ. Q } G (iii) K ~ ^ ~ (Associative property) JH ~ } OY Q ~ K ~ ^ ~ H ey LO_ ^. a, c d i fe J q U" _ JH ~ } OY a q~ }:, i J q _ JH ~ } OY ' ' j! 9 ' ' j ' c e a c ' d f j! ' ' d j e f

J^ Q } G LHS ' ' j RHS ' ' j ' j ' j ' 0 0 0 0 0 LHS! RHS Q ~ K ~ ^ ~ x[ H ^. OY [ OY ~ OY ~ OH JH ~ } OY O$ ^ ~ Q ~ ^ ~ H ^ K ~ ^ ~.. (e) q Q ^ ~ (Distriutive Property) : (i) _ H Q }H ~ H q Q ^ ~ JH ~ } OY Q }H ~ _ H q Q ^ ~ H ey k. (ii) a, a, c i e d f J q U" _ JH ~ } OY a c d q~ }:, 9 i J q _ JH ~ } OY + 9 j 9 LHS + 9 j RHS + 9 0 7 + j 8 + 7 7 9 0 + LHS RHS + e a c a f j + d + JQ. 9 Q }H ~ _ H q Q ^ ~ H ey O_. f " Q }H ~ H q Q ^ ~ JH ~ } OY Q }H ~ f " q Q ^ ~ H ey O_. c i e d f J q U" _ JH ~ } OY a c e a c a - d f j - d e f e f 0

" OM q~ }: 7, i J q JH ~ } OY J~ 7 - j 7 7 LHS 7 j RHS 7 7 8 7 0 j 9-9 7 0 70 70 70 - JQ. LHS RHS Q }H ~ f " q Q ^ ~ H ey O_. Q } G. i Z H. J. i) JH ~ } OY H OH. (A) 0 (B) (C) (D) ii) iii) - H OH q. (A) (A) - (B) (C) (D) - H. (B) iv) 7 H Q }H q. - - (C) (D) - (A) 7 (B) 7 (C) 7 (D) v) LO_ ^. (A) 0 (B) (C) (D). H O^ W _ q OH U ^ ~ H ey " ~. - 7 (i) - 7 j + + - 9 9 7 j (ii) + 7 7 9 + 8 j + + 9 8 j (iii) 8 + 7 7 + 8 (iv) 7 0 7 0 0 0 - j + - + -7 j (v) + 0 - j. H O^ W _ Q }H ~ U ^ ~ H ey " ~. (i) (ii) - j - - j

J^ Q } G (iii) 7 8 - - 8 j 7 j (iv) (v) 9 9 7 + 0 j + 7 0 7 7 7 8 j 8 j. H O^ W _ JH ~ } OY [ _ H, f ", Q }H ~, Q ~ H ^ ~ ik _. (i) i (ii) - i - 7. H O^ W _ JH ~ } OY _ H, f ", Q }H ~, Q ~ H K ~ ^ ~ ik _. (i), i - (ii), - i 9 7 0 6. JH ~ } OY Q }H ~ q Q ^ ~ L yoz H H iok. (i) - 8 + 9 7 j (ii) 7 - j.. ~O_ JH ~ } OY ^ Q JH ~ } OY H Q @ i ^ Q [ OY? Jq i JQ. i ^ Q ~ OH. P ~ OH, 0,,,. JQ. ~O_ [ OY H ~ OH ^ iq [ OY H ~ OH H Q K. i ^ [ OY H Q @ ^? ^ H ^. H x ~O_ ~ OH ^ JH ~ } OY LO_. L^ ~ } 0 i ^,,, g J JH ~ } OY H Q K. g\ x 0., 0., 0., g Jx 0 0 0 _ " K. W^ q^ Q 0 i ^,, J JH ~ } OY _ LO_ x e. W\ JH ~ } OY H Q 0., 0., 0.7 Jx _ " K.

" OM 0. 0. 0.7, J JH ~ } OY f H O^., J JH ~ } OY ^ U^ XH JH ~ } OY LO_ @ ^ ^? L k. J JH ~ } OY H ^. W^ q^ Q 0 i ^,, i J JH ~ } OY L x e. i ^ WOH Z JH ~ } OY H Q K <? H Q K. 0 0 Q i 0 0 Q " K. ^ D ~O_ JH ~ } OY ^ Q J< H JH ~ } OY " K. Q } G 00,,,,, 6, 7, 0 0 0 0 0 0 0, 0 0 00 00 8, 9 0 0 J JH ~ } OY " K. 0 00 Q i 0 ^ H x JH ~ } OY H Q 0 0 Q ". ^,,,, 00 00 J q k JH ~ } OY O^ K., 6, 7, 8, 00 00 00 00 9 00 H W _ @ ^ ~ OM ~Y Q i Q Q e H K. 0 i ^ Q OM ~Y ^ kq K Q H @H ^ ~ ijeok K.

Q } G J^ W^ q^ Q O_, O_ W@ H JO ~ J< H JH ~ } OY Q i OK K. W^ q^ Q u Oz, U ~O_ JH ~ } OY ^ < J< H JH ~ } OY H Q K. [ OY, ~ OH H O_, U ~O_ JH ~ } OY ^ < J OM H " JH ~ } OY " K. ) L yok ^ u a, J q W _ JH ~ } OY J H. a, ^ J< H JH ~ } OY q, q, q,... H Q ^ u H O^ W _ k. q ^a+ h q (a + q ) q (a + q ) W@ K K. a q a q q a q q q q Z_ " q, q OY J i O_. W^ q^ Q a, ^ q, q J JH ~ } OY q _ " J i O_ @ H O^ W _ k. q (q + ) q (q + ) W@ K K. a q q a q q q ~O_ OY ~ i Z C_ P OY ^ J i O_. ) " ~ H ^ u a, J q ~O_ JH ~ } OY J H. (i) ~O_ a ~ H..Q f @ ^ ~ ~ Q ~. W C_ ~ ^ XH OY LO_ a ^ _ XH JH ~ } OY LO_. (ii) ~ ^ U OM x _ ~O_ a i ~ 0 Q } OK @ ^ ~ JH ~ } OY O^ K. WOH Z OY O^ @ 00, 000... " ^ Q " \ Q } OK. H O^ W _ " ~ ~ ^ ^ ~ a, ^ " ~ ~ JH ~ } OY O^ K..

L^ ~ }., ^ XH JH ~ } OY H Q. ^ u : W _ k: a,, ^ Q JH ~ } OY q J H. 0 q ^a+ h + j 6 + 0 j q 0 j 0 XH JH ~ } OY JQ. " ~ H ^ u : W _ k: a, a, W@ " K. i, 0 0 0 " OM 6 ^ Q JH ~ } OY H Q @ i ~ 0 Q } OK. 0 0, 6 0 0 0 00 0 0 0, 00 00 00 00 00 60 ^ Q JH ~ } OY,,,,, 6, 7, 00 00 00 00 60 00 8 00 i 9. 00 L^ ~ }. - i ^ ~O_ JH ~ } OY H Q. W _ k: a -, q, q ~O_ JH ~ } OY J H. ~O_ JH ~ } OY - 0 i 6 0 q ^a+ h q + j - 6+ 0 j - 0 j - 0 q (a + q ) - + - 0 jj - + ^- h c m - - 0 0 j - 0 j - JQ. 0 Q xh : ~O_ JH ~ } OY W@ " K - 0 - < - < - < 0 0 Q } G

J^ Q } G J.. H O^ W _ JH ~ } OY [ ^ Q XH JH ~ } OY H Q. (i), (ii) -, (iii), 7 6 7 (iv) 7, 8 8. H O^ W _ JH ~ } OY [ ^ Q ~O_ JH ~ } OY H Q. (i) 7, (ii) 6, 9 (iii), (iv) -, 6. H O^ W _ JH ~ } OY [ ^ Q JH ~ } OY H Q. (i), (ii) 0 7, (iii) -, (iv), 8. _ O_ H ~ } Q H H iok @ H x L^ ~ } ijeok ^. (i) + (ii) 0 (iii) 7 (iv) -? L^ ~ } (i), (ii) i (iii) X H XH H H ey k. L^ ~ } (iv) ~O_ H H ey k. L^ ~ } (iv), U H " ^ @ i+ iok x e <? L^ ~ } (iv) U^ XH ^ ux \ OK x _ " ~ ~ a OK. L^ ~ } (i) ^ h (ii) j " ~ " ~ q O^. D _ @ O_ @ _ @ H H J iok @ XH + " ^ ux \ OK. Z_ O_ _ " Q J H H i+ iok H BODMAS JO^ ~. B - O_ H ~ }, O - H D - Q ~, M - Q }H ~, A- _ H S- f " O_ H ~ } H x s H io _ O H L yok @ ^ ~ H XH H i+ iok @ P H " k. Q L yok O H H O^ W _ q. s H iok _ O H ~ " " H J_ w O_ H ~ } ( ) ~< f < H ^ ~ } O_ H ~ } { } < H } O_ H ~ } [ ] H\ H K ~ O_ H ~ } 6

" OM H '' H H x O^ ~ '' H ~O_ ~@, ''0 H < Q XH O. '' k H Q @ O\ Q xoz O^. D '' H JQ '' Q } Oz Jx J~ L^ ~ } (i) '' H ~O_ ~@ Jx " ^. (ii) ''0 H < Q XH O 0 Jx " ^. (iii) '0 H Q 0 Jx " ^. XH \ H O> Z Q Q O H L yok _ q H e Q Q O H H " ^ @ i+ ioz O H yok. ^ ~ " u JO ~ O H H i+ iok. W^ q^ Q H q Q _ K. L^ ~ }. H H iok : L^ ~ }. H H iok : L^ ~ }. H H iok : + j 8 8 + j 8 + j 6 8 8 6. + H 8. 9 + H 8 9 j [ O_ H ~ } Y W _ ] 8 9 + 6 + + 7 6. 6 6 6 - j+ 8 ' - jb - ' j+ - j + ['' H Y W _ ] 8 B [q H e Q O_ H ~ } Y W _ ] - ' j+ 8 B - + j 8 B - + - + 9 9. 60 60 60 Q } G 7

J^ Q } G L^ ~ }.6 H H iok : 7 7 - $ ' - j. 6 - $ ' - j. - 6 7 $ - j. 6 - - 7 $. - 9 0 8 6 7 $ -. - 7 $ -. + 7 8 + 77. 68 68. i [" ZO H K. J. (i) (A) 0 (B) 6 (C) 0 6 (D) (ii) 7 (A) (B) 8 (C) 0 (D) 0 8 (iii) (iv) + 9 0 (B) (A) ' 8 (C) 8 (D) 6 (A) (B) (C) 0 (D) 7 0 (v) - j+ - j (A) 0 (B) (C) (D). H H iok : (i) (iii) 6 (v) (vii) ' 8 9 j (ii) H 6 ' $ H + 6 H 7 0 9 - ' j (iv) 8 8 0 j' + j 8 ' H + j 8 - - B. (vi) ' 7 j- 8 j ' j (viii) 8-6 j- ' ' + - - - 7 j ; E 8 8

" OM. : ~ OH Q H ey OY f ~ " @ D Q O^ OY f ~ Z@ " K < < ~ H < ^., Jx " K. WH _ P^ ~ i. ^ ~ } Q a a n ~ Q } OK Q K a n WH _ a U^ XH ^ < H " OY i n _ U^ XH ^ < H " OY JQ. a P^ ~ i n JO^ ~. x~ K n XH ^ < H ~ OH J~ x n JQ x.x.x...x n H ~ } OH i.e, x n x x x... x ( WH _ n, H O> ^ k) n ~ Q xh : x x. nxx K ^ _ O Z? 7 7 H H 7 H Jx K k" ^. WH _ P^ ~ 7 Jx Jx K k" ^. Q } G nxx L^ ~ } < ~ H @, H O^ W _ \ H Q xok. ~ XH OY J^ OY Q }H ~ f ~ P^ ~ x x x x ^-h ^-h ^-h ^- h - j j j j j j 9 9 6 j 6 a a a... m ~ a m a m L^ ~ }.7 H O^ W _ OY H Q ". (i) (ii) 8 (iii) (iv) 8 (v) 6 (i) (ii) 8

J^ Q } G (iii) (iv) 8 7 (v) 6 8.6 ~ OH x W _ ^ < H " OY ~ OH x~ K O H O^ W _ '' x qiok _. (i) (ii) x m n m n a a a, WH _ a XH " OY i m, n ~ OH Q. q~ } + 7 m n m nx L yo _. j j j j (a a a WH _ a, m, n ) Q ~ x a q~ } 6 6 a m 6 6 ( a m a a m n n m n a WH _ a 0! i m, n ^ < H ~ OH m > n n J x L yok _. WH _ a 6, m, n ) (iii) m n m n x ^a h a, WH _ m, n ^ < H ~ OH. q~ } ( ) + + + 8 Q } OK @ ^ ~ J^ e O^ K. i.e, ^ h 8. (iv) Q Q OY m o, 0 m ' m m m. m ' m Y J~ (x L yoz) " ~ H ^ u: m m m m m m m m 0 D ~O_ ^ L yoz m ' m m JQ. W _ L^ ~ } ^ ~ < x O^ ^. x a H x U^ XH JH ~ } OY J~, a 0. q~ } (i) 0 (ii) x~ OK : a ( x y ) z a ( y zx ) a ( z x ) y 0 j (iii) 0 (iv) 0 0 - j (v) ^- 00h 0

" OM (v) x } H H ey OY H q Q } OK @, P OY H ^ < H Q Q Q }H q H Q. q~ } (i) 6 (ii) (iii) 0 0 0 0 00 0 0. 0 6 0 6 6 6 8 j H 8 8 j. j H W^ q^ Q 6 H 6 W _ L^ ~ } ^ ~ S^ x O^ ^. x a XH " OY i m XH ~ OH J~ a (vi) X H Q OY Q } OK @ H Ok H H ~ } Q xok, m a JQ. (i) 7 ^ h ^7 7 7h ^ 7h ^ 7h ^ 7h ^ 7h (ii) j j j j j 0 j (iii) 0 ( ) j j j j j j j a, J q U" ~O_ ~ OH J~ a ^a h ^ah H Ok q^ Q O^ K. ~ ^a a a... m h ^... m h a a a... m m x 6 a m ^a m (i.e.,) a m ~ m ^ah m ~ ^ah m h, WH _ a, " OY i m ~ OH. m Q } G

Q } G J^ q~ } x x (i) ^ h x x (ii) 7 ^7 h 96 (vii) Q ~ H H ~ } Q xok. (i) (ii) x 7 q~ } (i) L^ ~ }.8 j 9 6 i j m j ca a m m j c m a WK @ a j a. Jx " K a m j a a a... m a m j am j m a a a... m... m ~ ~ a m m j am, WH _! 0, a, " OY, m XH ~ OH. a 7 7 j a7 (iii) j 6 H H iok : (i) (v) (ii) j 7 9 6 (ii) 0 ' 0 (iii) ^x 0 h (iv) ^ h 0 j (vi) ^ h (vii) ^ h (viii) p J~ p H q H Q. + 8 (i) 9 6 (ii) 0 ' 0 0 0 9 6 (iii) ^x 0 h ^h [a a 0 ] (iv) ^ h 0 8 0 [a a 0 ] ~

" OM (v) j (vi) ^ h 0 0 (vii) ^ h 6 96 ( H ) ^ h 6 8 96 (viii) W _ q : p p 6 8 p (WH _ W~ " P^ ~.) Q } G L^ ~ }.9 H Ok " x q H Q. (i) (ii) (iii) j (iv) 0 (v) - j (vi) 7 0 j (vii) 8 j B (viii) ' 9 8 j 8 j 8 j (i) +^ h (ii) (iii) (iv) 0 (v) (vi) (vii) (viii) j 8 000 6 - j - 0 0 7 j 7 8a j B 8 j B 6 j j 8 + 9 ' 9 8 j 8 j 8 j 8 j 8 j 9 9 8 j 8 j ( H ) 9 9 8 j 0 8 j L^ ~ }.0 6 P^ ~ Q Q Q e K. 6 Jx e. 6 ^ h

J^ Q } G L^ ~ }. H H iok. (i) ^ h ^ h (ii) ^ h ^ h ^ h (i) ^ h ^ h ^ h (ii) ^ h L^ ~ }. k OK. (i) x (i) W _ k x (ii) L^ ~ }. ^ h 6 6 6 8 6 6 6. 8 x x (ii) 8 j 8 j 8 j x x x x 8 j 8 j 6 8 j x + x 8 j 6 8 x+ x 6 x 6 ^ h ^ h H H iok : ^ h ^ h ^ h ^ h x 6. 6 (a W~ " P^ ~ ) j (a W~ " P^ ~ ) 6 6 8 + + 6 6 8 6 6 9 6 7. 9 9

" OM J.. H Ok " xh i [" Z H. m n a " k (i) a (A) a m n + a (B) a m- n (C) a m+ n (D) a mn (ii) p 0 " k (A) 0 (B) (C) (D) p (iii) 0 (A) (B) (C) 0 (D) 00 (iv) 6 " k (A) 6 (B) (C) - (D) 6 6 (v) H Q }H q (A) (B) (C) (D) 6 (vi) ^-h ^- (A) (B) (C) (D) 6 h " k (vii) ^- (A) (B) (C) h " k ^ h " k - (D) - 0 (viii) + (A) (B) (C) (D) (ix) (x) (A) (B) (C) (D) j " k ( ) 0 " k (A) (B) 0 (C) 0 (D). H H iok : 8 (i) ^-h ' ^-h (ii) (iv) c m (iii) ^- h 7 0 j j j (v) ^ ' h (vi) j 8 6 6 7 a- -c c- a / 0 (vii) y y y (viii) ^ph ^ph p (ix) 9 - - 8 j (x) / 0 8 9 / - + j 6 j. H O^ W _ " \ q H Q. 0 (i) ^ + h (ii) ^ h ' (iii) 0 (iv) ^ + + h (v) - 8 j B (vi) 7 0 7. j + + j j / Q } G

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J^ Q } G 600 H O> 00, ^ Q ~ Q L k. D O^ ~ H L* ~ O q 00 JQ. i H L^ ~ } ijeok ^. (ii) OY 6 f H < ^. OM ~Y UH } Q f H < ^ WH _, 6 H 66 ^ Q ~ Q 6 LO_ @ K ZO^ H Q 6, 66 ^ Q 6 LO_. H x J H ~ 66 ^ Q ~ q K K. H L* ~ O q 66 Q f H _. 00 Q }H Q f H x, 6 H L* ~ O q 600 JQ H x 700 H ^.. WOH H x L^ ~ } 7,68 f H x (a) ^ io _ L* ~ O q 7,60 () O^ iok _ L* ~ O q 7,600 (c) " iok _ L* ~ O q 8,000 (d) k" iok _ L* ~ O q 0,000 ^ O L* ~ O q q~ }: ^ O OY 6.79 f H < ^. (a) ~O_ ^ O ioz, 6.7 JQ. (zi JO H 9> H _ JQ ) 6.79-6.7 ( ~O_ ioz) () 6.79 ~O_ ^ O Q xoz,, H O> z OY H 7 ^ e" _. 6.79 6.7 ( XH ioz) 8

q~ }: ^ O OY 6.7 f H < ^. ~O_ ^ O io _ L* ~ O q 6.7 zi JO H H _ K. (a) () XH ^ O iok _ L* ~ O q 6.7 ~O_ ^ O. Wk H O> z OY H 7 J> " ^. 6.7-6.7 q~ }: ^ O OY.89 f H < ^ (i) XH ^ O iok _ L* ~ O q. JQ (ii) ~O_ ^ O iok _ L* ~ O q. JQ. (iii) _ ^ O iok _ L* ~ O q.8 JQ. (iv) < Q ^ O iok _ L* ~ O q.8 JQ. L^ ~ }. " OM H O^ W _ OY ^ Q ~ Q O_ ~ OH iok. : (a) 88.9 () 998.7 (c) 86.79 (d) 999.96 (a) 88.9-88 () 998.7-998 (" ^ \ ^ O Q OY H O>. H ~ Q ~ ^ ) (c) 86.79-87 (d) 999.96-000 [" ^ \ ^ O Q OY H O> Z. H ~ Q q O^ Z Q ] J.8. H O^ W _ " xx ~O_ ^ O iok : (i).68 (ii).6 H.g (iii) 9.97 g. (iv) 6.96 g. (v).06 g. (vi) 79.9 H.g ~ q ^ H Ok JO H Q H ~ q. 9 0,000 ^ Q ~ Q Q L* ~ O q ~ @ @ J xh _. L* ~ O ^ ux L yoz Ju ^ OY H Q. a. 000 + 8 7. 000 8 7 c. 000 x 8 7 d. 000 8 7. H O^ W _ " xx _ ^ O iok : (i) 0.08 g. (ii).7 H.g (iii) 8.96 (iv) 0.7 Q (v) 6.006 (vi) 00. Q } G 9

J^ Q } G. H O^ W _ OY Wz q^ Q L* ~ O q " : (i) 7 ^ ^ Q ~ Q. (ii) ^ ^ Q ~ Q (iii) 688 O^ ^ Q ~ Q. (iv) 76 k" ^ Q ~ Q. (v) 767 H ^ Q ~ Q (vi) 067809 H @ ^ Q ~ Q. H O^ W _ OY ^ Q ~ Q O_ ~ OH iok : (i).66 (ii) 777. (iii) 0.06 (iv) 0.8 (v) 99.77 (vi) 9999.967.9. OY P_ H @ q< ^, WO ^ * i J^ Q ~ Q Q }. D Q O^ L H eyok J^ " q+ < ~ H <. (a) ^ ~ } ~ O OY OY W@ " K. 0 + 0 + W^ q^ Q OY 7 _ " K. () 7 0 + 7 0 + 7 ^ ~ } Q a J JO H ~O_ JO H OY a, W@ " K. a 0 a + 0 a + WK @ a J k ~O_ a 0 + a 0 + a JO H OY a H ^. J OY f H. Wk _ JO H OY JQ. D OY W@ _ " K. 00 + 0 + 00 + 0 + ^ ~ } Q ac J JO H _ JO H OY a,, c W@ " K. ac 00 a + 0 + c 00a + 0 + c W^ q^ Q _ JO H OY ca i ca " K. ca 00c + 0a + ca 00 + 0c + a OY P@ (i) ~O_ JO H OY u " @ " }, < * ~O_ O H OY Q ~ @ H <. ^ K @ Q K <. " i O + } D H Ok @ qiok _. @ * Q Q K, K ^. 0

" OM " } i < * ^ O + } : Q } G " } H H x Q i qiok ^. < * a J OY Z H <. D OY 0 a+. J ~O_ JO H OY H H ~ O D JO H u " H OY a 0+ a. K. D ~O_ OY _ : ^0a+ h+ ^0+ ah a+ ^a+ h " } K @ Q " Z C_ H Q }H Q LO_. z [" yok Q ^a+ h O^ K. (i.e.) OY H ~O_ JO H " K. (c) W _ ^ ux Q i OK. ~ " K _ ^ H Q. XH H ^ < ~ H @. (i), 9,,, ( u ^ O^ ^ H O> 6 Z ) D ^ ux _ yoz ~ " K _ ^,, JQ. (ii) 00, 96, 9, 88,,,. ( u ^ O^ ^ H O> ) (iii) 7,,, 8,,,. (7 H Q }H ) (iv) 000, 00, 0,,,. ( u ^ O^ ^ Q ) (v),, 9, 6,,,. ( [ OY H ~ )

J^ (d) u [ OM ^ u H Ok W _ u [ PH ~ Q OM ^ ux u [ JO^ ~. Q } G u [ OM ^ ux Q i Oz 6 J_ ~ iok. " lh K ~ (Magic Square) H L \ H Q xok nxx x " lh K ~ JO^ ~. " lh K ~ u J_ ~, x ~ i H ~ Q OY ". D " lh K ~, " lh " 7 JQ. ^ > Q OY Q xok. " lh " ^ OY _ ~@ JQ. ^ > 9 xo. WOH 8 > xo L k. 9H H Ok Q < Q > i 9H Q < Q > L q. Jq (a),6,7,8 i 0,,, g\ ^ Q _ JQ. (),,,7 i,,,7 g\ ^ Q _ JQ H ^ Q U ~ OY -,-6,- i,9,,9 O\ q L yok K. g\ ^ Q _ JQ. U OY L yok < O^ Q x~ ~ OK. L^ ~ },,,7 i,,,7. H O^ W _ @ K ~ " < Q " _ yoz > w. H ~ ^ u OY ~ Q xo. _ yoz > Q OY " \ H Z^ ~ Q Q M m > xh ~.

" OM " lh K ~ ~ Q < ^ O_ 9 OQ ~ < } q Q 9 H. " \ x Q ~ ~ Q OK @ h _ Q "? " lh H H @ x $ O_ ~ L yoz u ~Y Q OY " 6 K @ Q J ~. u OY ~O_ ~ H O> Z Q L yok ~ ^. _ O_ 9 ~ Q Jx JO H L yoz J_, x ~ qq^ ~ OQ Q K ~ xo. L yoz JO H m L yok ~ ^. XH H ~ Q i O OY _ O H ~ OY. P OY u " Jk ~ H H ~ OY i ~ OY JQ. P ~O_ H ~ Q i O OY H Q? H ~ O q OK OY 87 OY $ H ~ Q i OK. 8 6 9 7 6 9 7 8 7 6 9 7 8 9 9 8 7 6 7 8 9 8 7 Q } G 87 O_ 6 ~ OY Q } OK. 87 87 87 87 87 87 87 87 87 6 78 78 87 OY " ~ ~ O Q J^ ~ JO H $ H ~ " ~ ~ ao^ O_ i } K @ Q xok K.

J^ J.9 Q } G. H O^ W _ ^ u io : (i) 0,, 0,,,. XH OY J H. ^ xh 9 _ (ii),,,,,. [" ~\ O K. ^ xh _. (iii) 8, 77, 70,,,. e K Q } OK. ^ x O_ f ". (iv).,., 6.6,,,. q ye ^ xx 6K yok. [" O_ h J H x OY (v),, 6, 0,,,. f ". (vi),,,,, 8,,,,,. q ye OY ZO? [" : k (D J H < (FIBONACCI) J H JO^ ~.) (vii), 8, 7, 6,,,.. h\ > " @ H. XH H u Q Q " ^ \ " @ ~ O_. h\ @ q ^ " @ ~ H ^. (a) H u " @ H Ok " ^ ~O_ " @ " ^. W@ Zx ~ ^ H h\ @ K ~? () h\ x y ~ " " H H <. ^ xh ~ " @ " ^ " @ H Ok " ^. W@ Zx ~ ^ H Q Q " ^ \ " @ ~ K ~?. XH " i P O_ H O^ W _ ^ u J ~. (a) k ~ P O_ LOK. " \ x H OK O_ " Zx P O_ L " H Q. () 0 J_ ~ LO_, Zx P O_ L " H Q. " Zx P O_ L " e H @ Q ^ ux Q i Ozu"? H Ok " xx io. J_ ~ 6 7 8 9 " P O_ 6 0

" OM JH ~ } OY H _ H f ", Q }H ~ O$ ^ ~ H ey O_. H x JH ~ } OY Q ~ O$ ^ ~ H ey O_. JH ~ } OY H _ H i Q }H ~ i K ~ ^ ~ H ey O_. JH ~ } OY _ H '0 JQ. JH ~ } OY Q }H ~ ' JQ. JH ~ } OY Q }H ~, _ H i f " q Q ^ ~ H ey O_. a H _ H q - a J^ q^ OQ a H _ q JQ. a H Q }H ~ q a JQ. ~O_ JH ~ } OY ^ J OM H " JH ~ } OY H. H U_ x : a, " OY i m, n ~ OH - a Q } G (i) a (ii) a m m n a a m+ n n ' a a m n Y -, WH _ a 0 (iii) a 0, WH _ a Y 0 (iv) a m (v) (vi) a (vii) - a ^a m h n a mn m a m m m, WH _ a Y 0 m ^ah m a m j WH _! 0 H OY O_ ^ ~ Q OY H L* ~ O q Z C_ W _ OY H O> ^ kq H ^ Q ~ Q O_.

H. ik. J~ $ i $. O H z. ik H K @ J k XH g}. P" /J _ x rq U^ XH ^ xx H K ^ ~. u XH ~ ^ Ok rq ^ x< XH ^ xx H O\. L^ ~ } H ez @. (i) q O_ h~ _ @ H ^ ~ _., H \ H ~ ^ H @ @ H Q _ _. (ii) (iii) " O@ @ Q k < H i } (size) (iv) ~ x " (school uniform) H ~ H Q _ H. ''H J q ~ O^ ~ O_ Q OK K. _, H }, ", K @ H ~ O ~ ", i } Q i Q } H Y< ''H i Q } JO^ ~. 6

Q ~ K H O^ 7 ~ Q u < ~ H H O^ W _ x~ K Q ~ K H O^. (i) " XH O^ XH O$ @ (closed figure) P H q OK Q " JO^ ~. (ii) K @ H XH O$ @ H ^ H " K @ H JO^ ~. @ K \ H K @ ( H ) H " O _ H K @ K @ H JO^ ~.. H Ok PH ~ Q i OK Q "? H ' w O^ peri JQ K @ meter JQ H K @. Q } G @. PH ~ XH ''$ (iii) $ 'O J k $ H ^ ao^ r J k " ~ ( OA) $ H " A r r K. }. $ ik H K @ H P r 7 r }, r H.7 JQ. Q xh : XH $ HO ^ ^ H } 60 0 XH J@ > f H x qq^ " ~ $ H u ioz " \ H K @ H i " H Q O 60 o A @. O r @. @.. O. " ~ " ik... A 7

Q } G J^. J~ $ i $.. J~ $ J " ~ " 7 ~ A h ~ u PH K O ^ x Q xozu"? K O ^ x PH ~ Z@ O_? Wk @.6 PH ~ O_. nxx h U x? @.6 nx< J~ $ O^ ~. ($ J~ Q ) $ " K ~O_ Q Q K K Q J~ $ JO^ ~. XH $ O_ Z@ J~ $ h O^ Q? XH $ PH ~ J@ H " A B P^ ~ Q H u iok. (a) @.7 () (a) () Q xh : J~ $ H HO ^ H } 80. @.8 J~ $ K @ H K @ H, P ($ ik ) { r } J~ $ " ", A r r+ r P rr r ( r ) r rr A r + + } r K. }... $ $ O^ ~O_ " XH ^ xh H \ O Q O_ @ H u iok. W C_ < Q Q K. u Q $ JO^ ~. @. K @ $ H u ioz OCA, OAD, ODB, OBC J< < Q $ K. Q xh : $ O^ HO ^ H } 90. @. @.9 A @.0 C O D @. B 8 8

(a) $ H K @ H K @ H, P ($ ik ) + r } H () r r + r rr P + r r r + $ H " ", A ($ " ) A r r K. } j } L^ ~ }. @. XH J~ $ H " ~ O.g. J~ K @ H i " H Q. J~ $ H " ~ r O.g. J~ $ H K @ H, P ( r + )r } P ( + ) 7 ( + ) 6 7 7 7 J~ $ H K @ H 7 O.g. J~ $ H ", A rr K. } A 08 K. O.g 7 L^ ~ }. $ H " ~ O.g. J~ K @ H i " H Q. $ " ~, r O.g. $ H K @ H, P $ ", A r + r j } c + m + 7 j P 8 + j 0 7 O.g. r A 7 r K. } 6. K. O.g. @. @. @.6 Q } G 9

Q } G J^ L^ ~ }. XH J~ $ H ~ z H H " g. P K @ H OK " @ XH g. ~.0 O JQ Y~ H Q. @.7 " : d g " ~, r 7 g. J~ $ H ~ K @ H OK " @ $ ik x H Q. J~ $ H K @ H, P ^r + h r } P + 7 7 j + 7 7 j 6 g. XH g. H OK " @ JQ Y~ ~. 0 6 g " @ JQ Y~ 6 0 ~.60 L^ ~ }. XH J~ $ H ~ @ 6g. fq H OK " _ k. @ H " H Q. H OK H _ J~ $ H K @ H ^r + h r 6 g. + r 7 j 6 g. + r 7 6 r 7 r 7 j 6 g. ( g. @ H " J~ $ H " r K. } A r 7 @ H ". 77 g. 7 7 77 g @.8 XH W H g @ K q^ Q u * H ~ OK _ k. J^ H g x K ~ H ~ Q i K ~ [ _ H Q. 60

H J.. i Z H (i) J~ $ H " XH $ "... ~@ O_. (A) ~O_ (B) < Q (C) XH J~ Q (D) XH Q (ii) J~ $ H K @ H... (A) r + j r } (B) ^r + h r } (C) r } (D) ^r + hr } (iii) XH $ H " ~ 7 O.g. J~ J~ $ "... (A) 77 g (B) g (C) 88 g (D) g (iv) $ H " K. O.g. J~ $ H "... (A) K. O.g. (B) K. O.g. (C) 7 K. O.g. (D) 6 K. O.g. (v) $ H " 8 O.g. J~ K @ H... (A) 0 O.g. (B) 0 O.g. (C) O.g. (D) O.g. (vi) XH $... $ H (A) (B) (C) (D) (vii) $... O LO_. (A) J~ Q (B) < yo@ XH O (C) _ O@ XH O (D) _ ~O_ O (viii) J~ $ O^ HO ^ H } (A) 90 (B) 70 (C) 80 (D) 60 (ix) $ O^ HO ^ H } (A) 90 (B) 80 (C) 70 (D) 0 (x) J~ $ H " 8 K. O.g. J~ $ H "... (A) K. O.g. (B) K. O.g. (C) 68 K. O.g. (D) 88 K. O.g.. H Ok " ~ J~ $ ", K @ H H Q. (i) O.g. (ii) 0. O.g. (iii) 6. g. (iv).9 g.. H Ok " J~ $ ", K @ H H Q. (i).8 O.g. (ii) 6 O.g. (iii) 8 O.g. (iv) g.. H Ok " ~ $ H K @ H, " H Q. (i) 98 O.g. (ii) 70 O.g. (iii) g. (iv)8 g.. @ O_ J~ $ ACB i $ BOC H " H Q.. 6. O.g. " ~ XH J~ $ H ~ @ H ^. ^ x K @ H OK " @ XH g @~ ~. K C ZO Y~ JQ? Q } G 6

J^. O H z Q } G @.9 PH ~ O_ h Uq Q xozuq? @.9 (a), J~ $ u [ J i k. @.9 (), K ~ O O ~ K ~ [ (trapezium) J i k. ~O_ H _ P Q K i H z U~ _ " \ < O H z JO^ ~. e u [, n~ K ~ J~ $ " ^ Q " \ U~ _ O H z JO\ Oz(Juxtaposition) OK _ q. H x L^ ~ } K ^? ~ OY PH ~ JO\ OK @... (a) () (c) (d) (e) ~O_ q+ u [ ~O_ O H } u [ i XH n~ K ~ P~ u [ K ~ [ (Quadrilateral) O O ~ K ~ [ (Trapezium) H + _ l (Hexagon) (a) l n ~M YO_ K U~ _ XH O$ @ l JO^ ~. ~ ~M YO_ K U~ _ @ ~ ~v z JO^ ~. _ [ Q ~ ~v z u [ Jx, i < Q [ Q ~ ~v z K ~ [ Jx JO^ ~ H ey z P Q Oz " \ H _ ~ OH H H " \ H Q O_. nxx Juxtaposition JO^ ~. D A A A D E F B E C C C B F B D @..0 l JQ _ H JO H O> Z [ Q ~ ~v z Jx J~ 6

H () H l (Regular polygon) l Jx [ i H } " J\ lx H l JO^ ~, L^ ~ } (i) XH [ u [, _ [ Q H l JQ. @. (ii) < Q [ Q H l XH K ~ Q.. Q } G (c) J H l (Irregular polygon) H ~M ~ O x lx J H l JO^ ~. @. (d) \ H ~ l (Concave polygon) H h XH H } 80 H O> Z Q Q lx \ H ~ l JO^ ~. (e) O H ~ l (Convex polygon) @. u JO ~ H } 80 H O> Q Q lx O H ~ l JO^ ~. l W q_ n K. [ OY 6 7 8 9 0 l ~ u [ K ~ l OK l + _ l l J+ l l ^ l @. q[ g _ Q fq H OK ". H Ok U PH ~ LO_ Z " P H q OK. Z) $ a) K ~ ) g 0g Q n~ K ~ _ ) 7g g Q n~ K ~ K ~ O H z x J H A Q LO_. " \ x e @ Q q lo. " \ H " i K @ H O 7 ~ Q u < ~ H H ~ H Q K. Wq H Ok \ H ~ _ q. 6

J^. O ~ " (A) (K. } ) K @ H (P) ( } ) Q } G. u [ h AB + BC + CA. O H } u [ ( q +Z +H ~ ) h... u [ (Equilateral triangle) k u [ (Isosceles triangle) q+ u [ (Scalene triangle) B B A A C C a ( -.7) AB+BC+CA a ; Z, h } a - h a + a - h ss ( a)( s )( s c ) s a + + c a AB + BC + CA ^a+ + ch 6. 7. K ~ [ (Quadrilateral) O ~ K ~ [ (Parallelogram) A D h d h C B d ( h+ h) AB + BC + CD + DA h (a + ) 8. n~ K ~ (Rectangle) l (l + ) 9. O O ~ K ~ [ (Trapezium) h (a+) AB + BC + CD + DA 0. ~ O (Rhomus) dd, d d. H ~. a. K ~ (Square) a a 6

H H Ok h z PH ~ xh q loz " \ x Q i K i OK. L^ ~ }. H Ok O H z H K @ H H Q. (i) E @. (ii) Q } G A D B C @.6 @.7 (i) @ O^ ABCD XH K ~ DEA XH J~! $ Q k. DEA K $ ik Q " $ " AD JQ. K ~ [ 7 O.g. J~ $ O H " 7 O.g. J~ $ H " ~, r 7 O.g. K ~ K @ H AB BC CD DEA + + +! P 7 + 7 + 7 + ( $ ik ) + r r + 7 7 P + O.g. @ H K @ H O.g. " O " J~ Q " + K ~ " A r r + a 7 7 7 + 7 77 + 9 O H z " O " 9. + 9 68. K. O.g. A B E D C 6

J^ Q } G (ii) ~ _ O H z ABCD K ~ i DEA u [. K ~ H [ O.g. O H z H K @ H AB + BC + CD + DE + EA + + + + 0 O.g. O H z H K @ H 0 O.g.. O H z H " K ~ " + u [ " a + a ^. 7h + 6 +.7 O H z " O " 6 + 6.98.98 O H z " -. 9 O.g. L^ ~ }.6 H O^ K " Q H K @ H " H Q. (i) (ii) @.8 @.9 (i) Wz @ XH n~ K ~, " " Q ~O_ J~ $ H ey k. n~ K ~ _, l O.g. n~ K ~ " _, O.g. J~ $ " O.g. J~ $ " ~, r O.g. O H z H K @ H AD+BC+ AEB 66!! + DFC + + ($ K @ H ) 8 + rr 8 + 7 8+. 8 + 6. 8

H (ii) z H K @ H.8 O.g.. z H " ABCD n~ K ~ " + J~ $ " l + rr + 7 " O " 8 +.. O.g. ADB, BEC i CFA J k _ " ~ ~ J~ $ ~ Q I, II,III Q J H. J~ $ I H " ~ r 0 O.g. J~ $ II H " ~ r 8 O.g. J~ $ III H " ~ r 6 O.g. III I II Q } G K " Q K @ H J~ $ I K @ H + J~ $ II K @ H + J~ $ III K @ H + ^r+ h + ^r+ h + ^r + h ^r + h^+ + h ^r + h + 7 j 6 6.7 7 K K Q K @ H - 6.7 O.g.. K K Q ", A J~ $ I " + J~ $ II " + J~ $ III " K " Q " - 78.7 O.g. L^ ~ } O_ Q xok Q, A r r r r + r + r 7 + 7 + 7 A 7 + 76 + 99 0 78. 7 O.g 7 7 7 7 BEC J~ $ " + CFA J~ $ " ADB J~ $ " 67

Q } G J^ L^ ~ }.7 70 g. _ g. " _ Q XH n~ K ~ H ~ XH Q ] XH 8g. _ Q ^ ~ F z H " @ H \ " _ k. Q ] " " O? ke" _ " O? E @.0 n~ K ~ _, l 70 g. n~ K ~ " _, g. ^ ~ H _ 8 g. K " Q AEF Q ] z H " K @ e *. nxx \ Jk XH $ x + OQ H K. $ " ~, r 8 g. AEF $ " r r K. } 8 8 66 g 7 z H " " 66 g. q ye ABCD n~ K ~ " AEF $ " n~ K ~ ABCD " l K. } 70 60 g ke " _ " 60 66 0 g. L^ ~ }.8 H O^ W _ @ ABCD O.g. [ H ey XH K ~. WO^ K " Q " H Q.. K ~ [, a O.g. @. $ " ~, r 7 O.g. K " Q " K ~ " - $ " a - ( r r ) 7 96 K " Q " K. O.g. 7 7 7 O.g 7 O.g 7/ O.g 7/ O.g @. 68

H L^ ~ }.9 O.g. " ~ Q XH ~ y H q $ H ~ K @ _ k. ^ xx XH K ~ Q ez [ H Q. $ " ~, r O.g. J^ ~ y H q K ~ Q K _ k. @. $ H K @ H K ~ H K @ H $ K @ H r r } O.g. 7 P 0 O.g. K ~ [ a J~ K ~ K @ H a } a 0 a O.g. K ~ [ O.g.. @. Q } G L^ ~ }.0 XH K ~ H < Q wz ABCD HO ^ Q Q < Q $ Q $tok K q. K " Q " H Q. K ~ [ H 8 O.g. JQ. H O^ ~ _ ABCD K ~ [ a 8 O.g. $ " ~, r 8 O.g. K " Q " K ~ " - $ " K " Q " 68 O.g. a - rr 8 8-7 78 66 @. 69

Q } G J^ L^ ~ }. XH H _ " ^ (athletic track) H " _ g _ 0 g H ^. ^ xh W~ " g " ~ Q ~O_ J~ $ H ~ L q. " ^ H " H Q. JO ~ $ H " ~, r g. @.6 " ^ " _ " e $ " ~, R g. + 9 g. R 9 g. " " ^ H " ~O_ J~ $ " ^ i n~ K ~ " ^ " JQ. ABCD i EFGH n~ K ~ " (l ) 0 60 g. J~ $ " ^ " (" e $ " - e $ " ) rr - rr j r^r - r h ^9 - h 7 ^9 + h^9 -h 7 8 696 g 7 " ^ H " 60 + 696 706 g. L^ ~ }. @.7 O_ PQSR J k XH ~ t (flower ed). OP g., OR O.g. J~ K " Q " H Q.. OP g. i OR g. ~ t " OQP $ " - OSR $ " r OP - r OR @.7 70

H r - r r ^ - h ^ + h ^-h 7 7 9. g. 7. ~ t " L^ ~ }. @.8 O_ K " Q " H Q. ABCD K ~ [ H 7 O.g JQ. K K x Q I,II,III,IV Jx @.9 Q i OK. P,Q,R i S J q ~ Q AB, BC,CD i DA ^ ao^. K ~ [, a 7 O.g J~ $ " ~, r 7 O.g I H " +III H " ABCD K ~ " P R HO ^ Q H ey ~O_ J~ $ " I " + III" 9 W^ q^ Q a - rr 7 7-7 7 7 77 - cm cm j. II H " + IV H " 9 - cm cm j. K K Q " ABCD K ~ " (I " + II " +III " + IV " ) 9 + j 9 8 O.g K K Q " 8 O.g L^ ~ }. XH ~ ~ H Ok q^ Q XH H ek. " H Q. J, K, L, M J q A O_ D ^ w ~ ~ Q ~. 77 @.8 @.9 Q } G @.0 7

J^ Q } G ~ _ q :AJ g., JF 7 g., KB 6 g., LE 9 g., MC 0 g., AK 0 g., AL g., AM g. i AD 0 g.. W _ > l " KBCM, LEFJ i ABK, MCD, DEL JFA J O H } u [ H ~ H J i k. > l " KBCM. H " A A ( KB + MC) KM ( 6+ 0) g. A 6 0 LEFJ. > l " " A A ( ) O H } u [ ABK " A A O H } u [ MCD A H " JF + LE JL ( 7+ 9) 7 g. A 6 7 6 AK KB g. A 0 6 0 A MC MD. 0 0 g. A O H } u [ DEL A H " A DL LE ^AD - ALh LE ^ 0 - h 9 g. A 8 9 6 E F (a KB, MC O ~ [ Z KM KB 6 g, MC 0 g, KM AM AK B 0 g. ) (a LE, JF O ~ [ JL Z JF 7 g., LE 9 g., JL AL AJ 7 g. ) C 7

O H } u [ JFA A 6 H " A 6 AJ JF 7 7. + + + + + g. H " O " A A A A A A6 0 6 0 6 7. + + + + + " O " 0. g. J.. H Ok ~ _ @ K @ H H Q. H Q } G (i) (ii) (iii) (iv) (v). H Ok ~ _ @ H " H Q. (i) (ii) (iii) (iv) (v) 7

J^. H O^ ~ _ @ K K Q " H Q Q } G (i) (ii) (iii) (iv) (v) (vi). W _ @ AC O.g., BC 0 O.g., i 'O ^ $ H HO ^ ~ K K " Q " H Q.. 0 g 6 g H Q n~ K ~ H ~ XH O^ g. _ Q ^ ~ XH P H \ K P " x Q " H Q. 6. H @ 00 g H Q XH K ~ H ~ XH H O^ g. " ~ Q $ H ~ @ K q ye Q " H Q. 7. H @ < Q $ H ~ L q. ^ O.g. " XH $ H ^. K " Q " H Q. 8. AB 0 O.g., BC O.g. H Q XH n~ K ~ H ~ H y O_ BC H " Q Q XH J~ $ H u iok _, q ye Q " H Q.. 7

H 9. XH K ~ H ~ K u ~ O_ 7 O.g. " ~ q k $ H ~ Q H u iok _ K q ye Q " H Q. 0. XH H H ~ ~ H O^ H Ok q^ Q W _ q. P H @ wz " H Q. (i) (ii) Q } G p K Q "? qq^ PH ~ Q P ~ ^ ~ < g ^ K K ^ ~ Q " _ q. XH p P P ~ ^ ~ K \ K @ ^ xh, ^ xh Z @? nx Zx u [ H? K ~ K @ H ( ^ ) ^ x J i $ K @ H ^ x H z k? 7

J^ Q } G D @ Uk K @ H H ey L k? $ H HO ^ H } 60. J~ $ K @ H ^r + h r } rr J~ $ " K ^ ~ } J~ $ HO ^ H } 80. $ K @ H r + r rr j }. $ " K ^ ~ }. $ HO ^ H } 90. O H @ K @ H ^ x Z H _ " JQ. n ~M YO_ K U~ _ XH O$ @ l JO^ ~. Jx [ i H } " lx H l JO^ ~. H ~M ~ O x lx J H l JO^ ~. 76

~M Q }. ik. u [ H }. u [ ~. ik < k ~ ^ K ki ~ q Q ~ Q i OK @ H.. 000 OII O^ Dl " ~ ~M Q } Ja $k K ~. J~ w nxx ~ ^ u J ~ " x~ } ^ O Q H K Q O Q OK ~. ~M Q } rq J< H q^ Q Y + k. H $u [ ~ < >, Q H ~ O, n~ K ~ h\ >, H ~ " ^ Q " \ x Q x <. ~M Q } Q H OQ L yok H @ ~ q _ x~ } XH J^ L^ ~ }. ~M Q } qq^ ~ OQ JQ uh, ~, WO[hiOQ, x~ } ~ OQ i < ~ i " ^ Q " \ H Q L Q _ K k. ~M Q } J ^ ~O_ w ^ 'l ' JQ q i '" \ JQ ''H K @ O_ U~ _ k. Q } XH Q " '' ~M Q } PH ~, H, i H } Q ioz K i OK. 7 ~ Q u O ~ ~Y, u~ QY, YO_ ~Y H }, P H }, UH O ~ H } Q ioz H <. JO H O_ u [ H } " J H } _ < ~ H x L<. H _, ~M Q } ~ H.. 00 OII ~M Q } ~ su P zozk ^ ux ~ OkOz '' H _ XH Q Q } ". ~ H.. 00 OII H H _ ~M Q } ZO K ~ H ioz, ^ xx H ^ u H Q K iok. D H '' H _ Ze" O\ Jx JO\ ~. ''XH H i Q ^ x x U Q H O> _ ^ kq LO_ Jx H _ e. 77

Q } G J^ H Ok J Q O < ~ H " \ e Q ~ K O^. i ~ J. @. x 8 J~. @. + ACD 90c J~ y H Q. + BCE i + ECD H Q. @.. XH u [ x ~O_ H } i 7 J~ _ H } H Q.. @. PQ RS J~. @. AB i CD ~Y 'O ^ x H H. YO_ OK H. x i y H Q. 6. @. AB CD J~ H Ok M m iok. (i) (ii) (iii) A P R x o O y o + EFB i + FGD... H }. + AFG i + FGD... H }. + AFE i + FGC... H }.. u [ H } (Properties of triangles) C @. N M x + x + @. B Q S XH O^ _ ~M YO_ K _ @ ' u [ ' JO^ ~. u [ J O H Q i OK ^ ~. ABC J U^ < u [ O^ A,B,C j~ Z^ ~ Q [ ~ Q a,, c Q i OK ^ ~. A A C x. 7 0 C D O y x x +0 D B @. A C G H B F E E @. @.6 D B 78

~M Q }... u [ ~ H u [ " \ [, H } P^ ~ Q ~O_ ~ H Q q lok ^ ~. [ P^ ~ OQ u [ ~ H : (a) [ u [ () k [ u [ (c) q+ [ u [ Q } G H } P^ ~ OQ u [ ~ H : (d) J H } u [ (e) O H } u [ (f) Jk H H } u [... u [ H } " H ^ ~ ^ O XH u [ x _ H } " 80. A > ^ O : ABC XH u [. o ~ O : + ABC + + BCA + + CAB 80 B @.7 x~ } : A j~ ^ ~ BC O ~ Q XY w. x~ } : L u H ~ } (i) BC < XY i AB XH u~ QY + ABC + XAB UH O ~ H } (ii) i + BCA + YAC UH O ~ H } (iii) + ABC + + BCA + XAB+ + YAC (i), (ii) _ (iv) ^+ ABC + + BCAh + + CAB ^+ XAB + + YACh + + CAB + BAC ~O_ " _ (v) + ABC + + BCA + + CAB 80 ~ ~Y H } 80. X > Y C 79

J^ Q } G (i) (ii) (iii) _ [ Q l< u [ JO\ ~. U l< < H ~ H @ ^ ~ u [ Q q lok K. XH l x JO ~ H } " (n ) 80 Jx fh iok K. WH _ n J k [ OY. @ [ OY s H ~ } u [ K ~ [ OK l H } " ^ O XH u [ O^ XH [ _ yoz _ U~ _ H } ^ xh Ja Y Q ~O_ JO ~ H } ". ^ O : ABC XH u [. BC, D ~ _ yok _ k. ~ O : + ACD + ABC + + CAB x~ } : L u (i) T ABC, + ABC + + BCA + + CAB80 0 (ii) + BCA + + ACD 80 0 (iii) + ABC + + BCA + + CAB + BCA + + ACD (iv) + ABC+ + CAB + ACD (v) H } + ACD J k JO ~ H } + ABC, + CAB ". B @.8 H ~ } u [ H } " XH ~ ~Y L P H } " (i), (ii) K Q (iii) O_ + BCA ~O_ " f " Q x~ OK _ k. A C D 80

~M Q } (i) XH u [ [ Z^ ~ Q H }. (ii) Ju _ " [ Z^ ~ Q Q H } Ju ^ H }. L^ ~ }. L^ ~ }. D ABC + A 7 0, + B 6 0 J~ + C H Q. D ABC + A+ + B+ + C 80 Jx o o 7 + 6 + + C 80 o 0 + + C 80 + C 80 0 + C 0. D ABC, u [ + A 70 o i AB AC. D ABC x q ye H } H Q. + B x i + C y J H. ^ O O O_, D ABC XH k [ u [ AC AB + B + C [ [ Z^ ~ Q H }.] x o y o D ABC + A+ + B+ + C 80 70 o + x o + y o 80 70 o + x o + x o 80 ( x 0 y 0 ) x 80 70 x 0 x 0 o. H + B i + C. L^ ~ }. XH u [ x H } H : : x+ u L q. P u [ x H } H Q. ^ O O O_ D ABC, + A: + B: + C : :. u [ Wz H } x, x i x J H. @.9 @.0 Q } G 8

J^ Q } G u [ x H } " 80 Jx. x + x + x x 80 x 80 0 H, u [ x H } 7,60 i. L^ ~ }. Wz @. x ABC u [ H } H Q. BD XH ~ ~Y ~ ~Y H } 80 Jx. x + 0 80 x 80 0 x 70 u [ x H }, ^ x ~O_ JO ~ a Y H } " Jx. x + y 0 70 + y 0 y 0 70 0 H, x 70 i y 0. L^ ~ }. Wz @. + DEC q H Q U u [ < <, H }, ^ xh Ja Y Q JO ~ H } " Jx. D ABC, + ACD + ABC+ + CAB 0 0 + ACD 70 + 0 0 0. @. @. J^ q^ OQ, + ACD + ECD 0 0. D ECD, O^ + ECD+ + CDE + + DEC 80 0 [ u [ x H } " ] 0 0 0 + + + DEC 80 0 + DEC 80 - + DEC 8 0 0 0 8

~M Q } T, T, T, T, T i T J u [ xi OK. " \ x ABC 6 u [ Q ~. A, B, C j~ Z^ ~ Q Q [ ~ Q a,, c Jx Q i OK. [ H ez, " \ q~ H Ok \ H O^ ~ K. T. O. T T T a ( O.g.) ( O.g.) c ( O.g.) (c+a) > X C/ C (a + ) > c X C/ C ( + c) > a X C/ C Q } G T T T 6 \ H O_ h Uq Q xok? ^ O XH u [ x ~O_ [ ", _ [ H O> Z Q O_ (nxx u [ J g H ~ } JO^ ~. (Triangle Inequality)) ik K @ ABC u [ O^ BC O.g., AB 8 O.g., AC 9 O.g.. (i) AB 8 O.g., AB + BC 0 O.g. (ii) BC O.g., BC + CA O.g. (iii) CA 9 O.g., CA + AB 7 O.g. W C_ + Q, (i) AB + BC > CA (ii) BC + CA > AB (iii) CA + AB > BC Jx O^ ~, u [ x U ~O_ [ " ", _ [ H O> Z Q LO_ x H Q <. L^ ~ }.6 H Ok " x u [ U~ ~ K [ Uq? (i) O.g., 7 O.g., 8 O.g. (ii) O.g., 0 O.g., O.g. (iii) 9 O.g., 7 O.g., 6 O.g. (i) (ii) Wz H O.g., 7 O.g., 8 O.g. WH _ + 7 > 8, 7 + 8 > i + 8 > 7. O.g., 7 O.g., 8 O.g. H XH u [ U~ ~ K. Wz H O.g., 0 O.g., O.g. O.g, O.g i O. g H Q u [ xi OK. J^ q^ Q H Ok H u [ xi OK. Z) O.g, 7 O.g, O.g a) O.g, 7 O.g, O.g ) O.g, 7 O.g O.g g ~ H Q q+. 8

J^ Q } G (iii) WH _ + 0 J k H O> ^ k H ^. ie, [ + 0 $ ] O.g.,0 O.g., O.g. [ u [ U~ ~ K. Wz H 9 O.g., 7 O.g., 6 O.g. 9 + 7 J k 6 H O> ^ k H ^. ie, [9+ 7 6, 9+ 7 $ 6] 9 O.g., 7 O.g. i 6 O.g. [ u [ U~ K. (i) c+ a ( c+ a ( - c a (ii) + c a ( a + c ( a- c (iii) a+ c ( c a + ( c- a e Q xoz U u [ < ~O_ [ _ [ H O> Q LO_.. ~ [" H Q : J. (i) H Ok " x u [H } Q L q Uq? (A),, 90 (B) 6, 8, 96 (C) 8, 6, 96 (D) 0,, 90 (ii) H Ok " x Uk? (A) [ u [ H } H ey LO_. (B) k [ u [ H } H ey LO_. (C) H } H ey L u [ [ u [ H ^ (D) q+ [ u [ H } H ey LO_. (iii) XH u [ x ~O_ H } 0, 0, x J~ x... (A) 90 (B) 00 (C) 0 (D) 0 (iv) H Ok H Uq u [ U~ ~ K? (A) O.g., O.g., 6 O.g. (B) O.g., O.g., O.g. (C) 8 O.g., O.g., O.g. (D) O.g., 6 O.g., O.g. (v) Wz ~O_ H } Q xoz, Uk O H } u [ U~ ~ K < H Q (A),66 (B) 6,6 (C) 6,8 (D) 68,. H Ok " xx k OK. XH u [ x H } (x ),(x 0) i (x + 0) J~ _ H } H Q.. D ABC O^, + A H B + C 08 J~ ABC + H H O> c Z Q L k. H } D H } H Q. 8

~M Q }. D ABC O^ + B i + C H } k YO_ ~Y O ^ YO_ OK H. + BOC 90c + + A Jx x~ OK. H Ok @ x i y q H Q. Q } G (i) (ii) (iii) 6. @ O^ x, y i z H } H Q.. u [ ~ ~M Q } Y " '' ~ J< q+ Q ioz H K <. ~ Q ioz H @ H Ok H $ K ^. k~ < @ ' f H. " \ x XH ^ x XH \ LOK. h Uq Q xok? XH < @, ~ H ^ xx i Q H " K k. H $ O_ ~O_ z X H PH ~ i X H H H ey L x Q xok. ^ ~ }OQ, ~O_ ~M Q } z X H PH ~, X H H H ey LO> " \ x '' ~ JO\ ~. H Ok ~? H ^? Jx ijeok : (a) X H q H ey a. () X H H @ x a @. (c) X H O_ x +qoq _. 8

J^ W _ H Ok x z Q xok ^. F U A E P T Q } G B C D @. @. ~O_ @ Q xok. Jq ~? Z@ ik _ Q? XH @ ~ H @ LOK ^ u (Superposition) x L yok ^.. " @ : @. > ~ f H. H ~ < ~ _ L yok K. " @ : P @. OH ~ O_, Q f O_ LOK. " @ :. @. @ i Q H " k. H ~O_ @ ~. Q R S ~ : XH x ~O_ @, XH ^ x XH \ LOz _ Jk i Q H _ K " \ x ~ JO\ ~. nxx / Q ~ K zok ^... (a) ~M YO_ ~ ~O_ ~M YO_ X H _ H ey L K " \ x '' ~ JO\ ~. O.g B C O.g D () A WH _, AB _ CD _. H AB / CD H } ~ ~O_ H } X H H H ey L K " \ x ~ H } JO\ ~. M P 0 o Q O 0 o N WH _ H } H. H + MON / + PQR. R 86

~M Q } (c) K ~ ~ [ H Q L ~O_ K ~ ~. WH _, ABCD K ~ [ PQRS K ~ [ _, ABCD K ~ / PQRS K ~ (d) $ ~ X H " ~ H ey ~O_ $ ~. Q } G Wz @, C $ " ~ C $ " ~ $ C / $ C H @ J@ H wz K H P^ ~ OQ ~O_ H Q K. g\ O_ h" q J~ K H O\ q. < Q ~ H ^ u [ ~ Q ioz H " x Ok H ^? H Ok ~O_ u [ Q xok. TABC x TPQR JQ A xp, B x Q i C x R LO_ @ XH ^ x XH \ LOzK, " \ j~, [, H } XH ^ xx XH \ Yz Q H ". O Ok Q W [ ~ K K. J ~ j~ J ~ [ J ~ H } A * P AB PQ + A + P B * Q BC QR + B + Q C * R CA RP + C + R 87

Q } G J^... u [ ~ XH u [ x _ [, _ H } ~ Q, ~ H u [ x _ [ _ H } Q L K " \ x ~ u [ JO^ ~. Q xh : ~O_ u [ ^ ~ x, " \ j~ Y H Q ". TABC / TPQR, J~, ~ TBAC / TQPR, TCBA / TRQP W qq^ ~ H Q " K. g\ x u (Anti clock wise) _ " K..... u [ ~ x ~O_ u [ ~ ", " \ 6 J ~ Q [ ( [ [, H } [ ) Jx. J~ H x O^ ~ ~O_ u [ ~? H ^?Jx H @ P u [ _ Q ije K. g\ < K Q W _ O [iyk. J@ O\ ^ ~ " < Q K qq^ K O: x~ } J ~ Q, ~ [ ~ K _ L q. D K H ~ Jx u [ ~ Q i OK @ L Q _. JOwH iok @ g Q H x ^ ~ } ^ ~. S J k [, A J k H }, R J k O H }, H J k H ~ zoz, K, H Ok q^ Q " K. (i)... (SSS) K (ii).h.. (SAS) K (iii) H..H. (ASA) K (iv) O.H.. (RHS) K (i). K ( [ [ [ K ) XH u [ x _ [ ~ Q, ~ H u [ x _ [ " " \ x ~ u [ JO^ ~. A P B C Q R 88

~M Q } ABC i PQR ~O_ u [, AB PQ, BC QR i CA RP. T ABC x > ~ Q i Oz, ^ xxt PQR LOz, JQ AB PQ, BC QR i AC PR LO_ @ H LOK. AB PQ & P A LO_ Q B LO_. J^ q^ OQ BC QR & R C LO_. W _, ~O_ u [ XH ^ xx XH \ i Q H L q. DABC / DPQR WH _ AB PQ,BC QR,CA RP Jx Q xok. nx< AB PQ BC CA Jx " K. QR RP x+ u ' H H x _ U Q? L^ ~ }.7 H Ok @, Wz u [... K O ^ ~ ~ JQ <?. Q } G T PQR i T XYZ [. PQ XY O.g., QR YZ. O.g. i RP ZX O.g.. T PQR T XYZ LOK. P X Q Y R Z LOz T PQR DPQR / D XYZ (... ^ ~ )., T XYZ i Q P H q OK. L^ ~ }.8 H @O PQSR XH O ~ K ~ [.PQ. O.g. PR. O.g. J~ TPQR TPSR JQ <? TPQR i TPSR ~O_ u [, PQ SR. O.g i, PR QS. O.g. PR PR (L _ [ ) DPQR / D RSP (...) DPQR _ D PSR (RSP i PSR " ~ H L q) 89

Q } G J^ (ii) H K XH u [ x U" x ~O_ [ i " \ ^ H } i H u [ x U" ~O_ [ i " \ ^ H } ", P u [ ~. A B D ABC i D PQR J ~O_ u [ f H O^. " \ AB PQ, AC PR i " \ ^ H } BAC " \ ^ H }O QPR. D ABC x > ~ O^ wz, D PQR AB PQ AC PR LO_ @ O^ ~ K. B j~ Q j~,c j~ R j~, BC [ QR [ iq i. D ABC, D PQR Yz Q H " _ k. H DABC / DPQR (iii) H H K O (H } [ H } ) XH u [ H ~O_ H } i XH [, ~ H u [ O^ e ~O_ H } i J ~ [ " P ~O_ u [ ~. A C Q P P R B D ABC i PQR WH _ BC QR, + B + Q, + C + R D J u [ f H. XH ^ x XH \ O^ ~ K ^ u ABC O^ ~ K _. W C_ B j~ Q j~, C j~ R j~ LO_. H A j~, P j~ LO_. D ABC, D PQR iq i ~ k. H DABC / D PQR. u [ ~, H, q ye Q _. (i.e.) AB PQ, AC PR i + A + P C Q +, + PQR + BCA, QRP R + iq H u ioz H y H ^ ~ H Ok K x~ OK. Z) K. a) H H K. 90

~M Q } zok @: ~ u [ x J ~ [ ~. nx< Q c.p.c.t.c. Jx ". WH O^ D O H L y. L^ ~ }.9 AB i CD 'O ^ k YO_ K H K q. AC BD Jx x~ OK. ^ O : AB, CD ^ ao^ 'O. AO OB i CO OD. BD ~ O : AC C x~ } : TAOC i TBOD ~O_ u [ J H. AO OB (^ O ) CO OD (^ O ) + AOC + BOD (j~ a Y H } ) T AOC / TBOD ( H } ) AC BD (c.p.c.t.c.) L^ ~ }.0 H @ D DAB i D CAB X H P^ ~ AB L q. DDAB / D CAB Jx x~ OK. TDAB i TCAB u [ f H. + DAB c+ 0c c + CBA (W _ k) + DBA + CAB 0c (W _ k) AB ~O_ u [ L _ [. DDBA / D CAB (H..H } ) H ~ H ~ JQ J~ Uq \ h? 'H ~ J k O H } u [ O Ok Oz XH ^. A C B A O @. B D B Q } G H ~ H ~ B C A ABC O H } u [ f H. + B XH O H }. O H } Z^ ~ Q [ H ~ JO^ ~. WH _ AC XH H ~. 9 C H ~ A

Q } G J^ (iv) O H } H ( O H } H ~ [ ) XH O H } u [ x H ~, XH [, ~ H O H } u [ x H ~ i [ ", P ~O_ u [ ~ JO^ ~. A D D ABC, DEF D ~O_ u [ J H x, WH _ + B + E 90 o H ~ AC H ~ DF (^ O O) [ AB [ DE (^ O O) XH ^ x XH \ O^ ~ K ^ u DABC / D DEF ~ Jx Q xok Q... ~ u [ Jx K C@ K x (not sufficient) x. (i) B H H H (H } H } H } ) C ~O_ u [ ~ Jx K C@ D } K ^. ZO^? H Ok u [ O_ H ~ } H Q O^. E F @, + A + P, + B + Q i + C + R J~ D ABC, D PQR H O> z kq L k. D ABC x PQR D LOz Jk i Q H _ ^. ABC _ PQR 9 D D. (ii) H ( [ [ H } ) XH q+ W qi. D ABC + B 0, AB.7 O.g. i AC O.g. H LO_ @ u [ xi OK. BC x X ~ _ yok. A HO ^ Q AC" ~ O.g. XH K w. Wk BX, C i D ^ YO_ OK.

~M Q } AD _ O.g.[aAC i AD X H $ " ~ ], D ABD J ~O_ u [ O_ + B L _ H }. AB L _ [ i ACAD O.g. (x~ } ) DABC AC [, AB [ D ABC i + B, DABD AD [, AB [, B BC, BD H ^. D ABC _ D ABD. + ~ Q ~. J~ L^ ~ }. XH u [ x [ Z^ ~ Q Q H } Jx x~ OK. ^ O : ABC u [ AB AC. ~ O : AB H Ja Y H } AC H Ja Y H } (i.e.) + C + B. x~ } : BCH AD J< O w. + ADB + ADC 90c x~ }: DABD i DACD ~O_ u [ O_, AD L _ [. H AB AC [ ABC XH k [ u [ ] + ADB + ADC 90c [ x~ } ] DADB / D ADC [ O H K ] + ABD + ACD [ c.p.c.t.c] ( H ) + ABC + ACB. + B + C. x~ OK _ k. nxx k [ u [ ^ O JO^ ~. L^ ~ }. XH u [ x H } Z^ ~ Q Q [ Jx x~ OK. ^ O : D ABC O^ + B + C. ~ O : AB AC. x~ } : BC H AD J< O w B A D C Q } G 9

Q } G J^ x~ }: + ADB + ADC 90 [x~ } ] + B + C [^ O ] AD L _ [ ADB / ADC JO^, AB AC. [ c.p.c.t.c] D D (H.H.. K ) H XH u [ H } Z^ ~ Q Q [. Wk k [ u [ ^ O q ~. L^ ~ }. Wz @ O^ AB AD i + BAC + DAC J~ K, " x J ~ Q q ye [. T ABC i T ADC, O^ AC L _ [ + BAC + DAC [^ O ] AB AD [^ O ] TABC q ye J ~ Q [. TABC / TADC? / TADC [.H. K ] BC DC, + ABC + ADC, + ACB + ACD. [y c.p.c.t.c] JQ <? L^ ~ }. D PQR XH k [ u [. PQ PR, QP S ~ _ yok _ k. i PT, H } Q U~ _ x H } k YO_ K K k. J~ + Q x o i PT < QR Jx x~ OK. ^ O : D PQR XH k [ u [ PQ PR. x~ } : H } + SPR PT k YO_ K K k. H + SPT + TPR xc. + Q + R. [ k [ u [ q ~ ] JO H O_, U u [ < < H } JO ~ a Y H } " Jx D PQR O^, H } + SPR + PQR + PRQ H xc + Q+ + R + Q+ + Q x o + Q x o + Q o + Q x. + A Q B 0 o 0 o B D P A D S x o x o C T C R 9

PT < QR Jx x~ OK @, PT, QR ~Y u~ QY SQK YO_ OK _ K k o + Q x Jx O^ x~ OK. H + SPT i PQR PT < QR. ~ [" zok. + J ~ H }. J. + SPTx. ~M Q } (i) D XYZ, J k [ u [ O^ XY YZ J~, H Ok " x U H }? (A) +X i +Y (C) +Z i +X (B) +Y i +Z (D) +X +Y i +Z (ii) D ABC i D DEF, + B + E,AB DE, BC EF. D ~O_ u [... K ^ ~ ~ JQ. (A).. (B) H.H.H (C).H. (D) H..H (iii) XH O^ Q ~O_ z ~ x K C@...H ey LO_. (A) X H H (B) X H ~ (C) X H ~ i X H H (D) X H H, " ~ ~ (iv) ABC u [ O^ + A 60 o, AB AC, J~ ABC XH ------ u [. (A) XH O H } (B) XH [ (C) XH k [ (D) XH q+ [ (v) ABC u [ O^ + A 90c J~ H ~... (vi) (A) AB (B)BC (C) CA (D) UnH ^. D PQR O^ PQ, PR ^ U~ _ H } (A) + P (B) + Q (C) R + (D) UnH ^. (vii) H @ O^ x q ---------- (A) 80 o (B) 00 o (C) 0 o (D) 00 o. @ ABC u [ O^. @ O^ x H Q. ABAC J~ x i y H Q. A A y 0 O 0 O C Q } G x O B x 0 x 0 C x+8 0 E B D 9

J^ Q } G. H Ok @ O^ D PQR i DSQR k [ u [ J~ x H Q. Q x O P 0 O S 70 O R. H Ok @ O^ BR PC + ACB + QRP <. i AB PQ J~ AC QR Jx x~ OK. P C Q A R 6. H Ok @ O^ AB BC CD, + A x o. J~ + DCF + A Jx x~ OK. F B 7. H Ok @ O^ AB BD, BCDC i DAC 0 o + J~ x, y, z H Q. D C x o z o x 0 A B D E 8. H Ok @ O^ ABCD XH O ~ K ~ [. AB, E ~ _ yok _ k H AB BE, AD, F ~ _ yok _ k. H ADDF. DFDC / D CBE Jx x~ OK. F 0 0 A B C 9. H Ok @ O^ D ABC x+ ABC H } BO k YO_ K K k. BO P ao^ H ^. P O_ BA i BC wz O Jx x~ OK. y o A D C D P O A B 0. PH O ZQ ~ K ~ " q L _ [ Q ~O_ u [ Q H x OK K q. SQ i SR ^ ao^ T, i SR RQ J~ SRT / QRT Jx x~ OK. E B E C 96

~M Q } Ä XH u [ x _ H } " 80. Ä XH u [ x [ _ yoz _ U~ _ H } ^ x ~O_ JO ~ a Y H } " Q. Ä u [ x U^ x ~O_ [ " _ [ H O> ^ kq LO_. Ä XH x ~O_ @, XH ^ x XH \ O^ ~ z _ iq i ~ K " \ x ~ @ (Congruent) J O \ ~. nxx / Q ~ K Q i ~. Ä XH u [ x _ [, _ H }, ~ H u [ x _ [, _ H } ", P u [ ~ u [ JO\ ~. Ä... K : XH u [ x _ [, ~ Q ~ H u [ x _ [ " P u [ ~ JO\ ~. Ä.H.. K : XH u [ x ~O_ [ " \ ^ H }, ~ Q ~ H u [ x ~O_ [ " \ ^ H } ", P u [ ~ JO^ ~. Ä H H K : XH u [ x ~O_ H } XH [ ~ Q, ~ H u [ x ~O_ H }, XH [ ", P u [ ~. Ä O H K : XH O H } u [ H ~, XH [ ~ Q ~ H O H } u [ H ~, XH [ " P u [ ~.. Q } G 97

J^ Q } G Q } O H $ ~ Y q^ ~ ~! ~ J< x rq K q+ L y <. WO\ L yok '' ~O_ ^ ~ ~O_ ~ Q LO_. H ~H XH ^ x H H \ ~ Q LO_. D q^ OQ K L^ ~ } K Q. PH Zy ~, u * H ~ O ~ Ok ZQ ~. O^ Q Zy ~ H ^ ~ XH ^ Q wzk h ~ K _ Q. P ~ H ~ K, P ~ z~ Q K ~ H ~, ZQ ~ *. W _, H $u, h x[ rq K _ Q ~ ~ Q i OK. 98 98

Q H ~M Q }. ik. ik. K ~ [. O K ~ [. O ~ K ~ [ ~ H Dl ~M Q } H Q H < } " i ^ H x~ }, i } H K @ L yoz < ~. " i < Q ih _ Q H ~M Q } L yoz < ~. J< H Q x~ } " ~ H ^ i $ dx xi Oz< ~. Q } H e Y XH \ ~M Q }. ~M Q }, ^ O ~M Q } i Q H ~M Q } Q s H io _. ~M Q } ^ O K K Q w _ @ H x~ } H ^ O ~M Q } qiok _. ~M Q } L H ~ } L yoz i @ wz " \ x~ } H Q H ~M Q } qiok _. H x ~M z x~ }, ^ ~ i " \ " H Q H Ok ~ Q < ~ H x <. D Q O^ H x H ~M Q } z ~M Q } ih ~ L yoz xi OK @ < ~ H < ^. Q [ H. 777-8] Q XH [~ < Q } ". Q k _ p l (p [ Q l) xi OK @ ijeok. WO^ p XH ^ } OH Q. J \ p i p ^ " k. p l xi OK _ x p XH ^ < OH '' ~ " \ OY (i.e.) p n + J^ q^ Q ^ x q ~ _ Jx H Q xk Q JQ. 99

Q } G J^. K ~ [.. ik K ~ [, K ~ [ H ^ ~ Q i 7 ~ Q u < ~ H x <. " \ x XH i Q ~ K H. @. A, B, C, D J ao^ X H J i q. U _ ao^ X H ~Y. AB, BC, CD, DA j~ ^ YO_ OK H. K ~ [ < Q [ Q @ Jx < ~ H x <. K ~ [ @. H < Q H } " 60 Jx e. < Q ~M YO_ K _ @ K ~ [ JO^. ^AB, ADh, ^AB,BCh, ^BC, CDh, ^CD, DAh J q P [. AC, BD H ~ JQ. K ~ [ ABCD EA, EB, EC i ED ( H EDAB, EABC, EBCD, ECDA) H } JQ. H } " EA + EB + EC + ED 60 Q xh : (i) K ~ [ K H Q ABCD H BCDA Jx ~ K. (ii) K ~, n~ K ~, K ~ [, O ~ K ~ [, O K ~ [ J q _ K ~ [ Q. (iii) XH K ~ [ O^ < Q j~, < Q [, < Q H } i ~O_ H ~ LO_... K ~ [ " ABCD J K ~ [ BD XH H ~ JQ. AE i FC J q j~ A i C O_ H ~ BD w _ O. @. K ~ ABCD H " ABD H " +BCD H ". BD AE + BD CF BD ^AE + CF h d (h + h ) K. } 00 00 @.

Q H ~M Q } WH _ BD d, AE h i CF h. K ~ [ " k Z^ \ j~ O_ H ~ w _ O > _ H " H ~ J~ Q }H Q. A d (h + h ) K ^ ~ } WH _ d J k H ~, h i h J k Z^ \ j~ O_ H ~ w _ O > Q. H y _ K q^ ^ ~ A d (h + h ) ik _.. K ~ [ x~ } K ~ [ x~ } Q i < ~ H < ^. K ~ [ xi OK @ W _ q~ O_ O^ Q u [ xi Oz ^ < j~ H Q K. u [ xi OK @ _ O H J ~. < j~ Q i OK @ WOH ~O_ H J ~. JQ K ~ [ xi OK @ S^ O H J ~. Q } G H Ok W _ q^ Q H Wz, XH K ~ [ xi OK K. (i) < Q [ i XH H ~. (ii) < Q [ i XH H } (iii) _ [, XH H ~ i XH H } (iv) _ [ i ~O_ H } (v) ~O_ [ i _ H }.. W _ < Q [ i XH H ~ K ~ [ xi OK @ L^ ~ }. H Ok H ABCD J K ~ [ xi Oz ^ x " H Q. AB O.g., BC 6 O.g., CD.6 O.g., DA O.g. i AC 8 O.g. W _ k: AB O.g., BC 6 O.g., CD.6 O.g. DA O.g. i AC 8 O.g.. K ~ [ x~ } x~ } H ^ : z @ wz Wz H Q i OK ^ : AB O.g. ~M YO_ w. ^ : A i B HO ^ Q f H x 8 O.g. 6 O.g. " ~ ~O_ K w. Jq C ^ YO_ OK H. 0 0 @.

J^ K ~ [ x~ } Q } G @. ^ : AC i BC w. ^ : A i C HO ^ Q f H x O.g.,.6 O.g. " ~ ~O_ K w, Jq D ^ YO_ OK H. ^ 6 : AD i CD w. ABCD H K ~ [. ^ 7 : B O_ BE AC i D O_ DF AC w. " \ H _ H. BE h O.g., DF h. O.g. AC d 8 O.g. " H OK @ K ~ [ ABCD d 8 O.g., h O.g. i h. O.g.. K ~ [ ABCD H " d ^h + h 0 0 h ^ 8 h^ +. h 8 6. 6 O.g... W _ < Q [ i XH H } K ~ [ xi OK @ L^ ~ }. AB6 O.g., BC O.g., CD O.g., DA. O.g., EABC 00 J H K ~ [ ABCD xi Oz " x " H Q. W _ q: AB 6 O.g., BC O.g.,CD O.g., DA. O.g. EABC 00.

K ~ [ x~ } Q H ~M Q } Q } G @. @.6 x~ } H ^ : z @ wz Wz H Q i OK. ^ : BC O.g. ~M YO_ w. ^ : BC g ^ Q B ^ ECBX 00 LO_ @ Q H } Q i OK. ^ : B HO ^ Q f H x 6 O.g. " ~ XH K w. Wk BC A ^ YO_ OK. ^ : C i A HO ^ Q f H x O.g. i. O.g. " ~ ~O_ K w. Jq D ^ YO_ OK H. ^ 6 : CD i AD w. ABCD H K ~ [ ^ 7 : B O_ BF AC i D O_ DE AC w BF, DE _ H. BF h O.g., DE h.7 O.g., AC d 7.8 O.g.. " H OK @ K ~ [ ABCD, d 7.8 O.g., h O.g., h.7 O.g.. K ~ [ ABCD H " d (h + h ) ^ 78. h ^ + 7. 7.8 7.. O.g. 0 0 h

Q } G J^..6 W _ _ [, XH H ~ i XH H } K ~ [ xi OK @ L^ ~ }. PQ O.g., QR 6 O.g., PR 7 O.g., PS O.g., EPQS 0 H K ~ [ PQRS xi Oz ^ x " H Q. W _ q: PQ O.g., QR 6 O.g., PR 7 O.g., PS O.g. i EPQS 0. K ~ [ x~ } @.7 @.8 x~ } H ^ : z @ wz Wz H Q i OK. ^ : PQ O.g. ~M YO_ w ^ : P i Q HO ^ Q f H x 7 O.g. i 6 O.g. " ~ ~O_ K w. Jq R ^ YO_ OK H <. ^ : PR i QR w ^ : PQ g ^ Q Q ^ PQT 0 0. LO_ @ Q H } w. ^ 6 : P HO ^ Q f H x O.g. " ~ XH K w. Wk QT S ^ YO_ OK. ^ 7 : PS w. H K ~ [ PQRS JQ. ^ 8 : Q O_ QX PR i S O_ SY PR w QX, SY _ H. QX h. O.g., SY h.9 O.g. PR d 7 O.g. 0 0

Q H ~M Q } " H OK @ K ~ [ PQRS, d 7 O.g., h. O.g., h.9 O.g. K ~ [ PQRS H " d (h + h ) ^ 7 h ^. +. 9 h 7 7. O.g...7 W _ _ [ i ~O_ H } K ~ [ xi OK @ L^ ~ }. AB 6. O.g., AD O.g., CD O.g., EBAC 0 i EABC 0 H K ~ [ ABCD xi Oz ^ x " H Q. W _ q: AB 6. O.g., AD O.g., CD O.g., EBAC 0 i EABC 0. K ~ [ x~ } Q } G @.9 @.0 x~ } H ^ : z @ wz Wz H Q i OK. ^ : AB 6. O.g. ~M YO_ w. ^ : AB g ^ Q A ^ EBAX 0 i AB g ^ Q B ^ EABY 0 LO_ @ H } Q i Oz Jq C ^ YO_ OK H. 0 0

J^ Q } G ^ : A i C HO ^ Q f H x O.g. i O.g. " ~ ~O_ K w. Jq D ^ YO_ OK H. ^ : AD i CD w ABCD H K ~ [ JQ. ^ 6 : D O_ DE AC i B O_ BC AC w. BC, DE H H. BC h. O.g., DE h. O.g. AC d O.g. " H OK @ K ~ [ ABCD d O.g., BC h. O.g., h. O.g. K ~ [ ABCD H " d ^h+ hh ^ h ^. +. h 8.. O.g...8 W _ ~O_ [ i _ H } K ~ [ xi OK @ L^ ~ }. AB 6 O.g., AD 6 O.g., E ABD, E BDC 0 i E DBC 0 H Q K ~ [ ABCD xi Oz ^ x " H Q. W _ k: AB 6 O.g., AD 6 O.g., EABD, EBDC 0 i EDBC 0. K ~ [ x~ } @. @. 06 06

07 07 Q H ~M Q } x~ } H ^ : z @ wz Wz H Q i OK. ^ : AB 6 O.g. ~M YO_ w. ^ : AB g ^ Q B ^ EABX LO_ @ H } Q i OK. 0 ^ : A HO ^ Q 6 O.g. " ~ XH K w. Wk BX D ^ YO_ OK. ^ : AD w. ^ 6 : BD g ^ Q B ^ EDBY 0 LO_ @ H } Q i OK. 0 ^ 7 : BD g ^ Q D ^ EBDZ 0 LO_ @ H } Q i OK. 0 ^ 8 : BY i DZ, C ^ YO_ OK H. ABCD H K ~ [ ^ 9 : A O_ AE BD i C O_ CF BD w. AE, CF _ H. AE h. O.g., CF h.8 O.g., BD d 8. O.g. " H OK @ K ~ [ ABCD d 8. O.g., h. O.g. i h.8 O.g. K ~ [ ABCD H " d ( h + h ) ^ 8. h ^. + 8. h 8. 8 O.g.. J. H Ok H K ~ [ xi Oz " \ " H Q.. AB O.g, BC 6 O.g, CD O.g., DA. O.g. AC 7 O.g.. AB 7 O.g., BC 6. O.g, AC 8 O.g, CD 6 O.g. DA. O.g.. AB 8 O.g, BC 6.8 O.g, CD 6 O.g, AD 6. O.g,E B 0.. AB 6 O.g, BC 7 O.g, AD 6 O.g, CD O.g, E BAC.. AB. O.g, BC 6. O.g, BD 7 O.g, AD O.g,E BAC 0. 6. AB 7 O.g, BC O.g, AC 6 O.g, CD O.g,, EACD. 7. AB. O.g, BC. O.g, AC 6. O.g, E CAD 80,EACD 0. 8. AB O.g, BD 7 O.g, BC O.g, E BAD 00, E DBC 60. 9. AB O.g, AC 8 O.g, EABC 00, EABD 0, E CAD 0. 0. AB 6 O.g, BC 6 O.g, E BAC 0, EACD 0, E CAD 00. Q } G

J^ Q } G. O K ~ [ (Trapezium).. ik H K ~ [ Q O K ~ [ i k K ~ [ Q i 7 ~ Q u < ~ H x <. " \ H ^ ~ Q i _ < ~ H x <. O K ~ [ H x~ K i " \ ^ ~ Q ~ K H. XH [ Z^ \ [ O ~ Q O_ K ~ [ O K ~ [ Jx JO^ ~... O K ~ [ " O K ~ [ EASY f H. @. W _ O K ~ [ YA H ~ Q w @ ^ ~ ~O_ u [ Q q lok K. XH u [ P^ ~ EA ( EA a } ) i H u [ P^ ~ YS ( YS } ) EA YS Jx e YF HA h } EAY H " ah. K.. YAS H " h K.. O K ~ [ H " EAY H " + YAS H " ah + h h (a + ) K.. Z ( O ~ [ " ) K.. O K ~ [ H " A h (a + ) K.. WH _ a i J k O ~ [ _. h J k O ~ [ ^ Q O ^ ~ Q. 08 08

Q H ~M Q }.. O K ~ [ xi OK @ ^ ~ } Q O K ~ [ xi OK _, O ~ [ x _ " [ P^ ~ Q K H x, W _ H ^ x g ^ u [ xi OK ^. JQ u [, ~O_ O ~ [ ^ J i O_. u [ H P^ ~ Z^ ~ Q J i j~, O K ~ [ H P^ ~ [ Z^ ~ Q O ~ [ O^ J i k. j~ O_ P^ ~ O ~ Q XH ~Y wk ^. O K ~ [ H < j~ D ~M YO_ O^ J i k. W _ H x H H ^ ~ < j~ Q i OK _ K k, ^ H j~ H @ O K ~ [ O^ ^. O K ~ [ xi OK @ < Q O H H. H O^ W _ K ~ L yoz O K ~ [ xi OK @: (i) _ [ i XH H ~ (ii) _ [ i XH H } (iii) ~O_ [ i ~O_ H } (iv) < Q [.. W _ _ [ i XH H ~ O K ~ [ xi OK @ L^ ~ }.6 AB, DC O ~ AB 0 O.g, BC O.g, AC 8 O.g, CD 6 O.g. H O K ~ [ ABCD xi Oz " \ " H Q. W _ k: AB, DC O ~ AB 0 O.g, BC O.g, AC 8 O.g,CD6 O.g. O K ~ [ x~ } @. Q } G @. 6. 09 09

Q } G J^ x~ } H ^ : z @ wz Wz H Q i OK. ^ : AB 0 O.g. ~M YO_ w. ^ : A i B HO ^ Q f H x ~O_ K w. Jq C ^ YO_ OK H. ^ : AC i BC w. ^ : BA O ~ Q CX w. ^ 6 : C HO ^ Q f H x 6 O.g. " ~ w Jk CX D ^ YO_ OK. ^ 7 : AD w. ABCD H O K ~ [ Q. ^ 8 : C O_ CE AB wz CE H _ H. CE h O.g. AB a 0 O.g, DC 6 O.g. " H OK @ O K ~ [ ABCD, a 0 O.g, 6 O.g i h O.g. O K ~ [ ABCD H " h (a + ) ^ h ^ 0 + 6 h 6 O.g... W _ _ [ i XH H } O K ~ [ xi OK @ L^ ~ }.7 PQ, SR O ~ PQ 8 O.g EPQR 70, QR 6 O.g i PS 6 O.g. H O ~ K ~ [ xi Oz " \ " H Q. W _ q: PQ, SR O ~ PQ 8 O.g, EPQR 70, QR 6 O.g i PS 6 O.g. 0 0 @.6

O K ~ [ x~ } Q H ~M Q } Q } G @.7 x~ } H ^ : z @ wz Wz H Q i OK. ^ : PQ 8 O.g. ~M YO_ w. ^ : PQ g ^ Q Q ^ EPQX 70 LO_ @ H } w. ^ : Q HO ^ Q 6 O.g. " ~ K w. Jk QX R ^ YO_ OK. ^ : RY QP O ~ Q w. ^ 6 : P HO ^ Q 6 O.g. " ~ K w. Jk RY S ^ YO_ OK. ^ 7 : PS w. PQRS H O K ~ [. ^ 8 : S O_ ST PQ w. ST H _ H. ST h.6 O.g, RS.9 O.g. PQ a 8 O.g. " H OK @ O K ~ [ PQRS a 8 O.g,.9 O.g i h.6 O.g. O K ~ [ PQRS H " h ^ a + h ^ 6. h ^ 8 + 9. h. 6. 9. O.g.

Q } G J^..6. W _ ~O_ [ i ~O_ H } O K ~ [ xi OK @ L^ ~ }.8 AB, DC O ~. AB 7 O.g, BC 6 O.g, EBAD 80 i EABC 70 H O ~ K ~ [ xi Oz " x " H Q W _ k: AB, DC O ~. AB 7 O.g, BC 6 O.g, EBAD 80 i EABC 70. O K ~ [ x~ } @.8 @.9 x~ } H ^ : z @ wz Wz H Q i OK. ^ : AB 7 O.g. ~M YO_ w. ^ : AB g ^ Q A ^ EBAX 80 LO_ @ H } Q i OK. ^ : AB g ^ Q B ^ EABY 70 LO_ @ H } Q i OK. ^ : B HO ^ Q f H x 6 O.g. " ~ K w. Jk BY C ^ YO_ OK. ^ 6 : CZ AB O ~ Q w. WkAX D ^ YO_ OK. ABCD H O ~ K ~ [ ^ 7 : C O_ CE AB w CE _ H CE h.6 O.g i CD O.g., AB a 7 O.g.

Q H ~M Q } " H OK @ O K ~ [ ABCD a 7 O.g, O.g. i h.6 O.g. O K ~ [ ABCD H " h^a+ h ^ 6. h ^ 7 + h. 6 0.8 O.g...7. W _ < Q [ O K ~ [ xi OK @ L^ ~ }.9 AB, DC O ~, AB 7 O.g, BC O.g, CD O.g i AD O.g H O K ~ [ xi Oz " \ " H Q. W _ k: AB, DC O ~ AB 7 O.g,BC O.g CD O.g. i AD O.g. O K ~ [ x~ } Q } G @.0 @. x~ } H ^ : z @ wz Wz H Q i OK. ^ : AB 7 O.g. ~M YO_ w. ^ : AB E Q i Oz, AE O.g. [a DC O.g ] JQ.

Q } G J^ ^ : B i E HO ^ Q f H x O.g. " ~ ~O_ K w. Jq C ^ YO_ OK H. ^ : BC i EC w. ^ 6 : C i A HO ^ Q f H x O.g. i O.g. " ~ ~O_ K w. Jq D ^ YO_ OK H. ^ 7 : AD i CD w. ABCD H O ~ K ~ [. ^ 8 : D O_ DF AB w. DF _ H. DF h.8 O.g. AB a 7 O.g, CD O.g. " H OK @ O K ~ [ ABCD, a 7 O.g, O.g i h.8 O.g. O K ~ [ ABCD H " h^a+ ^ 8. h ^ 7 + h 8.. h 6. O.g...8 k O K ~ [ @ 6. ABCD XH k O K ~ [. k O K ~ [ (i) O ~ H x [ JQ AD BC. (ii) EA EB. i EADC EBCD (iii) (iv) H ~ JQ AC BD AE BF, (DE AB, CF BA) XH k O K ~ [ xi OK @ _ O H J ~ i H Ok W _ ~O_ x O^ LO_. (i) XH [ Z^ \ [ O ~. (ii) O ~ H x [. @.

Q H ~M Q }..9. k O K ~ [ x~ } L^ ~ } 6.0 AB J k DC O ~ AB O.g, DC 7 O.g, AD BC 6 O.g H Q k O K ~ [ xi Oz " \ " H Q. W _ q : AB J k DC O ~. AB O.g, DC 7 O.g, AD BC 6 O.g. k O K ~ [ x~ } Q } G @. @. x~ } H ^ : z @ wz Wz H Q i OK. ^ : AB O.g. ~M YO_ w. ^ : AE 7 O.g LO_ @ Q AB E Q i OK. ( DC 7 O.g ) ^ : E i B HO ^ Q f H x 6 O.g. (AD EC 6 O.g ) " ~ ~O_ K w. Jq C ^ YO_ OK H. ^ : BC i EC w. ^ 6 : C i A HO ^ Q f H x 7 O.g. i 6 O.g. " ~ ~O_ K w. Jq D ^ YO_ OK H. ^ 7 : AD i CD w. ABCD H k O K ~ [. ^ 8 : D O_ DF AB w DF _ H. DF h.6 O.g. AB a O.g i CD 7 O.g.

Q } G J^ " H OK @ k O K ~ [ ABCD, a O.g, 7 O.g i h.6 O.g. k O K ~ [ ABCD H " h ^a+ J. ^ 6. h ^ + 7 h 6. 8 0. O.g. I. H Ok W _ H k K ~ [ PQRS xi OK. " \ H " H Q. SR PQ6.8 O.g QR7. O.g, PR 8. O.g RS 8 O.g. PQ II,. PQ II SR, PQ 8 O.g, QR O.g, PR 6 O.g RS. O.g.. PQ II SR, PQ 7 O.g, E Q 60,QR O.g, RS O.g.. PQ II SR, PQ 6. O.g, QR 7 O.g, E PQR 8, PS 9 O.g.. PQ II SR, PQ 7. O.g, PS 6. O.g, E QPS 00 E PQR. 6. PQ II SR, PQ 6 O.g,PS O.g, E QPS60 E PQR00. 7. PQ II SR, PQ 8 O.g, QR O.g, RS 6 O.g,SP O.g. 8. PQ II SR, PQ. O.g, QR. O.g, RS O.g SP O.g. II. H Ok W _ H k O K ~ [ ABCD xi Oz " \ " H Q.. AB J k DC O ~, AB 9 O.g, DC 6 O.g, ADBC O.g.. ABJ k DC O ~, AB0 O.g, DC6 O.g, AD BC7 O.g. h ph ~ f J< H K ~ [ ^ ~ e H x O_ i." \ ^ + Q K _ ~O_ ~M Q } H O^ W _ q. i) n~ K ~ H ~ XH ^ xh H \ k YO_ K. Jq n~ K ~ < Q Q Q q lok. ii) K ~ [ H ~ XH ^ xh XH \ O k YO_ K. 6 6

Q H ~M Q }. O ~ K ~ [ (Parallelogram)... ik O ~ K ~ [ Q ioz 7 ~ Q u < ~ H x <. Wq H Ok q^ Q x~ zo _. Z^ \ [ O ~ Q L K ~ [ O ~ K ~ [ Jx JO^ ~. @. W _ O ~ K ~ [ BASE Q xok " \ H ^ ~ e H K. Q } G (i) BA ES ; BE AS (ii) BA ES, BE AS (iii) (iv) Ja Y H } EBES EBAS; EEBA EESA H ~ XH ^ x< H \ k YO_ K OB OS; OE OA, H x BS! AE. (v) U ~O_ P H } " 80 W _ O ~ K ~ [ xi OK @ Q i < ~ H < ^... O ~ K ~ [ " O ~ K ~ [ FAME O_ Z ~ \ Q ( O H } u [ EFS) YO_ OK. D Q @ FAME H _ " JuH OK W C_ n~ K ~ ~ Q LO_ @ ( @.7 ) Q xok K. n~ K ~ H _ } i Z h } n~ K ~ H " A h } Jx e ^. WH _ O ~ K ~ [ FAME n~ K ~ Q i " _ k. H O ~ K ~ [ H " A h K. }. J k O ~ K ~ [ H P^ ~ Q i h J k O ~ [ ^ O ^ ~ Q. @.6 @.7 @. 7 7

Q } G J^.. O ~ K ~ [ x~ } @ y u [ Q q lok @ ^ ~ O ~ K ~ [ xi OK _. W _ q~ O_ O^ Q XH u [ xi OK. ^ < Q j~ H Q. O ~ K ~ [ xi OK @ _ O H J ~.. H O^ W _ H O ~ K ~ [ xi OK K. (i) (ii) (iii) (iv) ~O_ P [ i XH H } ~O_ P [ i XH H ~ ~O_ H ~ i " \ ^ Q H } XH [, XH H ~ i XH H }... W _ ~O_ P [ i XH H } O ~ K ~ [ xi OK @ L^ ~ }. AB 6 O.g, BC. O.g i EABC 80 H O ~ K ~ [ ABCD xi Oz ^ x " H Q. W _ q: AB 6 O.g, BC. O.g i EABC 80. O ~ K ~ [ x~ } @.8 @.9 8 8

Q H ~M Q } x~ } H ^ : z @ wz Wz H Q i OK. ^ : AB 6 O.g. ~M YO_ w. ^ : AB g ^ Q B ^ EABX 80 LO_ @ H } w. ^ : B HO ^ Q f H x. O.g. " ~ K w. Jq BX C ^ YO_ OK. ^ : C i A HO ^ Q f H x 6 O.g. i. O.g. " ~ ~O_ K wz, Jq D ^ YO_ OK H. ^ 6 : CD i AD w. ABCD H O ~ K ~ [ ^ 7 : C O_ CE AB w. CE _ CE h. O.g. AB 6 O.g. H " H OK @ O ~ K ~ [ ABCD 6 O.g i h. O.g. O ~ K ~ [ ABCD H " h 6.. O.g.... W _ ~O_ P [ i XH H ~ O ~ K ~ [ xi OK @ L^ ~ }. AB 8 O.g, AD 7 O.g, BD 9 O.g H O ~ K ~ [ ABCD xi Oz ^ x " H Q. W _ k: AB 8 O.g, AD 7 O.g, BD 9 O.g. O ~ K ~ [ xi OK @ Q } G @.0 @. 9 9

Q } G J^ x~ } H ^ : z @ wz Wz H Q i OK. ^ : AB 8 O.g ~M YO_ w. ^ : A i B HO ^ Q f H x " ~ 7 O.g. i 9 O.g. ~O_ K w. Jq D ^ YO_ OK H. ^ : AD i BD w. ^ : B i D HO ^ Q f H x 7 O.g. i 8 O.g. ~O_ K wz, Jq C ^ YO_ OK H. ^ 6 : CD i BC w. ABCD H O ~ K ~ [. ^ 7 : D O_ DE AB w, DE _ H. DE h 6.7 O.g. AB DC 8 O.g " H OK @ O ~ K ~ [ ABCD 8 O.g, h 6.7 O.g. O ~ K ~ [ ABCD H " h 8 6.7.6 O.g...6 W _ ~O_ P [ i XH H ~ O ~ K ~ [ xi OK @ L^ ~ }. AC 9 O.g, BD 7 O.g EAOB 0 JO^ AC i BD J q O ^ YO_ OK H. W _ k:ac9 O.g, BD 7 O.g,E AOB 0. O ~ K ~ [ x~ } @. @. 0 0

Q H ~M Q } x~ } H ^ : z @ wz Wz H Q i OK. ^ : AC 9 O.g. ~M YO_ w. ^ : AC H ^ ao^ O Q Q i OK. ^ : EAOY 0 LO_ @ O ^ ~ XY ~Y w. ^ : O HO ^ Q f H x. O.g. " ~ XY g ^ AC W~ " ~O_ K wz Jq OX D ^ i OY B ^ YO_ OK. ^ 6 : AB, BC, CD i DA w. ABCD H O ~ K ~ [. ^ 7 : D O_ DE AB w DE _ H. DE h O.g. AB 7 O.g. " H OK @ O ~ K ~ [ ABCD 7 O.g i h O.g. O ~ K ~ [ ABCD H " h 7 8 O.g...7. W _ XH [, XH H ~ i XH H } O ~ K ~ [ xi OK @ L^ ~ }. AB 6 O.g, EABC 80 i AC 8 O.g H O ~ K ~ [ xi Oz ^ x " H Q. W _ k:ab 6 O.g,E ABC 80, AC 8 O.g. O ~ K ~ [ xi OK @ Q } G @. @.

Q } G J^ x~ } H ^ : z @ wz Wz H Q i OK. ^ : AB 6 O.g LO_ @ ~M YO_ w. ^ : AB g ^ Q B ^ EABX 80 LO_ @ H } w. 0 ^ : A HO ^ Q 8 O.g. " ~ K w. Jq BX C ^ YO_ OK. ^ : AC w. ^ 6 : C HO ^ Q 6 O.g. " ~ XH K w. ^ 7 : A HO ^ Q f H x BC H _ " ~ Q f H x " ~ H K w. P ~O_ K D ^ YO_ OK H. ^ 8 : AD i CD w. ABCD H O ~ K ~ [. ^ 9 : C O_ CE AB w CE H _ H. CE h 6. O.g. AB 6 O.g. " H OK @ O ~ K ~ [ ABCD 6 O.g i h 6. O.g. O ~ K ~ [ ABCD H " h 6 6. 8. O.g. J. W _ H ABCD O ~ K ~ [ xi Oz ^ x " H Q. AB 7 O.g, BC O.g i E ABC 60.. AB 8. O.g, AD 6. O.g i E DAB 00.. AB 6 O.g, BD 8 O.g i AD O.g.. AB O.g, BC O.g, AC 7 O.g.. AC 0 O.g, BD 8 O.g, E AOB 00 JO^ AC i BD J q O ^ YO_ OK H. 6. AC 8 O.g, BD 6 O.g,E COD 90 JO^ AC i BD J q O ^ YO_ OK H. 7. AB 8 O.g, AC 0 O.g, E ABC 00. 8. AB. O.g, E DAB 0 i BD 7 O.g.

Q H ~M Q } < Q ~M YO_ K _ XH @ XH K ~ [ Q. XH K ~ [ xi OK @ S^ O H H. XH [ Z^ \ [ O ~ Q O_ K ~ [ O K ~ [ O^ ~. O K ~ [ xi OK @ < Q O H H. O K ~ [ O^ e J O ~ * Q L _, ^ xx k K ~ [ O^ ~. k K ~ [ xi OK @ _ O H H. Z^ \ * O ~ Q K ~ [ O ~ K ~ [ x JO^ ~. O ~ K ~ [ xi OK @ _ O H H. K ~ [ " A d (h + h ) K. }. WO^ d J k H ~, h i h J q Z^ \ j~ O_ H ~ w _ O > Q. O K ~ [ " A h^a+ h K.. WH _ a i J q O ~ [ _ Q. h J k O ~ [ ^ Q O ^ ~ Q. O ~ K ~ [ ", A h K. } WH _ J k O ~ K ~ [ P^ ~, h J k O ~ [ ^ Q O ^ ~ Q. L Q H ~ " K ~ H x O ~ Q z H i t H x OK n~ K ~ OQ ~ n~ K ~ JO^ ~. OQ ~ n~ K ~ [ _ x+ u ~ :.6 LO_. D x+ u x OQ ~ x+ u JO^ ~. OQ ~ n~ K ~ K K @ < O^ H ~ Q LO_. H. S^ O ^ H O w OQ ~ x+ u x H Q xi. Q } " Q 8 ~ } OK. 7 [ lx k w x H ~ H <. H x Jk te K H $ H x OK. xq _ + _ l (Mystic hexagon) : XH H + _ l Q Jx H ~ H _ ^ xx ''xq _ + _ l JO^ ~. Q } G

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