‡TÔ•‡TÔ•‡TÔ•‡TÔ•‹‹‹‹

Seõ” ›[ÒSeõ” ›[ÒSeõ” ›[ÒSeõ” ›[Ò www.amaromiffa.com

"The best angle from which to approach a problem is the Try-angle." #T“†¨<”U ‹Ó` KSõ�ƒ õ~’< Ô” eK?(ƒ)-Ô” ’¨<::$

eUu=e (Anonymous) S”Å`Å`ÁS”Å`Å`ÁS”Å`Å`ÁS”Å`Å`Á

ÃI ÙT` (article) u‡TÔ•‹‡TÔ•‹‡TÔ•‹‡TÔ•‹ (polygons) Là ¾T>Á}Ÿ<` ÙT` c=J”' #’Øw' SeS`“' ²«’Øw' SeS`“' ²«’Øw' SeS`“' ²«’Øw' SeS`“' ²«$ (point, line, and angle) uT>M `°e ¾ÙS`Ÿ<ƒ ÙT` }Ÿ�à ÙT` ’¨<:: 6’²=I ÙTa‹ ÅÓV #cÑL© ›T[cÑL© ›T[cÑL© ›T[cÑL© ›T[————$$$$ (›TaU (›TaU (›TaU (›TaU————)))) uT>M `°e ŸéõŸ<ƒ SêNõ ¾}¨cÆ “†¨<:: c=û' ›=c=ûc=û' ›=c=ûc=û' ›=c=ûc=û' ›=c=û

›=c=ú (v”É LÃ) ŸT>K¨< ¾¨L× nM c=ûc=ûc=ûc=û (same) ¾T>K¨<” nM 6“Ñ—K”:: c=û TKƒ Á¨< (¾²=Á’< ¯Ã’ƒ' ›”É ¯Ã’ƒ) TKƒ ’¨<:: ¾c=û }n^’> ÅÓV ›=c=û›=c=û›=c=û›=c=û (not same) c=J”' ƒ`Ñ<S<U ›”É ¯Ã’ƒ ÁMJ’' ¾}KÁ¾' 6¾pM TKƒ ’¨<::

same c=û not same ›=c=û

#›=c=û$ (not same) 6“ #›=cû$ 6”ÇÃU�~wI ›Å^' Ÿ›TaT>ó

¯LS¨‹ ›”Æ ¾J’¨<” ¾Ùu=Á” Q´w uÒ^ ƒ¨<òƒ Te}dc`” uê’< ¾T>é[[¨< #¨Á’@$ 6’@” Seõ” ›[Ò” #Å`Ó ›=cû$ 6”Ç=K˜ "MðK¡ ue}k`:: Ç\ Ó” u=K˜e U” †Ñ[˜; ÔÖ— ŸSvM ›Ãwe!

c<Ïd' +v' ò×' c=T¡c<Ïd' +v' ò×' c=T¡c<Ïd' +v' ò×' c=T¡c<Ïd' +v' ò×' c=T¡ c<Ïe (Ñó) ŸT>K¨< ¾¨L× nM c<Íc<Íc<Íc<Í (dia-) ¾T>K¨<” òMÖ<õ (prefix) 6“Ñ—K”:: Ÿ²=I òMÖ<õ ÅÓV c<Ïdc<Ïdc<Ïdc<Ïd (diagonal) ¾T>K¨<” nM 6“Ñ—K”::

}v (ÁÒÅK) ŸT>K¨< ¾*aU— nM +v+v+v+v (hypothenuse) ¾T>K¨<” nM 6“Ñ—K”:: ò×ò×ò×ò× (Ýõ) ¾T>K¨< ¾*aU— nM ÅÓV vertex ¾T>K¨<” ¾6”ÓK=²— nM Ã}"M“M::

UeT¡ (ÉÒõ) ŸT>K¨< ¾Ó6´ nM c=T¡c=T¡c=T¡c=T¡ (base)' c=T"©c=T"©c=T"©c=T"© (basic) ¾T>K<ƒ” nKA‹ 6“Ñ—K”:: c=T¡ TKƒ SW[ƒ TKƒ c=J”' c=T"© TKƒ ÅÓV SW[�© TKƒ ’¨<:: KUdK? ÁIM c=T"© cÑM (basic science) TKƒ SW[�© cÑM TKƒ ’¨<::

dia- c<Í vertex ò× diagonal c<Ïd base c=T¡ hypothenus +v basic c=T"©

TÊ' LÊ' ›=LÊTÊ' LÊ' ›=LÊTÊ' LÊ' ›=LÊTÊ' LÊ' ›=LÊ TÊTÊTÊTÊ (opposite) TKƒ u¨Ç=Á uŸ<M' u}n^’> uŸ<M' vhÑ` TKƒ ’¨<:: ÊL (Ô[u?ƒ) ŸT>K¨< ¾*aU— nM ÅÓV LÊLÊLÊLÊ (adjacent) ¾T>K¨<” nM 6“Ñ—K”:: LÊ TKƒ Ô[u?ƒ' Ÿ<� ÑÖU TKƒ ’¨<:: LÊ (adjacent) ŸT>K¨< nM ›=LÊ (nonadjacent) ¾T>K¨<” nM 6“Ñ—K”:: ›=LÊ TKƒ LÊ ÁMJ’ TKƒ ’¨<:: KUdK? ÁIM TÊ ›=LÊ ’¨<::

LÊ 6“ ›=LÊ ŸT>K<ƒ nKA‹ LÊ Ô•‹ (adjacent sides)' LÊ ²«¨‹ (adjacent angles)' ›=LÊ Ô•‹ (nonadjacent sides) ›=LÊ ²«¨‹ (nonadjacent angles) ' LÊ òר‹ (adjacent vertices)' ›=LÊ òר‹ (nonadjacent vertices) ¾SdcK<ƒ” nKA‹ 6“Ñ—K”::

opposite TÊ nonadjacent ›=LÊ adjacent LÊ

Å`†wÅ`†wÅ`†wÅ`†w

#Ú› Å`v$ (}hÑ[) ŸT>K¨< ¾*aU— nM Å`†w Å`†w Å`†w Å`†w (intersection) ¾T>K¨<” nM 6“Ñ—K”:: Å`†w TKƒ SÑ“—' Sq^[Ý TKƒ c=J”' Óc<U c=[v Å[†u (to intersect)' É`‹w (intersected)' Å[‰u= (intersecting)' É`†v (intersecting) 6ÁK ÃH@ÇM:: Ÿ²=IU Å`‰u= SeSa‹ (intersecting lines)' Å`†w ’Øw (intersection point) ¾SdcK<ƒ” nKA‹ 6“Ñ—K”::

intersection Å`†w intersecting Å`‰u= intersect Å[†u intersecting É`†v

Ÿ<MŸ<ƒ' Ÿ<MŸ<ƒ' Ÿ<MŸ<ƒ' Ÿ<MŸ<ƒ' \\\\”Ò”Ò”Ò”Ò ¢K¢K (Å[Å[) ŸT>K¨< ÁT[— nM Ÿ<MŸ<ƒŸ<MŸ<ƒŸ<MŸ<ƒŸ<MŸ<ƒ (parallel) ¾T>K¨<” nM

6“Ñ—K”:: Óc<U c=[v ¢K¢} (make or become parallel)' Ÿ<MŸ<ƒ' ¢M"D‹' Ÿ<M¢� 6ÁK ÃH@ÇM:: ¢K¢} TKƒ Ÿ<MŸ<ƒ ›Å[Ñ ¨ÃU Ÿ<MŸ<ƒ J’ TKƒ ’¨<::

aÒ (T6²”) ŸT>K¨< ¾*aU— nM \\\\”Ò”Ò”Ò”Ò (perpendicular) ¾T>K¨<” nM 6“Ñ—K”:: Óc<U c=[v [’Ñ (make or become perpendicular)' `”Ó' [“Ñ>' `’Ò 6ÁK ÃH@ÇM:: [’Ñ TKƒ \”Ò ›Å[Ñ ¨ÃU \”Ò J’ TKƒ ’¨<::

parallel Ÿ<MŸ<ƒ perpendicular \”Ò parallel (verb) ¢K¢} perpendicular (verb) [’Ñ

‡T' ‡TÔ” ‡T' ‡TÔ” ‡T' ‡TÔ” ‡T' ‡TÔ” Ÿ<T (g=) ŸT>K¨< ¾*aU— ‡T‡T‡T‡T (poly-) ¾T>K¨<” òMÖ<õ (prefix) 6“Ñ—

K”:: ‡T TKƒ w²< TKƒ ’¨<:: Ÿ²=IU ‡TÔ” (polygon)' ‡TÑê (polyhedron) ¾SdcK<ƒ” nKA‹ 6“Ñ“K”

poly- ‡T polyhedron ‡TÑê polygon ‡TÔ”

“•' “•M¡' “•S” “•' “•M¡' “•S” “•' “•M¡' “•S” “•' “•M¡' “•S” “•“•“•“• (¡w) ¾T>K¨< ¾*aU— nM peri- ¾T>K¨<” ¾6”ÓK=²— òMÖ<õ (prefix) Ã}"M“M:: Ÿ²=IU “•M¡“•M¡“•M¡“•M¡ (perimeter) ¾T>K¨<” nM 6“Ñ—K”:: 6”Ç=G<U “• ¾T>K¨<” òMÖ<õ ²S” ŸT>K¨< nM Ò` uT×S` “•S” “•S” “•S” “•S” (period) ¾T>K¨<” nM 6“Ñ—K”::

peri- “• period “•S” perimeter “•M¡

Ë`+U' ›=Ë`+U Ë`+U' ›=Ë`+U Ë`+U' ›=Ë`+U Ë`+U' ›=Ë`+U

u[}T (¾}KSÅ) ŸT>K¨< ¾*aU— nM Ë`+U Ë`+U Ë`+U Ë`+U (regular) ¾T>K¨<” nM 6“Ñ—K”:: Óc<U c=[v Ë[}S (regularize)' Ï`ƒU (regularized)' Ë`�T>' Ï`}T 6ÁK ÃH@ÇM::

¾Ë`+U }n^’> ›=Ë`+U›=Ë`+U›=Ë`+U›=Ë`+U (irregular) c=J”' Óc<U c=[v ›=Ë[}S (irregularize)' ›=Ï`ƒU (irregularized)' ›=Ë`�T> (irregularizer)' ›=Ï`}T (irregularization) 6ÁK ÃH@ÇM::

regular Ë`+U irregular ›=Ë`+U regularize Ë[}S irregularize ›=Ë[}S

úÇ' ççõúÇ' ççõúÇ' ççõúÇ' ççõ

úÇ (Öõ×ó) ŸT>K¨< ¾¨L× nM úÇúÇúÇúÇ (plane)' úÇ©úÇ©úÇ©úÇ© (planar) ¾T>K<ƒ” nKA‹ 6“Ñ—K”:: Óc<U c=[v ø¾Åø¾Åø¾Åø¾Å (to plane)' ýÃÉ' øÁÏ' ý¾Ç 6ÁK ÃH@ÇM:: úÇ TKƒ KØ ÁK (c}ƒ ÁK) TKƒ c=J”' ø¾Å TKƒ ÅÓV KØ ÁK J’ ¨ÃU KØ ÁK 6”Ç=J” ›Å[Ñ TKƒ ’¨<::

çõçõ (Öõ×ó) ŸT>K¨< ¾Ó6´ nM ççõççõççõççõ (surface) ¾T>K¨<” nM 6“Ñ—K”:: ççõ TKƒ Ñê TKƒ c=J”' Óc<U c=[v ççð (to surface)' êêõ' çéò' êçó (surfacing) 6ÁK ÃH@ÇM::

plane úÇ surface ççõ plane (to) úÇ© surface (to) ççð

c<`ò¡c<`ò¡c<`ò¡c<`ò¡' c<Íõ' c<Íõ' c<Íõ' c<Íõ

c<^ (Y°M) 6“ ðŸ= (Y°M) ¾T>K<ƒ” ¾*aU— nKA‹ uT×S` c<`ò¡c<`ò¡c<`ò¡c<`ò¡ (figure) ¾T>K¨<” nM 6“Ñ—K”:: c<`ò¡ TKƒ Y°M (picture)' UYM' ”Éõ 6”Å TKƒ c=J”' Óc<U c=[v f[ðŸ (to figure, figurate)' c<`õ¡ (figured)' f`óŸ= (figurater)' c<`ð" (figuration) 6ÁK ÃH@ÇM::

c<Ïe (Ñó) ŸT>K¨< ¾¨L× nM c<Í (dia) ¾T>K¨<” òMÖ<õ (prefix) 6“Ñ—K”:: ÃH@” òMÖ<õ ɉõ (graph, gram) ŸT>K¨< %EMÖ<õ (suffix) Ò` uT×S` ÅÓV c<Íõc<Íõc<Íõc<Íõ (diagram) ¾T>K¨<” nM 6“Ñ—K”:: c<Íõ TKƒ Y°M ¨ÃU ”Éõ TKƒ c=J”' Óc<U c=[v cËð (to diagram)' eÏõ' cÍò

(diagramatic)' eËó 6ÁK ÃH@ÇM:: Ÿ²=IU c<Íó© (diagramatic, diagramatical) ¾T>K¨<” nM 6“Ñ—K”:: figure c<`ò¡ diagram c<Íõ

figure (to) f[ðŸ diagram (to) cËð

¾‡TÔ” wÁ’@¾‡TÔ” wÁ’@¾‡TÔ” wÁ’@¾‡TÔ” wÁ’@ ‡TÔ”‡TÔ”‡TÔ”‡TÔ” (polygon) TKƒ Zeƒ ¨ÃU ŸZcƒ uLà ¾J’< T“†¨<U

¾TÃÅ^u‡ (non intersecting) (TKƒU ¾TÃÑ“–< ¨ÃU ¾TÃq^[Ö<) ¾kÖ? SeS` ®êq‹ (straight line segments) c=Ñ×ÖS< ¾T>SW`~ƒ úÇ© c<`ò¡ (planar figure) (TKƒU uc=û (same) úÇ Là ¾T>¨<M c<`ò¡) TKƒ ’¨<::

wÁ’@:wÁ’@:wÁ’@:wÁ’@: ‡TÔ” ‡TÔ” ‡TÔ” ‡TÔ” TKƒ Zeƒ ¨ÃU ŸZcƒ uLà ¾J’< T“†¨<U ¾TÃÅ^†u< ¾kÖ? SeS` ®êq‹ uSÑ×ÖU ¾T>SW`~ƒ úÇ© c<`ò¡ ’¨<::

¾}KÁ¿ ¯Ã’ƒ ‡TÔ•‹ c<`ò¡ 1 (figure 1) Là }f`õŸªM::

c<`ò¡ 1: ‡TÔ•‹ c<`ò¡ 2 Là ¾}f[ðŸ<ƒ c<Íö‹ (diagrams) Ó” ŸÁ”Ç”Ç”Æ c<Íõ Y` u}Ökc<ƒ ¾}KÁ¿ U¡’>Á„‹ du=Á ¾‡TÔ”” wÁ’@ (polygon definition) eKTÁ[Ÿ< (satisfy) (TKƒU eKTÁTEK<) ‡TÔ•‹ ›ÃÅK<U::

c<`ò¡ 2: ‡TÔ” ÁMJ’< c<Íö‹::

uT“†¨<U ‡TÔ” Là • ‡TÔ’<” ¾T>SW`~ƒ ¾kÖ? SeS` ®êq‹ ¾‡TÔ’< Ô•‹Ô•‹Ô•‹Ô•‹

(sides) ÃvLK<:: • ¾‡TÔ’< Ô•‹ ¾T>ÒÖS<v†¨< ’Øx‹ ¾‡TÑ<’< òר‹òר‹òר‹òר‹ (vertices)

ÃvLK<:: • c=û ò× (same vertex) ¾T>Ò\ ¾‡TÔ’< Ô•‹ (sides) LÊ Ô•‹LÊ Ô•‹LÊ Ô•‹LÊ Ô•‹

(adjacent sides) c=vK<' ¾TÃÒ\ƒ ÅÓV ›=LÊ Ô•‹›=LÊ Ô•‹›=LÊ Ô•‹›=LÊ Ô•‹ (nonadjacent

sides) ÃvLK<:: • ¾‡TÔ’< LÊ Ô•‹ (adjacent sides) ¾T>W\ƒ u‡TÔ’< u¨<eØ uŸ<M

¾T>¨<M ²« ¨<eש ²«¨<eש ²«¨<eש ²«¨<eש ²« (interior angle) c=vM' u‡TÔ’< u¨<ß uŸ<M ¾T>¨<M ²« ÅÓV ¨<Ý© ¨<Ý© ¨<Ý© ¨<Ý© ²«²«²«²« (exterior angle) ÃvLM::

• uc=û Ô” (same side) Ýö‹ Là ¾T>¨<K< ¾‡TÔ’< òר‹ LÊ LÊ LÊ LÊ òר‹ òר‹ òר‹ òר‹ (adjacent vertices) c=vK<' ¾Tè<K<ƒ ÅÓV ›=LÊ òר‹›=LÊ òר‹›=LÊ òר‹›=LÊ òר‹ (nonadjacent vertices) ÃvLK<::

• Ýö‡ ¾‡TÔ’< ›=LÊ òר‹ (TKƒU Ô[u?ƒ ÁMJ’< òר‹) ¾J’< ¾kÖ? SeS` ®êp ¾‡SÔ’< c<Ïdc<Ïdc<Ïdc<Ïd (diagonal) ÃvLM:: uK?L ›vvM ¾‡TÔ” c<Ïd TKƒ ¾‡TÔ’<” ›=LÊ òר‹ ¾T>ÁÑ“˜ ¾kÖ? SeS` ®êp TKƒ ’¨<::

c<`ò¡ 3' ¾‡TÔ” ¨<eש ²«' ¨<Ý© ²«“' c<Ïd

¾T“†¨<U ‡TÔ” (polygon) ¾G<K<U Ô•‹ (sides) `´S„‹ ÉU`

(TKƒU ‡TÔ’< ²<`Á¨<” ÁK¨< `´Sƒ) ¾‡TÔ’< “•M¡ “•M¡ “•M¡ “•M¡ (perimeter) ÃvLM::

wÁ’@:wÁ’@:wÁ’@:wÁ’@: ‡TÔ” “•M¡ “•M¡ “•M¡ “•M¡ TKƒ ¾‡TÔ’< ¾G<K<U Ô•‹ `´S„‹ ÉU`' ¨ÃU ÅÓV ‡TÔ’< ²<`Á¨<” ÁK¨< `´Sƒ TKƒ ’¨<::

¾‡TÔ•‹¾‡TÔ•‹¾‡TÔ•‹¾‡TÔ•‹ eÁT@¨‹eÁT@¨‹eÁT@¨‹eÁT@¨‹

‡TÔ•‹ (polygons) ¾Ô•‰†¨<” (sides) lØ` SW[ƒ uTÉ`Ó uT>kØK¨< c”Ö[» Là u}SKŸ}¨< G<’@� Ãc¾TK<::

¾Ô” lØ^†¨< Ÿ 3 6eŸ 10 ¾J’<ƒ ‡TÔ•‹ eV‹ ¾}Ñ–<ƒ eK? (Zeƒ)'

^u? (›^ƒ)' HT@ (›Ueƒ)' dÈ (eÉeƒ)' du? (cvƒ)' dT@ (eU’ƒ)' �c? (²Ö˜)' ›c? (›e`) ¾T>K<ƒ” ¾lØ` òMÖ<ö‹ (number prefixes) Ô” ŸT>K¨< nM Ò` uT×S` ’¨<::

¾Ô” lØ^†¨< Ÿ 11 6eŸ 19 ¾J’<ƒ ‡TÔ•‹ eV‹ ¾}Ñ–<ƒ HÈ (›”É)' ¡K? (G<Kƒ)' eK? (Zeƒ)' ^u? (›^ƒ)' HT@ (›Ueƒ)' dÈ (eÉeƒ)' du? (cvƒ)' dT@ (eU’ƒ)' �c? (²Ö˜) uT>K<ƒ ¾lØ` òMÖ<ö‹ (number

prefixes) òƒÑ@ #Ÿ<$ ¾T>K¨<” òÅM uSÚS` ’¨<:: ÃI òÅM ¾}Ñ–¨< ÅÓV Ÿ<Å” (›e`) ŸT>K¨< ¾*aU— nM ’¨<::

¾Ô” lØ\ 20 ¾J’¨< ‡TÔ” eU L+TÔ” L+TÔ” L+TÔ” L+TÔ” c=J”' L+T TKƒ ÅÓV u¨L× HÁ TKƒ ’¨<::

¾Ô” lØ^†¨< Ÿ 21 6eŸ 29 ¾J’<ƒ ‡TÔ•‹ eV‹ #Ç=$ ¾T>K¨<”“ Ç=ÓÅT (HÁ) ŸT>K¨< ¾*aU— nM ¾}Ñ–¨<” òÅM HÈ (›”É)' ¡K? (G<Kƒ)' eK? (Zeƒ)' ^u? (›^ƒ)' HT@ (›Ueƒ)' dÈ (eÉeƒ)' du? (cvƒ)' dT@ (eU’ƒ)' �c? (²Ö˜) uT>K<ƒ ¾lØ` òMÖ<ö‹ (number prefixes) òƒÑ@ uSÚS` ÃÑ—K<:: KUdK? ÁIM Ç=HT@Ô” Ç=HT@Ô” Ç=HT@Ô” Ç=HT@Ô” TKƒ HÁ ›”É Ô•‹ ÁK<ƒ ‡TÔ” TKƒ c=J”' Ç=¡K?Ô”Ç=¡K?Ô”Ç=¡K?Ô”Ç=¡K?Ô” TKƒ ÅÓV HÁ G<Kƒ Ô•‹ ÁK<ƒ ‡TÔ” TKƒ ’¨<::

vÖnLÃ ›’ÒÑ` ÅÓV ’ Ô•‹ (n sides) ÁK<ƒ ‡TÔ” ’’’’----Ô”Ô”Ô”Ô” (n-gon) ÃvLM::

M¿ ‡TÔ•‹M¿ ‡TÔ•‹M¿ ‡TÔ•‹M¿ ‡TÔ•‹ • G<K<U Ô•‡ (sides) 6Ÿ<M ¾J’< ‡TÔ” 6Ÿ<MÔ“U ‡TÔ”6Ÿ<MÔ“U ‡TÔ”6Ÿ<MÔ“U ‡TÔ”6Ÿ<MÔ“U ‡TÔ” (equilateral

polygon) ÃvLM:: • G<K<U ²«¨‡ (angles) 6Ÿ<M ¾J’< ‡TÔ” 6Ÿ<M²«ÁU ‡TÔ”6Ÿ<M²«ÁU ‡TÔ”6Ÿ<M²«ÁU ‡TÔ”6Ÿ<M²«ÁU ‡TÔ”

(equiangular polygon) ÃvLM::

¾‡TÔ•‹ eV‹¾‡TÔ•‹ eV‹¾‡TÔ•‹ eV‹¾‡TÔ•‹ eV‹ ¾Ô” ¾Ô” ¾Ô” ¾Ô” lØ`lØ`lØ`lØ`

Name eUeUeUeU ¾Ô” ¾Ô” ¾Ô” ¾Ô” lØ`lØ`lØ`lØ`

Name eUeUeUeU

3 triangle eK?Ô” 11 Ÿ<HÈÔ” 4 quadrilaterial ^u?Ô” 12 dodecagon Ÿ<¡K?Ô” 5 pentagon HT@Ô” 13 Ÿ<eK?Ô” 6 hexagon dÈÔ” 14 Ÿ<^u?Ô” 7 heptagon du?Ô” 15 Ÿ<HT@Ô” 8 octagon dT@Ô” 16 Ÿ<dÈÔ” 9 nonagon �c?Ô” 17 Ÿ<du?Ô” 10 decagon ›c?Ô” 18 Ÿ<dT@Ô” 19 Ÿ<�c?Ô” 20 icosagon L+TÔ” 21 Ç=HÈÔ” 22 Ç=¡K?Ô” 23 Ç=eK?Ô” ’ ’-Ô”

• 6Ÿ<MÔ“U 6“ 6Ÿ<M²«ÁU ¾J’ ‡TÔ” Ë`+U ‡TÔ”Ë`+U ‡TÔ”Ë`+U ‡TÔ”Ë`+U ‡TÔ” (regular

polygon) ÃvLM:: uK?L ›vvM Ï`+U ‡TÔ” TKƒ G<K<U Ô•‡ 6Ÿ<M ¾J’<' 6”Ç=G<U G<K<U ²«¨‡ 6Ÿ<M ¾J’< ‡TÔ” TKƒ ’¨<::

• Ë`+U ÁMJ’ ‡TÔ” ›=Ë`+U ‡TÔ”›=Ë`+U ‡TÔ”›=Ë`+U ‡TÔ”›=Ë`+U ‡TÔ” (irregular polygon) ÃvLM::

Ë`+U ‡TÔ•‹ (regular polygons) c<`ò¡ 4 Là }f`õŸªM:: Ë`+U eK?Ô” (regular triangle) (TKƒU Ze~U Ô•‡ 6Ÿ<M ¾J’<' 6”Ç=G<U Ze~U ²«¨‡ 6Ÿ<M ¾J’< eK?Ô”) ›w³—¨<” Ñ>²? ¾T>�¨k¨< 6Ÿ<MÔ“U eK?Ô”6Ÿ<MÔ“U eK?Ô”6Ÿ<MÔ“U eK?Ô”6Ÿ<MÔ“U eK?Ô” (equlaterial triangle) uT>M eÁT@ ’¨<:: Ë`+U ^u?Ô”Ë`+U ^u?Ô”Ë`+U ^u?Ô”Ë`+U ^u?Ô” (regular quadrilaterial) (TKƒU ›^~U Ô•‡ 6Ÿ<M ¾J’<' 6”Ç=G<U ›^~U ²«¨‡ 6Ÿ<M ¾J’< ^u?Ô”) ÅÓV ›w³—¨<” Ñ>²? ¾T>�¨k¨< "M„“"M„“"M„“"M„“ (square) uT>M eÁT@ ’¨<::

c<`ò¡ 4: Ë`+U ‡TÔ•‹ eK?Ô•‹eK?Ô•‹eK?Ô•‹eK?Ô•‹

eK?Ô” (triangle) TKƒ Zeƒ Ô•‹ (sides) ÁK<ƒ ‡TÔ” (polygon) TKƒ ’¨<:: eK?Ô•‹ ¾²«¨‰†¨<” (angles) 6c?„‹ (values) 6“ ¾Ô•‰†¨<” `´S„‹ SW[ƒ uTÉ`Ó 6”ÅT>Ÿ}K¨< Ãc¾TK<::

• G<K<U Ô•‡ 6Ÿ<M ¾J’< eK?Ô” 6Ÿ<6Ÿ<6Ÿ<6Ÿ<MÔ“U eK?Ô” MÔ“U eK?Ô” MÔ“U eK?Ô” MÔ“U eK?Ô” (equilaterial triangle) ÃvLM::

• G<K<U ²«¨‡ 6Ÿ<M ¾J’< eK?Ô” 6Ÿ<M²«ÁU eK?Ô” 6Ÿ<M²«ÁU eK?Ô” 6Ÿ<M²«ÁU eK?Ô” 6Ÿ<M²«ÁU eK?Ô” (equiangular triangle) ÃvLM::

M¿ ‡TÔ•‹M¿ ‡TÔ•‹M¿ ‡TÔ•‹M¿ ‡TÔ•‹ equilaterial polygon 6Ÿ<MÔ“U ‡TÔ” equiangular polygon 6Ÿ<M²«ÁU ‡TÔ” regular polygon Ë`+U ‡TÔ” irregular polygon ›=Ë`+U ‡TÔ”

• u=Á”c G<K~ Ô•‡ 6Ÿ<M ¾J’< eK?Ô” 6Ÿ<M6Ó^U eK?Ô” 6Ÿ<M6Ó^U eK?Ô” 6Ÿ<M6Ó^U eK?Ô” 6Ÿ<M6Ó^U eK?Ô” (isosceles

triangle) ÃvLM:: • T“†¨<U Ô•‡ 6Ÿ<M ÁMJ’< eK?Ô” 6Ÿ<Mu=e eK?Ô” 6Ÿ<Mu=e eK?Ô” 6Ÿ<Mu=e eK?Ô” 6Ÿ<Mu=e eK?Ô” (scalen triangle)

ÃvLM:: • G<K<U ²«¨‡ Ÿ 90

o Á’c< eK?Ô” g<M eK?Ô” g<M eK?Ô” g<M eK?Ô” g<M eK?Ô” (acute triangle) ÃvLM::

• ›”Æ ²«¨< Ÿ 90o ¾uKÖ eK?Ô” ´`ØØ eK?Ô´`ØØ eK?Ô´`ØØ eK?Ô´`ØØ eK?Ô” (obtuse triangle) ÃvLM::

• ›”Æ ²«¨< ›<h‰ ²« (right angle) ¾J’ (TKƒU 90o ¾J’) eK?Ô” ›<h‰ ›<h‰ ›<h‰ ›<h‰

eK?Ô” eK?Ô” eK?Ô” eK?Ô” (right triangle) ÃvLM::

¾eK?Ô” ¯¾eK?Ô” ¯¾eK?Ô” ¯¾eK?Ô” ¯Ã’„‹Ã’„‹Ã’„‹Ã’„‹ equilaterial triangle 6Ÿ<MÔ“U eK?Ô” equiangular traingle 6Ÿ<M²«ÁU eK?Ô” isosceles triangle 6Ÿ<M6Ó^U eK?Ô” scalen triangle 6Ÿ<Mu=e eK?Ô” acute triangle g<M eK?Ô” obtuse triangle ´`ØØ eK?Ô” right triangle ›<h‰ eK?Ô”

c<`ò¡ 5: ¾eK?Ô” ¯Ã’„‹

u6Ÿ<M6Ó^U eK?Ô” (isosceles triangle) LÃ' 6Ÿ<M ¾J’<ƒ G<K~ Ô•‹ ¾eK?Ô’< 6Óa‹ 6Óa‹ 6Óa‹ 6Óa‹ (legs) c=vK<' fe}—¨< Ô” ÅÓV ¾eK?Ô’< c=T¡ c=T¡ c=T¡ c=T¡ (base) ÃvLM::

u›<h‰ eK?Ô” (right traingle) Là ÅÓV K›<h‰ ²« (right angle) }n^’> ¾J’¨< Ô” ¾eK?Ô’< +v+v+v+v (hypothenuse) c=vM' ¾k\ƒ G<K~ Ô•‹ ÅÓV ¾eK?Ô’< 6Óa6Óa6Óa6Óa‹ (legs) ÃvLK<::

c<`ò¡ 6: 6Ÿ<M6Ó^U eK?Ô” 6“ ›<h‰ eK?Ô”::

¾T“†¨<U eK?Ô” (triangle) fe~U ¨<eש ²«¨‹ (interior angles) ÉU` G<MÑ>²?U 180o ’¨<:: uK?L ›vvM Ze~ ¾eK?Ô” ¨<eש ²«¨‹ uòSKA‹ (symbols) ∠ G' ∠ K' 6“ ∠N u=¨ŸK<

∠ G + ∠ K + ∠N , 180o ÃJ“M TKƒ ’¨<::

NpNpNpNp: ¾T“†¨<U eK?Ô” ¾Ze~U ¨<eש ²«¨‹ ÉU` G<MÑ>²?U 180o ’¨<::

KUdK? ÁIM ¾6Ÿ<MÔ“U eK?Ô” (equilaterial triangle) Ze~U ²«¨‹ 6Á”ǔdž¨< 60o “†¨<::

c<`ò¡ 7: ¾T“†¨<U eK?Ô” ¾Ze~U ²«¨‹ ÖpLL ÉU` 180o ’¨<:: ^u?^u?^u?^u?Ô•‹Ô•‹Ô•‹Ô•‹

^u?Ô” (quadrilaterial) TKƒ ›^ƒ Ô•‹ (sides) ÁK<ƒ ‡TÔ” (polygon) TKƒ ’¨<:: ^u?Ô•‹ ¾²«¨‰†¨<” (angles) 6c?„‹ (values) 6“ ¾Ô•‰†¨<” `´S„‹ SW[ƒ uTÉ`Ó 6”ÅT>Ÿ}K¨< Ãc¾TK<::

• G<K<U ²«¨‡ ›<h‰ ²«¨‹ (right angles) ¾J’< (TKƒU 90o ¾J’<) ^u?Ô”

\\\\”Ñ@^w”Ñ@^w”Ñ@^w”Ñ@^w (rectangle) ÃvLM:: \”Ñ@^w ¾T>K¨< nM ¾}Ñ–¨< \”Ò

(perpendicular) 6“ ^w° (›^ƒ) ŸT>K<ƒ nKA‹ ’¨<' ¾\”Ñ@^w LÊ Ô•‹ (adjacent sides) (TKƒU Ô[u?ƒ Ô•‹) \”Ò “†¨<“::

• G<K<U Ô•‡ (sides) 6Ÿ<M ¾J’<“ G<K<U ²«¨‡ ›<h‰ ²«¨‹ (right angles) ¾J’< (TKƒU 90

o ¾J’<) ^u?Ô” "M„“"M„“"M„“"M„“ (square) ÃvLM:: eK²=IU "M„“

¾\”Ñ@^w (rectangle) M¿ [ÑÉ (special case) ’¨< TKƒ ’¨<:: uK?L ›vvM G<K<U "M„“¨‹ (²«¨‰†¨< ›<h‰ eKJ’<) \”Ñ@^x‹ c=J’<' G<K<U \”Ñ@^x‹ Ó” Ô•‰†¨< ¾ÓÉ 6Ÿ<M SJ” eKK?v†¨< "M„“¨‹ ›ÃÅK<U TKƒ ’¨<::

• G<K<U Ô•‡ 6Ÿ<M ¾J’< ^u?Ô” ›‰^w›‰^w›‰^w›‰^w (rhombus) ÃvLM:: ›‰^w ¾T>K¨< nM ¾}Ñ–¨< ›‰ 6“ ^u?Ô” ŸT>K<ƒ nKA‹ ’¨<:: eK²=IU "M„“ ¾›‰^w (rhombus) M¿ [ÑÉ (special case) ’¨< TKƒ ’¨<:: uK?L ›vvM G<K<U "M„“¨‹ (Ô•‰†¨< 6Ÿ<M eKJ’<) ›‰^x‹ c=J’<' G<K<U ›‰^x‹ Ó” ²«¨‰†¨< ¾ÓÉ ›<h‰ SJ” eKK?v†¨< "M„“¨‹ ›ÃÅK<U TKƒ ’¨<:

• }n^’> Ô•‡ Ÿ<MŸ<ƒ (parallel) ¾J’< ^u?Ô” Ÿ<MŸ?^wŸ<MŸ?^wŸ<MŸ?^wŸ<MŸ?^w (parallelogram) ÃvLM:: Ÿ<MŸ?^w ¾T>K¨< nM ¾}Ñ–¨< Ÿ<MŸ<ƒ (parallel) 6“ ^u?Ô” ŸT>K<ƒ nKA‹ ’¨<:: eK²=IU \”Ñ@^w (rectangle)' "M„“ (square)' 6“ ›‰^w (rhombus) G<K<U Ÿ<MŸ?^x‹ “†¨< TKƒ ’¨<' }n^’> Ô•‰†¨< Ÿ<MŸ<ƒ “†¨<“::

• G<Kƒ Ô•‡ w‰ Ÿ<MŸ<ƒ (parallel) ¾J’< ^u?Ô” â?³^wâ?³^wâ?³^wâ?³^w (trapezium or

trapezoid) ÃvLM:: â?³^w ¾T>K¨< nM ¾}Ñ–¨< â?³ (table) 6“ ^u?Ô” ŸT>K<ƒ nKA‹ ’¨<::

o ›=Ÿ<MŸ<ƒ (nonparallel) ¾J’<ƒ ¾â?³^w Ô•‹ ¾â?³^u< 6Óa6Óa6Óa6Óa‹‹‹‹ (legs) c=vK<' Ÿ<MŸ<ƒ ¾J’<ƒ Ô•‹ ÅÓV ¾â?³^u< c=Tc=Tc=Tc=T¢‹¢‹¢‹¢‹ (bases) ÃvLK<::

o ›=Ÿ<MŸ<ƒ (nonparallel) ¾J’<ƒ Ô•‡ 6Ÿ<M ¾J’< â?³^w 6Ÿ<M6Ó^U â?³^w6Ÿ<M6Ó^U â?³^w6Ÿ<M6Ó^U â?³^w6Ÿ<M6Ó^U â?³^w (isosceles trapezium) ÃvLM::

o G<K~ ²«¨‡ ›<h‰ ²«¨‹ (right angles) ¾J’< (TKƒU 90o ¾J’<)

â?³^w ›<h‰ â?³^w›<h‰ â?³^w›<h‰ â?³^w›<h‰ â?³^w (right trapezium) ÃvLM:: • 6Ÿ<M ¾J’< G<Kƒ Ø”É LÊ Ô•‹ (adjacent sides) (TKƒU Ô[u?ƒ Ô•‹)

ÁK<ƒ ^u?Ô” Jv Jv Jv Jv (kite) ÃvLM:: Jv (kite) ¾T>K¨< nM ¾}Ñ–¨< Jvà (›V^) ŸT>K¨< ’¨<:: Jv TKƒ u}¨Ö[ Ú`p }W`„ uÑSÉ uSÁ´ u’@óM (air) Là ¾T>”dðõ ¾MЋ (6”Ç=G<U Áªm¨‹) Sݨ‰ ’¨<::

o G<K~ }n^’> ²«¨‡ ›<h‰ ²«¨‹ (right angles) ¾J’< (TKƒU 90o

¾J’<) Jv ›<h‰ Jv›<h‰ Jv›<h‰ Jv›<h‰ Jv (right kite) ÃvLM::

¾^u?Ô” ¯¾^u?Ô” ¯¾^u?Ô” ¯¾^u?Ô” ¯Ã’„‹Ã’„‹Ã’„‹Ã’„‹ rectangle \”Ñ@^w square "M„“ rhombus ›‰^w parallellogram Ÿ<MŸ?^w trapezium (trapezoid) â?³^w kite Jv

¾â?³^w ¯Ã’„‹ c<`ò¡ 8 Là }f`õŸªM:: ÃH@” ¯Ã’ƒ c<`ò¡ ÅÓV ¾â?³^w ³õ¾â?³^w ³õ¾â?³^w ³õ¾â?³^w ³õ (quadrilaterial tree) ÃvLM:: Ÿ²=I ¾^u?Ô” ³õ Là 6”ÅU”SKŸ}¨<

›‰^w (rhombus)' \”Ñ@^w (rectangle)' 6“ "M„“ (square) Ze~U ¾Ÿ<MŸ?^w (parallellogram) ¯Ã’„‹' TKƒU ¾Ÿ<MŸ?^w M¿ [ÑÊ‹ (special cases) “†¨<:: "M„• ÅÓV ¾\”Ñ@^w M¿ [ÑÉ ’¨<:: 6”Ç=G<U "M„“ ¾›‰^w M¿ [ÑÉ ’¨<::

c<`ò¡ 8: ¾â?³^w ³õ::

Np:Np:Np:Np: G<K<U "M„“¨‹ \”Ñ@^x‹ c=J’<' G<K<U \”Ñ@^x‹ Ó” "M„“¨‹ ›ÃÅK<U:: Np:Np:Np:Np: G<K<U "M„“¨‹ ›‰^x‹ c=J’<' G<K<U ›‰^x‹ Ó” "M„“¨‹ ›ÃÅK<U:: NpNpNpNp: \”Ñ@^x‹' "M„“¨‹' 6“ ›‰^x‹ G<K<U Ÿ<MŸ?^x‹ “†¨<' }n^’> Ô•‰†¨< Ÿ<MŸ<ƒ “†¨<“:

6Ÿ<M6Ó^U â?³^w (isosceles trapezium) 6“ ›<h‰ Jv (right kite) c<`ò¡ 9 Là Kw‰†¨< }f`õŸªM::

c<`ò¡ 9: 6Ÿ<M6Ó^U â?³^w 6“ ›<h‰ Jv::

¾T“†¨<U ^u?Ô” (quadrilaterial) ›^~U ¨<eש ²«¨‹ (interior angles) ÉU` G<MÑ>²?U 360o ’¨<:: uK?L ›vvM ›^~ ¾^u?Ô” ¨<eש ²«¨‹ uòSKA‹ (symbols) ∠ G' ∠ K' ∠N 6“ ∠S u=¨ŸK<

∠ G + ∠ K + ∠N + ∠S , 360o ÃJ“M TKƒ ’¨<::

Np:Np:Np:Np: ¾T“†¨<U ^u?Ô” ¾›^~U ¨<eש ²«¨‹ ÉU` G<MÑ>²?U 360o ’¨<::

KUdK? ÁIM ¾\”Ñ@^w (rectangle) ›^~U ²«¨‹ 6Á”ǔdž¨< 90o “†¨<:: 6”Ç=G<U ¾"M„“ (square) ›^~U ²«¨‹ 6Á”ǔdž¨< 90o “†¨<::

c<`ò¡ 10: ¾T“†¨<U ^u?Ô” ¾›^~U ²«¨‹ ÖpLL ÉU` 360o ’¨<:: Ӕѓ' Ñ”Ñ>dӔѓ' Ñ”Ñ>dӔѓ' Ñ”Ñ>dӔѓ' Ñ”Ñ>d Ñ’Ñ’ (ð^' �²u' Ö[Ö[) ŸT>K¨< ÁT[— nM ӔѓӔѓӔѓӔѓ (theory) ¾T>K¨<” nM 6“Ñ—K”:: Óc<U c=[v Ñ’Ñ’ (theorize)' Ó”Ó” (theorized)' єҘ (theorist)' Ó”Ó’ƒ (theorization)' Ӕѓ© (theoretical) 6ÁK ÃH@ÇM::

Ӕѓ (theory) ŸT>K¨< nM Ñ”Ñ>d Ñ”Ñ>d Ñ”Ñ>d Ñ”Ñ>d (theorem) ¾T>K¨<” nM 6“Ñ—K”:: Óc<U c=[v Ñ’Ñc' Ó”Óe' Ñ”Òi' Ó”Ñd 6ÁK ÃH@ÇM:: Ñ’Ñc TKƒ Ñ”Ñ>d kS[ TKƒ ’¨<::

theory Ӕѓ theorem Ñ”Ñ>d theorize Ñ’Ñ’ theorem (verb) Ñ’Ñc

`¡’d' eMu^`¡’d' eMu^`¡’d' eMu^`¡’d' eMu^

T>`Ÿ’@c< (Te[Í) ŸT>K¨< ¾*aU— nM [Ÿ’c [Ÿ’c [Ÿ’c [Ÿ’c (demonstrate) ¾T>K¨<” Óe 6“Ñ—K”:: Óc<U c=[v [Ÿ’c' `¡”e (demonstrated)' [¡“i (demonstrator,

demonstrative)' `¡’d (demonstrating, demonstration) 6ÁK ÃH@ÇM:: Ÿ²=IU [¡’e (demonstration)' [¡“i' `¡’d© (demonstrative) ¾T>K<ƒ” nKA‹ 6“Ñ—K”:: [Ÿ’c TKƒ uTe[Í ›d¾ TKƒ c=J”' [¡’e TKƒ ÅÓV uTe[Í ¾�¾ TKƒ ’¨<::

#uY°M ›w^^$ ŸT>K¨< N[Ó cKu[cKu[cKu[cKu[ (illustrate) ¾T>K¨<” Óe 6“Ñ—K”:: Óc<U c=[v cKu[' eKw` (illustrated)' cMv] (illustrator, illustrative)' eMu^ (illustrating, illustration) 6ÁK ÃH@ÇM:: Ÿ²=IU cMu` (illustration)' cMv]' cMu^© (illustrative)' cMu[— (illustrious) ¾T>K<ƒ” nKA‹ 6“Ñ—K”:: cKu[ TKƒ uY°M ÑKç' uUdK? ›e[Ç TKƒ c=J”' cMu` TKƒ ÅÓV uY°M ¾}ÑKç TKƒ ’¨<::

demonstrate [Ÿ’c illustrate cKu[ demonstration `¡’d illustation eMu^

"M„“ Ñ”Ñ>d"M„“ Ñ”Ñ>d"M„“ Ñ”Ñ>d"M„“ Ñ”Ñ>d "M„“ Ñ”Ñ>d"M„“ Ñ”Ñ>d"M„“ Ñ”Ñ>d"M„“ Ñ”Ñ>d (square theorem) ¾›<h‰ eK?Ô” (right triangle) Ô•‹ (sides) ÁLD†¨<” `´S„‹ ¾T>Á³UÉ Ñ”Ñ>d (theorem) ’¨<:: ÃI” Ñ”Ñ>d (theorem) ›G<” Ówê (Egypt) uUƒvK¨< ›Ñ` Là ›³»“ “³» ¾’u\ƒ �LLp G`uà‹”G`uà‹”G`uà‹”G`uà‹” (pyramids) ¾Ñ’u<ƒ Ø”�¨<Á•‡ ’<v¨<Á”’<v¨<Á”’<v¨<Á”’<v¨<Á” (Nubians) ucò¨< ÃÖkS<uƒ 6”Å’u` ›T@]n©¨< Ñ>¨`Ñ>e ËUeÑ>¨`Ñ>e ËUeÑ>¨`Ñ>e ËUeÑ>¨`Ñ>e ËUe (George James) #¾}c[k p`e: ¾Ó]¡ õMeõ“ #¾}c[k p`e: ¾Ó]¡ õMeõ“ #¾}c[k p`e: ¾Ó]¡ õMeõ“ #¾}c[k p`e: ¾Ó]¡ õMeõ“ ¾}c[k ¾’<v õMeõ“ ’¾}c[k ¾’<v õMeõ“ ’¾}c[k ¾’<v õMeõ“ ’¾}c[k ¾’<v õMeõ“ ’¨<¨<¨<¨<$ ($ ($ ($ (Stolen Legacy: Greek Philosophy is Stolen Egyptian Philosophy) uT>M `°e ¾éð¨<” É”p SêNõ ›”wx S[ǃ ÉLM:: ƒ“”ƒ SÖ?¨‡ ¯Ã” ›¨<× ›¨<aä¨<Á” Ó” Ñ”Ñ>d¨<” u�]¡ KS•\ 6”"D” u`ÓÖ˜’ƒ uTÃ�¨k¨<“ Ó]"© ’¨< uT>vK¨< uú�Ô^e (Pythagoras) eU cÃS¨<ƒ ¾ú�Ô^e Ñ”Ñ>d¾ú�Ô^e Ñ”Ñ>d¾ú�Ô^e Ñ”Ñ>d¾ú�Ô^e Ñ”Ñ>d (Pythagorean theorem) ÃK<�M::

’à‹ Lõ]n¨<Á” "L†¨< SW[}u=e ”kƒ ¾}’d Ñ”Ñ>d¨<” ¾ú�Ô^e Ñ”Ñ>d u=K<ƒU' 6— ›õ]n¨<Á” Ó” ¾’c<” SW[}u=e 6”„ 𔄠6”Å ÑÅM TT>„ Te}Òvƒ ¾Kw”U:: K²=I ’¨< Ñ”Ñ>d¨<” "M„“ Ñ”Ñ>d"M„“ Ñ”Ñ>d"M„“ Ñ”Ñ>d"M„“ Ñ”Ñ>d ÁMŸ<ƒ::

KT”—¨<U "M„“ Ñ”Ñ>d unLƒ c=�}ƒ ¾›<h‰ eK?Ô” +v+v+v+v (hypothenuse) "M„“ (square) ŸG<K~ ¾eKÔ’< 6Óa‡6Óa‡6Óa‡6Óa‡ (legs) """"M„“¨‹M„“¨‹M„“¨‹M„“¨‹ (squares) ÉU` Ò` 6Ÿ<M ’¨< ÃK“M::

"M„“ Ñ”Ñ>d"M„“ Ñ”Ñ>d"M„“ Ñ”Ñ>d"M„“ Ñ”Ñ>d:::: ¾›<h‰ eK?Ô” +v "M„“ Ÿ6Óa‡ "M„“¨‹ ÉU` Ò` 6Ÿ<M ’¨<::

uK?L ›vvM ¾›<h‰ eK?Ô” (right triangle) +v (hypothenuse) `´Sƒ N u=J”“' ¾G<K~ ¾eK?Ô’< 6Óa‡ `´S„‹ ÅÓV G 6“ K u=J’<'

NNNN2222 , G , G , G , G2222 + K K K K2222 ÃJ“M TKƒ ’¨<:: ÃIU G<’@� c<`ò¡ 11 LÃ }SM¡…M::

c<`ò¡ 11: "M„“ Ӕѓ

u"M„“ Ñ”Ñ>d SW[ƒ' ¾›<h‰ eK?Ô” 6Óa‹ `´S„‹ (TKƒU G 6“ K) Ÿ}cÖ<' ¾+v¨<” `´Sƒ (N)

22 KG,N +

uT>K¨< kS` (formula) TÓ–ƒ ÉLM: ¾›<h‰¨< eKÔ” +v `´Sƒ (N) 6“ Á”Æ 6Ó` `´Sƒ (G ¨ÃU K) Ÿ}cÖ< ÅÓV' ¾K?L¨<” 6Ó` `´Sƒ

22 KN,G − ¨ÃU 22 GN,K −

uT>K<ƒ kSa‹ TÓ–ƒ ÉLM:: KUdK? ÁIM ¾›<h‰¨< eK?Ô” 6Óa‹ `´S„‹ G , 3 6“ K , 4 u=J’<' ¾eK?Ô’< +v `´Sƒ

543 22 ,,,,,,,,NNNN ++++ ÃJ“M:: ¾eK?Ô’< +v `´Sƒ N , 13 u=J”“ ¾eK?Ô’< ›”Å—¨< 6Ó` `´Sƒ G , 12 u=J” ÅÓV' ¾eK?Ô’< K?L—¨< 6Ó` `´Sƒ

51213 22 ,,,,,,,,KKKK −−−− ÃJ“M:: "M„“ Ñ”Ñ>d (square theorem) c<`ò¡ 12 Là }cMwbM (illustrated)' TKƒU uY°M }ÑMéDM ¨ÃU }w^`…M:: ÃI cMu` (illustration) (TKƒU ¾Y°M SÓKÝ ¨ÃU Tw^`Á) 6”ÅT>cKw[¨<' `´S„‰†¨< ¾›<h‰ eKÔ’< (right triangle) 6Óa‹ (legs) `´S„‹ ¾J’< "M„“¨‹ (squares) ÁLD†¨< eó„‹ (areas) ÉU`' `´S~ ¾›<h‰ eK?Ô’< +v (hypothenuse) `´Sƒ ¾J’ "M„“ (square) "K¨< eóƒ (area) Ò` 6Ÿ<M ’¨<::

c<`ò¡ 12: ¾"M„“ Ñ”Ñ>d eMu^::

"M„“ eK?‰”"M„“ eK?‰”"M„“ eK?‰”"M„“ eK?‰” u"M„“ Ñ”Ñ>d (square theorem) SW[ƒ ¾T“†¨<U ›<h‰ eK?Ô” (right

triangle) 6Óa‹ (legs) `´S„‹ G 6“ K u=J’<“' ¾eK?Ô’< +v (hypothenuse) `´Sƒ ÅÓV N u=J”' `´S„‹ G' K' 6“ N

NNNN2222 , G , G , G , G2222 + KKKK2222

uT>M 6Ÿ<M� (equation) óSÇK<:: ÃI” 6Ÿ<M� ¾T>Á[Ÿ< (satisfy) (TKƒU ¨<’ƒ ¾T>ÁÅ`Ñ< ¨ÃU ¾T>ÁTEK<) T“†¨<U feƒ ¨”¨”¨”¨”� Ë`+¢‹ lØa‹� Ë`+¢‹ lØa‹� Ë`+¢‹ lØa‹� Ë`+¢‹ lØa‹ (positive real numbers) (G' K' N) ÅÓV "M„“"M„“"M„“"M„“ eK?‰” eK?‰” eK?‰” eK?‰” (square triplets,

Pythagorean triplets) }wK¨< ÃÖ^K<:: uK?L ›vvM T“†¨<U Zeƒ ¨”� Ë`+¡ lØa‹ G' K' 6“ N u6Ÿ<M� N2 , G2 + K2 ¾T>³SÆ ŸJ’' lØa‡ "M„“ eK?‰” ÃvLK< TKƒ ’¨<::

wÁ’@wÁ’@wÁ’@wÁ’@: u6Ÿ<M� N2 , G2 + K2 ¾T>³SÆ T“†¨<U Zeƒ ¨”� Ë`+¡ lØa‹ (G' K' N) "M„“ eK?‰”"M„“ eK?‰”"M„“ eK?‰”"M„“ eK?‰” ÃvLK<::

KUdK? ÁIM (3'4'5) "M„“ eK?‰” “†¨<' 6”Ç=G<U (5'12'13) "M„“

eK?‰” “†¨<:: u}ÚT] ÅÓV (7'24'25)' (9'40'41) 6“ (11'60'61) 6Á”ǔdž¨< "M„“ eK?‰•‹ “†¨<:: %Mq Sdõ`ƒ ¾K?L†¨< "M„“ eK?‰•‹ ›K<:: Ÿ’²=IU ¨<eØ Øm„‡ uT>Ÿ}K¨< c”Ö[» Là }²`´[ªM::

"M„“ eK?‰•‹"M„“ eK?‰•‹"M„“ eK?‰•‹"M„“ eK?‰•‹

6Ó`6Ó`6Ó`6Ó` 6Ó`6Ó`6Ó`6Ó` +v+v+v+v 6Ó`6Ó`6Ó`6Ó` 6Ó`6Ó`6Ó`6Ó` +v+v+v+v 3 4 5 8 15 17 6 8 10 12 16 20 5 12 13 7 24 25 9 12 15 15 20 25

¾ð`T ¾SÚ[h Ñ”Ñ>d¾ð`T ¾SÚ[h Ñ”Ñ>d¾ð`T ¾SÚ[h Ñ”Ñ>d¾ð`T ¾SÚ[h Ñ”Ñ>d Ÿ"M„“ eK?‰” (square tiplets, Pythagorean triplets) Ò` ¾}³SÅ ulØulØulØulØ` ӔѓӔѓӔѓӔѓ (Number theory) ¨<eØ �ªm ¾J’ Ñ”Ñ>d (theorem) ›K:: ÃIU Ñ”Ñ>d ð`d©¨< K=p â?Øae ²ð`Tâ?Øae ²ð`Tâ?Øae ²ð`Tâ?Øae ²ð`T (Pierre de Fermat) u— ›q×Ö` u 1629 ¯.U ¾Ñ’Ñc¨< ¾SÚ[h Ñ”Ñ>d¨< eKJ’ ¾ð`T ¾SÚ[h Ñ”Ñ>d¾ð`T ¾SÚ[h Ñ”Ñ>d¾ð`T ¾SÚ[h Ñ”Ñ>d¾ð`T ¾SÚ[h Ñ”Ñ>d (Fermat's Last

Theorem) uSvM Ã�¨nM:: uð`T ¾SÚ[h Ñ”Ñ>d SW[ƒ Ë`+¡ (integer) ’ Ÿ 2 ¾T>uMØ ŸJ’

(TKƒU ’ > 2 ŸJ’)

N’ , G’ + K’ ’ > 2 ¾T>K¨<” 6Ÿ<M� (equation) K=Á[Ÿ< (satisfy) ¾T>‹K< (TKƒU ¨<’ƒ K=ÁÅ`Ñ< ¨ÃU K=ÁTEK< ¾T>‹K<) Zeƒ ¨”� Ë`+¢‹”¨”� Ë`+¢‹”¨”� Ë`+¢‹”¨”� Ë`+¢‹” (positive integers) (G' K' N) TÓ–ƒ ›Ã‰MU:: KUdK? ÁIM ’ , 3 u=J”'

N3 , G3 + K3

¾T>K¨<” 6Ÿ<M� K=Á[Ÿ< ¾T>‹K< Zeƒ Ë`+¡ lØa‹” (G' K' N) TÓ–ƒ ›Ã‰MU:: uK?L ›vvM lØ^†¨< ›=n”ä (infinite) (TKƒU ¾ƒ¾KK?) ¾J’ "M„“ eK?‰” "M„“ eK?‰” "M„“ eK?‰” "M„“ eK?‰” (square triplets) u=•\U' dM„“ eK?‰”dM„“ eK?‰”dM„“ eK?‰”dM„“ eK?‰” (cubic triplets) Ó” ›”ÉU ¾KU TKƒ ’¨<:: u}Sddà G<’@� ’ , 4 u=J”

N4 , G4 + K4 ¾T>K¨<” 6Ÿ<M� K=Á[Ÿ< ¾T>‹K< Zeƒ Ë`+¡ lØa‹” (G' K' N) TÓ–ƒ ›Ã‰MU:: uK?L ›vvM lØ^†¨< ›=n”ä (infinite) (TKƒU ¾ƒ¾KK?) ¾J’ "M„“ eK?‰” "M„“ eK?‰” "M„“ eK?‰” "M„“ eK?‰” (square triplets) u=•\U' ^w„“ eK?‰”^w„“ eK?‰”^w„“ eK?‰”^w„“ eK?‰” (quartic triplets) Ó” ›”ÉU ¾KU TKƒ ’¨<:: vÖnLà ›’ÒÑ`' uð`T ¾SÚ[h Ñ”Ñ>d SW[ƒ

N’ , G’ + K’

¾T>K¨< 6Ÿ<M� K=[" (satisfied) ¾T>‹K¨< ’ , 1 ¨ÃU ’ , 2 ŸJ’ w‰ ’¨< TKƒ ’¨<::

¾ð`T ¾SÚ[h Ñ”Ñ>d: ¾ð`T ¾SÚ[h Ñ”Ñ>d: ¾ð`T ¾SÚ[h Ñ”Ñ>d: ¾ð`T ¾SÚ[h Ñ”Ñ>d: Ë`+¡ ’ > 2 ŸJ’' 6Ÿ<M�” N’ , G’ + K’ K=Á[Ÿ< ¾T>‹K< Zeƒ ¨”� Ë`+¢‹ (G' K' N) ¾K<U::

U”U 6”"D” ð`T Ñ”Ñ>d¨<” u=Ñ’Óc¨<U T[ÒÑݨ<” (proof) Ó”

›McÖU ’u`:: u²=IU U¡’>Áƒ ÃI” c=Á¿ƒ kLM ¾T>SeM Ñ”Ñ>d KT[ÒÑØ (prove) ›ÁK? �LLp N=dwc=’—¨‹ (mathematicians) ÁpT†¨<” ÁIM u=Ø\“ u=KñU K=d"L†¨< ›M‰KU ’u`:: uSÚ[h Ó” u— ›q×Ö` u1987 ¯.U (TKƒU ð`T Ñ”Ñ>d¨<” ŸÑ’Ñc Ÿ 358 ¯S�ƒ u%EL) 6”É`Áe ªÃMe6”É`Áe ªÃMe6”É`Áe ªÃMe6”É`Áe ªÃMe (Andrew Wiles) ¾}vK¨< 6”ÓK=³© N=dwc=’— Ÿw²< É"U“ Ø[ƒ u%EL K=Á[ÒÓÖ¨< ‹LDM::

H>"H>"H>"H>"

H>"H>"H>"H>" (glossary) TKƒ ¾nLƒ ´`´` TKƒ c=J”' nK< ¾}Ñ–¨< H>¢ (õ‹) ŸT>K¨< ¾*aU— nM ’¨<:: Ÿ²=IU H>"©H>"©H>"©H>"© (glossarial)' H>ŸH>ŸH>ŸH>Ÿ———— (glossarian) ¾T>K<ƒ” nKA‹ 6“Ñ—K”:: Óc<U c=[v H¾Ÿ (prepare glossary)' Iá' HÁŸ=' I¾" 6ÁK ÃH@ÇM:: H¾Ÿ TKƒ H>" ›²ÒË TKƒ c=J”' ¾6”ÓK²— ›‰ Ó” ¾K¨<U::

acute angle - g<M ²« acute triangle - g<M eK?Ô” adjacent - LÊ adjacent (non) - ›=LÊ adjacent sides - LÊ Ô“‹ adjacent vertices - LÊ òר‹ base - c=T¡ base (verb) - cSŸ' eU¡' cTŸ=' /e/S"' eUŸƒ basic - c=T"©

basic mathematics - c=T¡ (c=T"©) N=dwc=” deca (deka) - ›c?' ¢Í decade - Ÿ<S”' Ÿ<S“© decagon - ›c?Ô” demonstrate - [Ÿ’c' `¡”e' [¡“i' `¡’d demonstrating - `¡’d demonstration - [¡’e' `¡’d demonstrative - [¡“i' `¡’d© demonstrator - [¡“i

diagonal - c<Ïd diagram - c<Íõ diagram (to) - cËð' eÏõ' cÍò' eËó diagramatic - c<Íó©' cÍò diagramatical - c<Íó©' cÍò dodeca - Ÿ<¡K? dodecagon - Ÿ<¡K?Ô” dodecahedron - Ÿ<¡K?Ñê equi - 6Ÿ<M equiangular - 6Ÿ<M²«ÁU equiangular polygon - 6Ÿ<M²«ÁU ‡TÔ” equiangular triangle - 6Ÿ<M²«ÁU eK?Ô” equilateral - 6Ÿ<MÔ“U equilateral polygon - 6Ÿ<MÔ“U ‡TÔ” equilateral triangle - 6Ÿ<MÔ“U eK?Ô” exterior - ¨<Ý© exterior angle - ¨<Ý© ²« Fermat's last theorem - ¾ð`T ¾SÚ[h Ñ”Ñ>d figuration - c<`ð" figurative - c<`ò"© figurativeness - c<`ò"©’ƒ figure - c<`ò¡ figure (to) - f[ðŸ' c<`õ¡' f`óŸ=' c<`ð" figure-head - c<`ò¡d hepta - du? heptagon- du?Ô” hexa - dÈ hexagon - dÈÔ” hypothenuse - +v icosa - L+T icosagon - L+TÔ” icosahedron - L+TÑê illustrate - cKu[' eMw`' cMv]' eMu^ illustrating - eMu^ illustration - cMu`' eMu^ illustrative - cMv]' cMu^© illustrator - cMv] illustrious - cMu[— interior - ¨<eש interior angle - ¨<eש ²«

intersect - Å[†u' É`‹w'Å`‰u=' É`†v intersecting - Å`‰u=' É`†v intersecting lines - Å`‰u= SeSa‹ intersection - Å`†w irregular - ›=Ë`+U irregular polygon - ›=Ë`+U ‡TÔ” irregularize - ›=Ë[}S' ›=Ï`ƒU' ›=Ë`�T>' ›=Ï`}T isosceles - 6Ÿ<M6Ó^U isosceles trapezium - 6Ÿ<M6Ó^U â?³^w isosceles triangle - 6Ÿ<M6Ó^U eK?Ô” kite - Jv n-gon - ’-Ô” nona - �c? nonagon - �c?Ô” obtuse - ´`ØØ obtuse angle - ´`ØØ ²« obtuse triangle - ´`ØØ eK?Ô” octa - dT@ octagon - dT@Ô” opposite - TÊ parallel - Ÿ<MŸ<ƒ

parallel (make) - ¢K¢}' Ÿ<MŸ<ƒ' ¢M"D‹' Ÿ<M¢� parallel lines - Ÿ<MŸ<ƒ SeSa‹ parallelism - Ÿ<MŸ<ƒ’ƒ' Ÿ<MŸ<ƒõ“ parallelist - Ÿ<MŸ<ƒð“˜ parallelogram - Ÿ<MŸ?^w penta- HT@ pentagon - HT@Ô” peri - “• perimeter - “•M¡ period - “•S” periodic - “•S”' “•S“©

periodic motion - “•S” (“•S“©) Ocƒ periodic table - “•S” â?³ periodical - “•N?ƒ

perpendicular - \”Ò perpendicular (make, become) - [’Ñ' `”Ó' [“Ñ>' `’Ò perpendicular bisector - \”Ò (©) ¡K?ÑTi perpendicular distance - \”Ò ``kƒ

perpendicular lines - \”Ò SeSa‹ planar - úÇ© planar figure - úÇ© c<`ò¡ planar mirror - úÇ(©) Se�¨ƒ planar surface - úÇ© ççõ plane - úÇ plane (verb) - ø¾Å' ýÃÉ' øÁÏ' ý¾Ç poly - ‡T polygon - ‡TÔ” polyhedron - ‡TÑê Pythagorean theorem (square theorem) - "M„“ Ñ”Ñ>d Pythagorean triplets (square triplets) - "M„“ eK?‰” quadrilaterial - ^u?Ô” rectangle - \”Ñ@^w rectangular - \”Ñ@^v© regular - Ë`+U regular polygon - Ë`+U ‡TÔ” regularize - Ë[}S' Ï`ƒU' Ë`�T>' Ï`}T rhombus - ›‰^w right - ›<h‰ right angle - ›<h‰ ²« right kite - ›<h‰ Jv right trapezium - ›<h‰ â?³^w right triangle - ›<h‰ eK?Ô” same - c=û same (not) - ›=c=û scalen - 6Ÿ<Mu=e scalen triangle - 6Ÿ<Mu=e eK?Ô”

square - "M„“' "M„“© square theorem (Pythagorean theorem) - "M„“ Ñ”Ñ>d square triplets (Pythagorean triplets) - "M„“ eK?‰” superfical - ççó© (ççð—) superficiality - ççó©’ƒ (ççð—’ƒ) superficialness - ççó©’ƒ (ççð—’ƒ) surface - ççõ surface (verb) - ççð' êêõ' çéò' êçó surface area - ççõ eóƒ surface tension - ççõ ¨<Ø[ƒ surfacing - êçó' çéò theorem - Ñ”Ñ>d theorem (to) - Ñ’Ñc' Ó”Óe' Ñ”Òi' Ó”Ñd theoremic - Ñ”Ñ>d© theoremical - Ñ”Ñ>d© theoremist - Ñ”Òi' Ñ”Ñ>c— theoretical - Ӕѓ© theorist - єҘ theorize - Ñ’Ñ’' Ó”Ó”' єҘ' Ӕѓ theory - Ӕѓ' Ӕѓ© trapezium (trapezoid) - â?³^w trapezoid (trapezium) - â?³^w

tri- eK? triangle - eK?Ô” vertex - ò×

eL’uu<˜ ›¢‚›¢‚›¢‚›¢‚ (thanks):: uK?L ÙT` (article) 6eŸU”Ñ—˜ É[e ÅI“ Ãc”w~' ›wN?` (God) - ¾Ùu=Á ›UL¡ - ÃÖwp¨::

Seõ” ›[Ò ²’ÑÅ Ÿ<i ¾ƒw~ kw` Åc?' dLÃi::

mesfin.arega@gmail.com