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合同

面積が同じで、かつ、形も同じ図形は、合同(ごうどう)になる。

例えば、\( \bigtriangleup ABC \) と\( \bigtriangleup DEF \) は、合同になる。

面積も形も、同じなので、合同になる。

\begin{xy} (0,0)*{A}="A", (0,-27)*{C}="A", (15,-27)*{B}="A", (2,-14)*{6}="A", (8,-20.5)*{4}="A", {(0,-3) \ar @{-}(0,-23)}, {(0,-23) \ar @{-}(15,-23)}, {(0,-3) \ar @{-}(15,-23)}, {(2,-21) \ar @{-}(2,-23)}, {(2,-21) \ar @{-}(0,-21)}, (30,0)*{D}="A", (30,-27)*{F}="A", (45,-27)*{E}="A", (32,-14)*{6}="A", (38,-20.5)*{4}="A", {(30,-3) \ar @{-}(30,-23)}, {(30,-23) \ar @{-}(45,-23)}, {(30,-3) \ar @{-}(45,-23)}, {(32,-21) \ar @{-}(32,-23)}, {(32,-21) \ar @{-}(30,-21)}, \end{xy}

しかし、\( \bigtriangleup ABC \) と\( \bigtriangleup GHI \) は、合同ではない。

面積は同じだが、形は同じではないので、合同ではない。

\begin{xy} (0,0)*{A}="A", (0,-27)*{C}="A", (15,-27)*{B}="A", (2,-14)*{6}="A", (8,-20.5)*{4}="A", {(0,-3) \ar @{-}(0,-23)}, {(0,-23) \ar @{-}(15,-23)}, {(0,-3) \ar @{-}(15,-23)}, {(2,-21) \ar @{-}(2,-23)}, {(2,-21) \ar @{-}(0,-21)}, (30,-10)*{G}="A", (30,-27)*{I}="A", (53,-27)*{H}="A", (32,-18)*{3}="A", (42,-26)*{8}="A", {(30,-13) \ar @{-}(30,-23)}, {(30,-23) \ar @{-}(55,-23)}, {(30,-13) \ar @{-}(55,-23)}, {(32,-21) \ar @{-}(32,-23)}, {(32,-21) \ar @{-}(30,-21)}, \end{xy}

合同の記号

合同は、数学の記号で、\( \equiv \) と、書く。

例えば、合同な図形、\( \bigtriangleup ABC \) と\( \bigtriangleup DEF \) は、\( \bigtriangleup ABC \equiv \bigtriangleup DEF \) と、書く。

合同の記号( \( \equiv \) )は、等号( = )に、さらに横線を1本追加し、3本の横線になっている。

\begin{xy} (20,10)*{\bigtriangleup ABC \equiv \bigtriangleup DEF}="A", (0,0)*{A}="A", (0,-27)*{C}="A", (15,-27)*{B}="A", (2,-14)*{6}="A", (8,-20.5)*{4}="A", {(0,-3) \ar @{-}(0,-23)}, {(0,-23) \ar @{-}(15,-23)}, {(0,-3) \ar @{-}(15,-23)}, {(2,-21) \ar @{-}(2,-23)}, {(2,-21) \ar @{-}(0,-21)}, (30,0)*{D}="A", (30,-27)*{F}="A", (45,-27)*{E}="A", (32,-14)*{6}="A", (38,-20.5)*{4}="A", {(30,-3) \ar @{-}(30,-23)}, {(30,-23) \ar @{-}(45,-23)}, {(30,-3) \ar @{-}(45,-23)}, {(32,-21) \ar @{-}(32,-23)}, {(32,-21) \ar @{-}(30,-21)}, \end{xy}

また、合同な図形は、頂点の順番を、対応させて書かなければいけない。

例えば、\( \bigtriangleup ABC \equiv \bigtriangleup DEF \) は正しいが、\( \bigtriangleup ABC \equiv \bigtriangleup DFE \) は間違いだ。

きちんと頂点を対応させて\( \bigtriangleup ACB \equiv \bigtriangleup DFE \) と書けば、正しくなる。




<例題 \( \Large 1 \) >\( \bigtriangleup ABC \) と合同な3角形を見つけ、合同の記号で表しなさい。なお、対応する頂点の順番には、気をつけなさい。

\begin{xy} (0,0)*{A}="A", (0,-27)*{C}="A", (15,-27)*{B}="A", (2,-14)*{6}="A", (8,-20.5)*{4}="A", {(0,-3) \ar @{-}(0,-23)}, {(0,-23) \ar @{-}(15,-23)}, {(0,-3) \ar @{-}(15,-23)}, {(2,-21) \ar @{-}(2,-23)}, {(2,-21) \ar @{-}(0,-21)}, (30,0)*{G}="A", (45,-27)*{I}="A", (45,0)*{H}="A", (42.5,-14)*{6}="A", (38,-6)*{4}="A", {(30,-3) \ar @{-}(45,-3)}, {(30,-3) \ar @{-}(45,-23)}, {(45,-3) \ar @{-}(45,-23)}, {(43,-3) \ar @{-}(43,-5)}, {(45,-5) \ar @{-}(43,-5)}, \end{xy} \begin{xy} (0,0)*{J}="A", (0,-20)*{L}="A", (21,-20)*{K}="A", (2,-8)*{4}="A", (11,-14)*{6}="A", {(0,-3) \ar @{-}(0,-16)}, {(0,-16) \ar @{-}(23,-16)}, {(0,-3) \ar @{-}(23,-16)}, {(2,-14) \ar @{-}(2,-16)}, {(2,-14) \ar @{-}(0,-14)}, (31,-27)*{O}="A", (45,-27)*{N}="A", (45,0)*{M}="A", (42.5,-14)*{6}="A", (38,-20.5)*{4}="A", {(30,-23) \ar @{-}(45,-3)}, {(30,-23) \ar @{-}(45,-23)}, {(45,-3) \ar @{-}(45,-23)}, {(43,-23) \ar @{-}(43,-21)}, {(43,-21) \ar @{-}(45,-21)}, \end{xy}

<解答 \( \Large 1 \) >
\( \bigtriangleup ABC \equiv \bigtriangleup IGH \)
\( \bigtriangleup ABC \equiv \bigtriangleup KJL \)
\( \bigtriangleup ABC \equiv \bigtriangleup MON \)
(3角形の頂点の順番に気をつけましょう)

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