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>E直角3角形の合同条件

ふつうの3角形と直角3角形

直角3角形は、3角形に含まれる。だから、ふつうの3角形の合同条件でも、もちろん証明できる。

しかし、直角3角形の特別な合同条件を、利用すると、もっとすばやく証明ができる。

直角3角形の合同条件

直角3角形の合同条件は、2つある。


合同条件1:直角3角形の、斜辺と1鋭角が、それぞれ等しい。

\begin{xy} (10,-8.5)*{\circ}="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)}, (18,-12.5)*{}="E"; (17,-16.5)*{}="F"; "E"; "F" **\crv{(17,-14)}; (40,-8.5)*{\circ}="A", {(30,-3) \ar @{-}(30,-16)}, {(30,-16) \ar @{-}(53,-16)}, {(30,-3) \ar @{-}(53,-16)}, {(32,-14) \ar @{-}(32,-16)}, {(32,-14) \ar @{-}(30,-14)}, (48,-12.5)*{}="E"; (47,-16.5)*{}="F"; "E"; "F" **\crv{(47,-14)}; \end{xy}

合同条件2:直角3角形の、斜辺と他の1辺が、それぞれ等しい。

\begin{xy} (10,-8.5)*{\circ}="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)}, {(10,-17) \ar @{-}(10,-15)}, (40,-8.5)*{\circ}="A", {(30,-3) \ar @{-}(30,-16)}, {(30,-16) \ar @{-}(53,-16)}, {(30,-3) \ar @{-}(53,-16)}, {(32,-14) \ar @{-}(32,-16)}, {(32,-14) \ar @{-}(30,-14)}, {(40,-17) \ar @{-}(40,-15)}, \end{xy}

直角3角形の合同条件は、斜辺とあと1つと、覚えよう。



<例題 \( \Large 1 \) >

以下から合同な直角3角形を見つけ、合同の記号で表しなさい。また、合同条件も、書きなさい。

\begin{xy} (0,0)*{\small A}="A", (0,-26)*{\small C}="A", (15,-26)*{\small B}="A", (10,-13)*{\small 7}="A", (8,-25.5)*{\small 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,-8)*{\small D}="A", (30,-26)*{\small F}="A", (53,-26)*{\small E}="A", (43,-21)*{\small 20 ^\circ}="A", (42,-14)*{\small 7}="A", {(30,-11) \ar @{-}(30,-23)}, {(30,-23) \ar @{-}(55,-23)}, {(30,-11) \ar @{-}(55,-23)}, {(32,-21) \ar @{-}(32,-23)}, {(32,-21) \ar @{-}(30,-21)}, (48,-19)*{}="E"; (47,-23.5)*{}="F"; "E"; "F" **\crv{(47,-21)}; \end{xy} \begin{xy} (0,0)*{\small G}="A", (0,-19)*{\small I}="A", (21,-19)*{\small H}="A", (13,-8)*{\small 7}="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)*{\small L}="A", (45,-27)*{\small K}="A", (45,0)*{\small J}="A", (37,-11)*{\small 7}="A", (38,-25.5)*{\small 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} \begin{xy} (25,-0)*{\small M}="A", (25,-18)*{\small O}="A", (0,-18)*{\small N}="A", (13,-13)*{\small 20 ^\circ}="A", (12,-6)*{\small 7}="A", {(25,-3) \ar @{-}(25,-15)}, {(25,-15) \ar @{-}(0,-15)}, {(25,-3) \ar @{-}(0,-15)}, {(23,-13) \ar @{-}(23,-15)}, {(23,-13) \ar @{-}(25,-13)}, (7,-11)*{}="E"; (8,-15.5)*{}="F"; "E"; "F" **\crv{(8,-13)}; (55,0)*{\small P}="A", (32,-19)*{\small Q}="A", (53,-19)*{\small R}="A", (45,-6)*{\small 7}="A", {(55,-3) \ar @{-}(55,-16)}, {(55,-16) \ar @{-}(32,-16)}, {(55,-3) \ar @{-}(32,-16)}, {(53,-14) \ar @{-}(53,-16)}, {(53,-14) \ar @{-}(55,-14)}, \end{xy}

<解答 \( \Large 1 \) >
\( \bigtriangleup ABC \equiv \bigtriangleup JLK \)
合同条件:直角3角形の、斜辺と他の1辺が、それぞれ等しい。

\( \bigtriangleup DEF \equiv \bigtriangleup MNO \)
合同条件:直角3角形の、斜辺と1鋭角が、それぞれ等しい。

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