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平行線と角の問題の解法 | 夢を叶える塾
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平行線と角の問題の解法

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平行線と角の性質を使って角を求める問題です。

基本問題

(1)次の図の\angle{CBA}を求めなさい。
平行線と角_1
BAmが作る鋭角=45^{\circ}
BClが作る鋭角=50^{\circ}
l//m

(2)次の図の\angle{BDA}を求めなさい。
平行線と角_3
DAmが作る鋭角=65^{\circ}
BClが作る鋭角=45^{\circ}
\angle{CBD}=60^{\circ}
l//m

解き方

手掛かりは「平行線」であること位ですね。
そのため、平行線の性質を使います。
しかし折れてしまっているのでこの折れている部分にも平行線の性質を使えるようにしましょう。
折れている点に平行線を引く」ことがポイントです。

解説

(1)次の図の\angle{CBA}を求めなさい。
問題の情報はすべて図に書き込んでくださいね。

解き方のとおり、Bを通る平行線を引きましょう。
平行線と角_2

これで平行線の錯角や同位角が等しいという性質を使えます

平行線l,nの錯角が等しくなる性質を利用して、
\angle{CBD}=50^{\circ}
平行線m,nの錯角が等しくなる性質を利用して、
\angle{ABD}=45^{\circ}

知りたい角である\angle{CBA}の部分の角がわかりました。
\angle{CBA}=\angle{CBD}+\angle{ABD}=50^{\circ}+45^{\circ}=95^{\circ}

(2)次の図の\angle{BDA}を求めなさい。
解き方のとおり、B,Dを通る平行線を引きましょう。
平行線と角_4

これで平行線の錯角や同位角が等しいという性質を使えます

平行線l,n_2の錯角が等しくなる性質を利用して、
\angle{CBF}=45^{\circ}
平行線m,n_1の錯角が等しくなる性質を利用して、
\angle{ADE}=65^{\circ}

\angle{CBD}=60^{\circ}でした。
\angle{CBD}=\angle{CBF}+\angle{DBF}なので、ここにわかった角を代入して
60^{\circ}=45^{\circ}+\angle{DBF}となるので、\angle{DBF}=15^{\circ}になります。

平行線n_1,n_2の錯角が等しくなる性質も利用できますね
\angle{BDE}=\angle{DBF}=15^{\circ}

知りたい角である\angle{BDA}の部分の角がわかりました。
\angle{BDA}=\angle{BDE}+\angle{ADE}=15^{\circ}+65^{\circ}=80^{\circ}

応用問題

次の図の\angle{BDA}を求めなさい。
平行線と角_5
DAmが作る鋭角=70^{\circ}
BClが作る鋭角=40^{\circ}
\angle{CBD}=15^{\circ}
l//m

解き方

応用問題ですが基本問題と同じです。
飛び出ちゃってますが、同じ考えで解くことができます。

解説

次の図の\angle{BDA}を求めなさい。
解き方のとおり、B,Dを通る平行線を引きましょう。
平行線と角_6

これで平行線の錯角や同位角が等しいという性質を使えます

平行線l,n_2の錯角が等しくなる性質を利用して、
\angle{CBF}=40^{\circ}
平行線m,n_1の錯角が等しくなる性質を利用して、
\angle{ADE}=70^{\circ}

\angle{CBD}=15^{\circ}でした。
\angle{CBF}=\angle{CBD}+\angle{DBF}なので、ここにわかった角を代入して
40^{\circ}=15^{\circ}+\angle{DBF}となるので、\angle{DBF}=25^{\circ}になります。

平行線n_1,n_2の錯角が等しくなる性質も利用できますね
\angle{BDE}=\angle{DBF}=25^{\circ}

\angle{ADE}を作る角が\angle{BDA}以外すべてわかりました。
\angle{ADE}=\angle{BDA}+\angle{BDE}にその値を代入して、
70^{\circ}=\angle{BDA}+25^{\circ}となるので、\angle{BDA}=45^{\circ}になります。

なお、ほとんど基本問題(2)そのままです

終わりに

平行線の性質は数多くあります。
図形の性質を組み合わせた応用問題の中でこの性質が使われることもよくあります。

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