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‫وأص‬ ‫وأصفها‬،‫األبعاد‬ ‫ثنائية‬ ‫األشكال‬ ‫أتعرف‬
‫نفها‬
‫الزاوية‬ ‫القائم‬ ‫المثلث‬
‫الزاوية‬ ‫الحاد‬ ‫المثلث‬
‫الزاوية‬ ‫المنفرج‬ ‫المثلث‬
‫الضلعين‬ ‫المتطابق‬ ‫المثلث‬
‫األضالع‬ ‫المتطابق‬ ‫المثلث‬
‫األضالع‬ ‫المختلف‬ ‫المثلث‬
‫ى‬َ‫ل‬ِ‫إ‬ ٌ‫ة‬َ‫َطير‬‫ش‬ ْ‫ت‬َ‫م‬ِِّ‫س‬ُ‫ق‬
‫ا‬َ‫م‬ ِ‫ن‬ْ‫ي‬َ‫ف‬ْ‫ص‬ِ‫ن‬
ُ‫ل‬ِِّ‫ث‬َ‫م‬ُ‫ي‬ ‫ذي‬َّ‫ل‬‫ا‬ ُ‫ل‬ْ‫ك‬َّ‫ش‬‫ال‬
ُ‫ه‬
‫َّط‬‫ش‬‫ال‬ َ‫ن‬ِ‫م‬ ٍ‫ف‬ْ‫ص‬ِ‫ن‬ ُّ‫ل‬ُ‫ك‬
‫؟‬ِ‫ة‬َ‫ير‬
‫عن‬ ‫عبارة‬ ‫كانت‬ ‫الشطيرة‬ ‫أن‬ ‫الشكل‬ ‫من‬ ‫الواضح‬
‫مثلثين‬ ‫نتج‬ ‫القطر‬ ‫عند‬ ‫تقسيمها‬ ‫وعند‬،‫مربع‬
.
ِ‫ت‬‫ثا‬َّ‫ل‬َ‫ث‬ُ‫م‬‫ال‬ َ‫ن‬ِ‫م‬ ٌ‫ة‬َّ‫د‬ِ‫ع‬ ٌ‫ع‬‫وا‬ْ‫ن‬َ‫أ‬ َ‫ناك‬ُ‫ه‬
.
ُ‫نيف‬ْ‫ص‬َ‫ت‬ َ‫ك‬ِ‫ن‬‫ا‬ْ‫ك‬‫إم‬ِ‫ب‬َ‫و‬
َ‫ز‬ ِ‫ت‬‫ياسا‬ِ‫ق‬ ‫َلى‬‫ع‬ ‫ًا‬‫د‬‫ما‬ِ‫ت‬ْ‫ع‬‫ا‬ ِ‫ت‬‫ثا‬َّ‫ل‬َ‫ث‬ُ‫م‬‫ال‬
‫واياها‬
.
‫مثال‬
‫وايا‬َّ‫للز‬ ِ‫ة‬َ‫ب‬‫س‬ِِّ‫ن‬‫بال‬ ُ‫صنيف‬َّ‫ت‬‫ال‬
1
َ‫ب‬‫س‬ِِّ‫ن‬‫بال‬ َ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬‫ال‬ ِ‫ف‬ِِّ‫ن‬َ‫ص‬
‫لزواياه‬ ِ‫ة‬
.
ُ‫م‬‫ال‬ َّ‫إن‬َ‫ف‬ ،ً‫ة‬َ‫ج‬ ِ
‫ر‬َ‫ف‬ْ‫ن‬ُ‫م‬ ً‫ة‬َ‫ي‬ِ‫و‬‫زا‬ َ‫ناك‬ُ‫ه‬ َّ‫ن‬َ‫أ‬ ‫ما‬ِ‫ب‬
ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ُ‫ج‬ ِ
‫ر‬َ‫ف‬ْ‫ن‬ُ‫م‬ َ‫ث‬َّ‫ل‬َ‫ث‬
.
َ‫ح‬ ِ‫ت‬‫ا‬َ‫ث‬َّ‫ل‬َ‫ث‬‫الم‬ َ‫ف‬ِِّ‫ن‬َ‫ص‬ُ‫ن‬ ‫أن‬ ‫ا‬ً‫ض‬‫ي‬َ‫أ‬ ُ‫ن‬ِ‫ك‬ْ‫م‬ُ‫ي‬
‫العها‬ْ‫ض‬َ‫أ‬ ِ‫ل‬‫وا‬ْ‫ط‬َ‫أ‬ َََ‫س‬
.
‫مثال‬
‫ا‬َ‫ي‬‫ا‬َ‫و‬َّ‫للز‬ ِ‫ة‬‫سب‬ِِّ‫ن‬‫بال‬ ُ‫صنيف‬َ‫ت‬‫ال‬
ِ‫واألضالع‬
.
2
َ‫ب‬ْ‫س‬ِِّ‫ن‬‫بال‬ َ‫ث‬َ‫ل‬َّ‫ث‬ُ‫م‬‫ال‬ ِ‫ف‬ِِّ‫ن‬‫ص‬
ُ‫ه‬‫ا‬َ‫ي‬‫وا‬َ‫ز‬ِ‫ل‬ ِ‫ة‬
‫ه؟‬ِ‫ع‬َ‫ال‬ْ‫ض‬‫وأل‬
َ‫أ‬ ‫منها‬ ٍِّ‫ل‬‫ك‬ َ
‫قياس‬ ‫ا‬َ‫ي‬‫وا‬َ‫ز‬ ُ‫الث‬َ‫ث‬ ُ‫ه‬َ‫ل‬ ُ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬‫ال‬ ‫ا‬ِ‫ذ‬‫ه‬
ْ‫ن‬ِ‫م‬ ُّ‫ل‬َ‫ق‬
90
‫ا‬ً‫ذ‬‫إ‬ ،
‫وايا‬َّ‫الز‬ ُّ‫د‬‫حا‬ ُ‫ث‬َّ‫ل‬‫فالمث‬
.
َ‫أل‬‫ا‬ ُ‫ق‬‫تطاب‬ُ‫م‬ َ‫و‬ُ‫ه‬‫ف‬ ‫ا‬ً‫ذ‬‫إ‬ ،ٌ‫طابقة‬َ‫ت‬ُ‫م‬ ِ‫ه‬ِ‫ع‬‫ال‬ْ‫ض‬َ‫أ‬ ُّ‫ل‬ُ‫ك‬‫و‬
‫ا‬ً‫ض‬ْ‫ي‬َ‫أ‬ ِ‫الع‬ْ‫ض‬
.
‫إلى‬ ‫أتي‬َ‫ي‬ ‫ا‬ِّ‫م‬ِ‫م‬ ٍ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬ َّ‫ل‬ُ‫ك‬ ْ
‫ف‬ِِّ‫ن‬َ‫ص‬
:
َ‫أ‬ ،‫وايا‬َّ‫الز‬ ِِّ‫د‬‫حا‬
ِ‫ج‬ ِ
‫ر‬َ‫ف‬ْ‫ن‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ِ‫م‬ِ‫ئ‬‫قا‬ ْ‫و‬
‫ا‬ ِ‫تطابق‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ن‬ْ‫ي‬َ‫ع‬ْ‫ل‬‫ض‬َّ‫ل‬‫ا‬ ِ‫ابق‬َ‫ط‬‫ت‬ُ‫م‬ ‫إلى‬َ‫و‬ ،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬
ِ‫ف‬ِ‫ل‬َ‫ت‬ْ‫خ‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫الع‬ْ‫ض‬َ‫أل‬
ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬
.
1
‫المثلث‬ ‫هذا‬
‫به‬
‫من‬ ‫أكبر‬ ‫قياسها‬ ‫زاوية‬
90
‫منفرج‬ ‫فهو‬ ً‫ا‬‫ْإذ‬‫؛‬
‫الزاوية‬
.
‫وبه‬
‫متطاب‬ ‫فهو‬ ً‫ا‬‫متساويين؛إذ‬ ‫ضلعين‬ ً‫ا‬‫أيض‬
‫ق‬
‫الضلعين‬
.
‫إلى‬ ‫أتي‬َ‫ي‬ ‫ا‬ِّ‫م‬ِ‫م‬ ٍ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬ َّ‫ل‬ُ‫ك‬ ْ
‫ف‬ِِّ‫ن‬َ‫ص‬
:
َ‫أ‬ ،‫وايا‬َّ‫الز‬ ِِّ‫د‬‫حا‬
ِ‫ج‬ ِ
‫ر‬َ‫ف‬ْ‫ن‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ِ‫م‬ِ‫ئ‬‫قا‬ ْ‫و‬
‫ا‬ ِ‫تطابق‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ن‬ْ‫ي‬َ‫ع‬ْ‫ل‬‫ض‬َّ‫ل‬‫ا‬ ِ‫ابق‬َ‫ط‬‫ت‬ُ‫م‬ ‫إلى‬َ‫و‬ ،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬
ِ‫ف‬ِ‫ل‬َ‫ت‬ْ‫خ‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫الع‬ْ‫ض‬َ‫أل‬
ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬
.
2
‫من‬ ‫أقل‬ ‫منها‬ ‫كل‬ ‫قياس‬ ‫زوايا‬ ‫ثالث‬ ‫له‬ ‫المثلث‬ ‫هذا‬
90
̊
،
‫الزوايا‬ ‫حاد‬ ‫فالمثلث‬ ً‫ا‬‫إذ‬
.
،‫متطابقة‬ ‫أضالعه‬ ‫وكل‬
‫فه‬ ً‫ا‬‫إذ‬
‫و‬
‫األضالع‬ ‫متطابق‬
.
‫إلى‬ ‫أتي‬َ‫ي‬ ‫ا‬ِّ‫م‬ِ‫م‬ ٍ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬ َّ‫ل‬ُ‫ك‬ ْ
‫ف‬ِِّ‫ن‬َ‫ص‬
:
َ‫أ‬ ،‫وايا‬َّ‫الز‬ ِِّ‫د‬‫حا‬
ِ‫ج‬ ِ
‫ر‬َ‫ف‬ْ‫ن‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ِ‫م‬ِ‫ئ‬‫قا‬ ْ‫و‬
‫ا‬ ِ‫تطابق‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ن‬ْ‫ي‬َ‫ع‬ْ‫ل‬‫ض‬َّ‫ل‬‫ا‬ ِ‫ابق‬َ‫ط‬‫ت‬ُ‫م‬ ‫إلى‬َ‫و‬ ،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬
ِ‫ف‬ِ‫ل‬َ‫ت‬ْ‫خ‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫الع‬ْ‫ض‬َ‫أل‬
ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬
.
3
‫المثلث‬ ‫هذا‬
‫به‬
‫يساوي‬ ‫قياسها‬ ‫زاوية‬
90
̊
‫قائم‬ ‫فالمثلث‬ ً‫ا‬‫إذ‬،
‫الزاوية‬
.
‫األضالع‬ ‫مختلف‬ ‫فهو‬ ً‫ا‬‫إذ‬،‫مختلفة‬ ‫أضالعه‬ ‫وأطوال‬
.
4
‫تحدث‬
ٍ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬ ْ‫ي‬َ‫ع‬ْ‫ل‬ ِ
‫ض‬ ْ‫ي‬َ‫ل‬‫و‬ُ‫ط‬ ُ‫ع‬‫مجمو‬ َ‫كان‬ ‫إذا‬
ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬ ِ‫متطابق‬
6
،‫سم‬
َ‫ب‬‫إجا‬ ْ‫ح‬َ‫ْر‬‫ش‬‫ا‬ ‫؟‬ِ‫ث‬ِ‫ل‬‫ا‬ِّ‫ث‬‫ال‬ ِ‫ع‬ْ‫ل‬ِّ ِ
‫الض‬ ُ‫ل‬‫طو‬ ‫ما‬َ‫ف‬
‫ك‬َ‫ت‬
‫أضالعه‬ ‫فجميع‬ ً‫ا‬‫إذ‬ ،‫األضالع‬ ‫متطابق‬ ‫المثلث‬ ‫أن‬ ‫بما‬
‫يساوي‬ ‫ضلعيه‬ ‫مجموع‬ ‫أن‬ ‫بما‬ ‫متساوية‬
6
‫بقسمة‬ ً‫ا‬‫إذ‬ ،‫سم‬
6
‫على‬
2
‫الضلع‬ ‫طول‬ ً‫ا‬‫إذ‬ ‫الضلع‬ ‫طول‬ ‫على‬ ‫نحصل‬
‫يساوي‬
3
‫الضلع‬ ‫طول‬ ‫إذا‬، ‫األضالع‬ ‫متطابق‬ ‫أنه‬ ‫بما‬ ‫سم‬
‫يساوي‬ ‫الثالث‬
3
‫سم‬
.
‫إلى‬ ‫أتي‬َ‫ي‬ ‫ا‬ِّ‫م‬ِ‫م‬ ٍ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬ َّ‫ل‬ُ‫ك‬ ْ
‫ف‬ِِّ‫ن‬َ‫ص‬
:
ِ‫م‬ِ‫ئ‬‫قا‬ ْ‫و‬َ‫أ‬ ،‫وايا‬َّ‫الز‬ ِِّ‫د‬‫حا‬
،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ِ‫ج‬ ِ
‫ر‬َ‫ف‬ْ‫ن‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬
ْ‫خ‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬ ِ‫تطابق‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ن‬ْ‫ي‬َ‫ع‬ْ‫ل‬‫ض‬َّ‫ل‬‫ا‬ ِ‫ابق‬َ‫ط‬‫ت‬ُ‫م‬ ‫إلى‬َ‫و‬
ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬ ِ‫ف‬ِ‫ل‬َ‫ت‬
.
5
‫من‬ ‫أقل‬ ‫منها‬ ‫كل‬ ‫قياس‬ ‫زوايا‬ ‫ثالث‬ ‫له‬ ‫المثلث‬ ‫هذا‬
90
̊
ً‫ا‬‫إذ‬،
‫الزوايا‬ ‫حاد‬ ‫فالمثلث‬
.
‫أضالعه‬ ‫كل‬ ‫أن‬ ‫وبما‬
‫متساوية‬
‫فهو‬ ‫إذا‬،
‫األضالع‬ ‫متطابق‬
.
‫إلى‬ ‫أتي‬َ‫ي‬ ‫ا‬ِّ‫م‬ِ‫م‬ ٍ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬ َّ‫ل‬ُ‫ك‬ ْ
‫ف‬ِِّ‫ن‬َ‫ص‬
:
ِ‫م‬ِ‫ئ‬‫قا‬ ْ‫و‬َ‫أ‬ ،‫وايا‬َّ‫الز‬ ِِّ‫د‬‫حا‬
،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ِ‫ج‬ ِ
‫ر‬َ‫ف‬ْ‫ن‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬
ْ‫خ‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬ ِ‫تطابق‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ن‬ْ‫ي‬َ‫ع‬ْ‫ل‬‫ض‬َّ‫ل‬‫ا‬ ِ‫ابق‬َ‫ط‬‫ت‬ُ‫م‬ ‫إلى‬َ‫و‬
ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬ ِ‫ف‬ِ‫ل‬َ‫ت‬
.
6
‫المثلث‬ ‫هذا‬
‫به‬
‫من‬ ‫أقل‬ ‫منها‬ ‫كل‬ ‫قياس‬ ‫زوايا‬ ‫ثالث‬
90
̊
،
‫الزوايا‬ ‫حاد‬ ‫مثلث‬ ‫فهو‬ ً‫ا‬‫إذ‬
.
‫مختلفة‬ ‫أضالعه‬ ‫جميع‬ ‫أن‬ ‫وبما‬
‫في‬
‫األضالع‬ ‫مختلف‬ ‫مثلث‬ ‫فهو‬ ً‫ا‬‫إذ‬،‫الطول‬
.
‫إلى‬ ‫أتي‬َ‫ي‬ ‫ا‬ِّ‫م‬ِ‫م‬ ٍ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬ َّ‫ل‬ُ‫ك‬ ْ
‫ف‬ِِّ‫ن‬َ‫ص‬
:
ِ‫م‬ِ‫ئ‬‫قا‬ ْ‫و‬َ‫أ‬ ،‫وايا‬َّ‫الز‬ ِِّ‫د‬‫حا‬
،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ِ‫ج‬ ِ
‫ر‬َ‫ف‬ْ‫ن‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬
ْ‫خ‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬ ِ‫تطابق‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ن‬ْ‫ي‬َ‫ع‬ْ‫ل‬‫ض‬َّ‫ل‬‫ا‬ ِ‫ابق‬َ‫ط‬‫ت‬ُ‫م‬ ‫إلى‬َ‫و‬
ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬ ِ‫ف‬ِ‫ل‬َ‫ت‬
.
‫المثلث‬ ‫هذا‬
‫به‬
‫يساوي‬ ‫قياسها‬ ‫زاوية‬
90
̊
‫قائم‬ ‫مثلث‬ ‫فهو‬ ً‫ا‬‫إذ‬،
‫الزاوية‬
.
،‫الطول‬ ‫في‬ ‫مختلفة‬ ‫أضالعه‬ ‫جميع‬ ‫أن‬ ‫وبما‬
‫األضالع‬ ‫مختلف‬ ‫مثلث‬ ‫فهو‬ ً‫ا‬‫إذ‬
.
7
8
ُ‫م‬َّ‫ل‬ُّ‫س‬‫ال‬ ُ‫ه‬ُ‫ع‬َ‫ن‬ْ‫ص‬َ‫ي‬ ‫ذي‬َّ‫ل‬‫ا‬ َ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬‫ال‬ ِ‫ف‬ ِ
‫ص‬
‫؟‬ُ‫ط‬ِ‫ئ‬‫الحا‬َ‫و‬
‫والحائط‬ ‫السلم‬ ‫يصنعه‬ ‫الذي‬ ‫المثلث‬ ‫أن‬ ‫بما‬
‫به‬
‫تس‬ ‫زاوية‬
‫اوي‬
90
̊
‫الزاوية‬ ‫قائم‬ ‫مثلث‬ ‫فهو‬ ً‫ا‬‫إذ‬،
.
‫أضالعه‬ ‫جميع‬ ‫أن‬ ‫وبما‬
‫األضالع‬ ‫مختلف‬ ‫مثلث‬ ‫فهو‬ ً‫ا‬‫إذ‬،‫الطول‬ ‫في‬ ‫مختلفة‬
.
9
‫ذ‬َّ‫ل‬‫ا‬ َ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬‫ال‬ ِ‫ف‬ِِّ‫ن‬َ‫ص‬
ُ‫ه‬ُ‫ع‬َ‫ن‬ْ‫ص‬َ‫ت‬ ‫ي‬
ِ‫ن‬ْ‫ب‬ُ‫ج‬‫ال‬ َِِ‫ل‬‫قا‬ ُّ‫واف‬َ‫ح‬
‫الجبن‬ َ‫قال‬ ‫واف‬ ‫تصنعه‬ ‫الذي‬ ‫المثلث‬ ‫أن‬ ‫بما‬
‫به‬
‫قي‬ ‫زاوية‬
‫اسها‬
‫أكبر‬
‫من‬
90
‫الزاوية‬ ‫منفرج‬ ‫مثلث‬ ‫فهو‬ ً‫ا‬‫إذ‬ ،‫درجة‬
.
‫أن‬ ‫وبما‬
‫به‬
‫مثلث‬ ‫فهو‬ ً‫ا‬‫إذ‬،‫الطول‬ ‫في‬ ‫متساويين‬ ‫ضلعين‬
‫متطابق‬
‫الضلعين‬
.
10
ُ‫ياس‬ِ‫ق‬‫ال‬
:
َ‫تطابق‬ُ‫م‬ ‫ا‬ً‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬ ُ‫ح‬‫با‬َ‫ص‬ ْ‫ت‬َ‫م‬َ‫س‬َ‫ر‬
ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬
.
ُ‫غ‬ُ‫ل‬ْ‫ب‬َ‫ي‬ ِ‫ه‬‫في‬ ِ‫ن‬ْ‫ي‬َ‫ع‬ْ‫ل‬ ِ
‫ض‬ ِ‫ل‬‫طو‬ ُ‫ع‬‫مو‬ْ‫ج‬َ‫م‬ َ‫كان‬ ‫إذا‬
12
،‫سم‬
‫؟‬ِ‫ث‬ِ‫ل‬‫ا‬ِّ‫ث‬‫ال‬ ِ‫ع‬ْ‫ل‬ِّ ِ
‫الض‬ ُ‫ل‬‫طو‬ ‫ما‬َ‫ف‬
ً‫ا‬‫إذ‬،‫األضالع‬ ‫متطابق‬ ‫صباح‬ ‫رسمته‬ ‫الذي‬ ‫المثلث‬ ‫أن‬ ‫بما‬
‫ضلعين‬ ‫مجموع‬ ‫أن‬ ‫بما‬،‫الطول‬ ‫في‬ ‫متساوية‬ ‫أضالعه‬ ‫جميع‬
‫يساوي‬
12
‫سم‬
‫بقسمة‬ ‫الضلع‬ ‫طول‬ ‫ايجاد‬ ‫يمكن‬
12
‫على‬
2
ً‫ا‬‫إذ‬
‫يساوي‬ ‫الثالث‬ ‫الضلع‬ ‫طول‬
6
‫سم‬
.
ُ‫ياس‬ِ‫ق‬‫ال‬
:
َّ‫ض‬‫ال‬ َ‫طابق‬َ‫ت‬ُ‫م‬ ‫ا‬ً‫ث‬َّ‫ل‬‫ث‬ُ‫م‬ ُ‫طان‬ْ‫ل‬ُ‫س‬ َ‫م‬َ‫س‬َ‫ر‬
ِ‫ن‬ْ‫ي‬َ‫ع‬ْ‫ل‬
.
‫ساوي‬ُ‫ي‬ ِ‫ه‬ِ‫ع‬‫ال‬ْ‫ض‬َ‫أ‬ ِ‫د‬َ‫ح‬َ‫أ‬ ُ‫ل‬‫طو‬ َ‫كان‬ ‫إذا‬
5
ُ‫ل‬‫طو‬َ‫و‬ ،‫سم‬
‫ر‬َ‫خ‬‫اآل‬
3
‫؟‬ِ‫ث‬ِ‫ل‬‫ا‬ِّ‫ث‬‫ال‬ ِ‫ع‬ْ‫ل‬ِّ ِ
‫الض‬ ُ‫ل‬‫طو‬ ‫ا‬َ‫م‬َ‫ف‬ ،‫سم‬
11
‫فالمثل‬ ً‫ا‬‫إذ‬،‫الضلعين‬ ‫متطابق‬ ‫سلطان‬ ‫رسمه‬ ‫الذي‬ ‫المثلث‬ ‫أن‬ ‫بما‬
‫ث‬
‫به‬
‫في‬ ‫متساويين‬ ‫ضلعين‬
‫الطولوبما‬
‫ضلع‬ ‫هناك‬ ‫أن‬
5
‫يكون‬ ‫أن‬ ‫يمكن‬ ‫سم‬
‫يساويه‬ ‫الثالث‬ ‫الضلع‬
‫الثالث‬ ‫الضلع‬ ‫طول‬ ‫يصبح‬ ‫وبالتالي‬
5
‫سم‬
.
‫أويمكن‬
‫طوله‬ ‫الذي‬ ‫الضلع‬ ‫الثالث‬ ‫الضلع‬ ‫يساوي‬ ‫أن‬
3
‫سم‬
‫وبالتالي‬،
‫يساوي‬ ‫الثالث‬ ‫الضلع‬ ‫طول‬ ‫يكون‬
3
‫سم‬
.
ْ‫م‬َ‫م‬‫ال‬ ُ‫مة‬ ِ
‫عاص‬ ُ‫ياض‬ِِّ
‫الر‬
ِ‫ة‬َ‫ك‬َ‫ل‬
ِ‫د‬‫و‬ُ‫ع‬ُّ‫س‬‫ال‬ ِ‫ة‬َّ‫ي‬‫ب‬َ‫ر‬َ‫ع‬‫ال‬
ِ‫ة‬َّ‫ي‬
.
ْ‫ن‬ِ‫وم‬
ِ‫ة‬‫المملك‬ ‫دن‬ُ‫م‬
ِ‫ل‬‫اح‬َّ‫س‬‫ال‬ ‫على‬ ُ‫م‬‫َّما‬‫د‬‫ال‬
ِ‫ل‬‫َّما‬‫ش‬‫ال‬ ‫في‬ ‫وتبوك‬ ،‫قي‬ ِ
‫َّر‬‫ش‬‫ال‬
ِِّ‫ي‬‫الغرب‬
.
12
ِ‫ة‬‫الخريط‬ ‫على‬ ْ
‫ف‬ِِّ‫ن‬‫ص‬
ِ
‫ص‬َ‫ي‬ ‫الذي‬ َ‫ث‬َّ‫ل‬‫ث‬ُ‫م‬‫ال‬
َ‫ن‬ْ‫ي‬َ‫ب‬ ُ‫ل‬
‫و‬ ِ
‫ِض‬‫ا‬‫ي‬ ِِّ‫والر‬ ِ‫َّمام‬‫د‬‫ال‬
‫تبوك‬
‫اه‬َ‫ي‬‫لزوا‬ ِ‫ة‬َ‫ب‬‫س‬ِِّ‫ن‬‫بال‬
ِ‫ه‬ِ‫ع‬‫ضال‬َ‫أل‬‫و‬
‫بين‬ ‫يصل‬ ‫الذي‬ ‫المثلث‬ ‫أن‬ ‫بما‬
‫الدمام‬
‫يح‬ ‫وتبوك‬ ‫والرياض‬
‫توي‬
‫من‬ ‫أكبر‬ ‫قياسها‬ ‫زاوية‬ ‫علي‬
90
̊
‫الزاوية‬ ‫منفرج‬ ‫المثلث‬ ً‫ا‬‫إذ‬،
.
‫األض‬ ‫مختلف‬ ‫مثلث‬ ‫الطول؛فهو‬ ‫في‬ ‫مختلفة‬ ‫أضالعه‬ ‫أن‬ ‫بما‬
‫الع‬
.
ٌ‫ة‬َ‫ح‬‫تو‬ْ‫ف‬َ‫م‬ ٌ‫ة‬َ‫ل‬‫أ‬َ‫س‬َ‫م‬
:
ِ‫م‬ ‫ا‬ً‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬ ْ‫م‬ُ‫س‬ْ‫ار‬
‫أتي‬َ‫ي‬ ‫ا‬ِّ‫م‬ِ‫م‬ ٍ‫ع‬ ْ‫و‬َ‫ن‬ ِِّ‫ل‬ُ‫ك‬ ْ‫ن‬
:
13
ْ‫ض‬َ‫أل‬‫ا‬ ‫ف‬ِ‫ل‬َ‫ت‬ْ‫خ‬ُ‫م‬َ‫و‬ ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ‫م‬ِ‫ئ‬‫قا‬
ِ‫الع‬
.
ٌ‫ة‬َ‫ح‬‫تو‬ْ‫ف‬َ‫م‬ ٌ‫ة‬َ‫ل‬‫أ‬َ‫س‬َ‫م‬
:
ِ‫م‬ ‫ا‬ً‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬ ْ‫م‬ُ‫س‬ْ‫ار‬
‫أتي‬َ‫ي‬ ‫ا‬ِّ‫م‬ِ‫م‬ ٍ‫ع‬ ْ‫و‬َ‫ن‬ ِِّ‫ل‬ُ‫ك‬ ْ‫ن‬
:
14
ِ
ِّ‫ض‬‫ال‬ ِ‫تطابق‬ُ‫م‬َ‫و‬ ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ‫ج‬ ِ
‫ر‬َ‫ف‬ْ‫ن‬ُ‫م‬
‫لعين‬
.
‫تط‬ُ‫م‬‫ال‬ ِ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬ْ‫ل‬ِ‫ل‬ ُ‫ن‬ِ‫ك‬ْ‫م‬ُ‫ي‬ ْ‫ل‬َ‫ه‬
ْ‫ن‬َ‫أ‬ ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬ ِ‫ابق‬
ِّ ِ
‫ض‬َ‫و‬ ‫؟‬ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ َ‫ج‬ ِ
‫ر‬َ‫ف‬ْ‫ن‬ُ‫م‬ َ‫كون‬َ‫ي‬
َ‫ك‬َ‫ت‬َ‫ب‬‫إجا‬ ْ‫ح‬
.
15
‫منفرج‬ ‫يكون‬ ‫أن‬ ‫األضالع‬ ‫المتطابق‬ ‫للمثلث‬ ‫يمكن‬ ‫ال‬
‫طول‬ ‫زاد‬ ‫المثلث‬ ‫في‬ ‫الزاوية‬ ‫قياس‬ ‫زاد‬ ‫كلما‬ ‫ألن‬،‫الزارية‬
‫أك‬ ‫تكون‬ ‫المنفرجة‬ ‫الزاوية‬ ‫أن‬ ‫وبما‬،‫لها‬ ‫المقابل‬ ‫الضلع‬
‫بر‬
‫يكون‬ ‫لها‬ ‫المقابل‬ ‫الضلع‬ ‫فإن‬،‫اآلخرين‬ ‫الزاويتين‬ ‫من‬
‫أكبر‬
‫اآلخرين‬ ‫الضلعين‬ ‫من‬
.
‫المتط‬ ‫للمثلث‬ ‫زاوية‬ ‫أي‬ ‫وقياس‬
‫ابق‬
‫تساوي‬ ‫األضالع‬
60
̊
.

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أنواع المثلث بالنسبة لاضلاعه وزواياه.ppt

  • 1.
  • 2. ‫وأص‬ ‫وأصفها‬،‫األبعاد‬ ‫ثنائية‬ ‫األشكال‬ ‫أتعرف‬ ‫نفها‬ ‫الزاوية‬ ‫القائم‬ ‫المثلث‬ ‫الزاوية‬ ‫الحاد‬ ‫المثلث‬ ‫الزاوية‬ ‫المنفرج‬ ‫المثلث‬ ‫الضلعين‬ ‫المتطابق‬ ‫المثلث‬ ‫األضالع‬ ‫المتطابق‬ ‫المثلث‬ ‫األضالع‬ ‫المختلف‬ ‫المثلث‬
  • 3.
  • 4. ‫ى‬َ‫ل‬ِ‫إ‬ ٌ‫ة‬َ‫َطير‬‫ش‬ ْ‫ت‬َ‫م‬ِِّ‫س‬ُ‫ق‬ ‫ا‬َ‫م‬ ِ‫ن‬ْ‫ي‬َ‫ف‬ْ‫ص‬ِ‫ن‬ ُ‫ل‬ِِّ‫ث‬َ‫م‬ُ‫ي‬ ‫ذي‬َّ‫ل‬‫ا‬ ُ‫ل‬ْ‫ك‬َّ‫ش‬‫ال‬ ُ‫ه‬ ‫َّط‬‫ش‬‫ال‬ َ‫ن‬ِ‫م‬ ٍ‫ف‬ْ‫ص‬ِ‫ن‬ ُّ‫ل‬ُ‫ك‬ ‫؟‬ِ‫ة‬َ‫ير‬ ‫عن‬ ‫عبارة‬ ‫كانت‬ ‫الشطيرة‬ ‫أن‬ ‫الشكل‬ ‫من‬ ‫الواضح‬ ‫مثلثين‬ ‫نتج‬ ‫القطر‬ ‫عند‬ ‫تقسيمها‬ ‫وعند‬،‫مربع‬ .
  • 5. ِ‫ت‬‫ثا‬َّ‫ل‬َ‫ث‬ُ‫م‬‫ال‬ َ‫ن‬ِ‫م‬ ٌ‫ة‬َّ‫د‬ِ‫ع‬ ٌ‫ع‬‫وا‬ْ‫ن‬َ‫أ‬ َ‫ناك‬ُ‫ه‬ . ُ‫نيف‬ْ‫ص‬َ‫ت‬ َ‫ك‬ِ‫ن‬‫ا‬ْ‫ك‬‫إم‬ِ‫ب‬َ‫و‬ َ‫ز‬ ِ‫ت‬‫ياسا‬ِ‫ق‬ ‫َلى‬‫ع‬ ‫ًا‬‫د‬‫ما‬ِ‫ت‬ْ‫ع‬‫ا‬ ِ‫ت‬‫ثا‬َّ‫ل‬َ‫ث‬ُ‫م‬‫ال‬ ‫واياها‬ .
  • 6. ‫مثال‬ ‫وايا‬َّ‫للز‬ ِ‫ة‬َ‫ب‬‫س‬ِِّ‫ن‬‫بال‬ ُ‫صنيف‬َّ‫ت‬‫ال‬ 1 َ‫ب‬‫س‬ِِّ‫ن‬‫بال‬ َ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬‫ال‬ ِ‫ف‬ِِّ‫ن‬َ‫ص‬ ‫لزواياه‬ ِ‫ة‬ . ُ‫م‬‫ال‬ َّ‫إن‬َ‫ف‬ ،ً‫ة‬َ‫ج‬ ِ ‫ر‬َ‫ف‬ْ‫ن‬ُ‫م‬ ً‫ة‬َ‫ي‬ِ‫و‬‫زا‬ َ‫ناك‬ُ‫ه‬ َّ‫ن‬َ‫أ‬ ‫ما‬ِ‫ب‬ ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ُ‫ج‬ ِ ‫ر‬َ‫ف‬ْ‫ن‬ُ‫م‬ َ‫ث‬َّ‫ل‬َ‫ث‬ .
  • 7. َ‫ح‬ ِ‫ت‬‫ا‬َ‫ث‬َّ‫ل‬َ‫ث‬‫الم‬ َ‫ف‬ِِّ‫ن‬َ‫ص‬ُ‫ن‬ ‫أن‬ ‫ا‬ً‫ض‬‫ي‬َ‫أ‬ ُ‫ن‬ِ‫ك‬ْ‫م‬ُ‫ي‬ ‫العها‬ْ‫ض‬َ‫أ‬ ِ‫ل‬‫وا‬ْ‫ط‬َ‫أ‬ َََ‫س‬ .
  • 8. ‫مثال‬ ‫ا‬َ‫ي‬‫ا‬َ‫و‬َّ‫للز‬ ِ‫ة‬‫سب‬ِِّ‫ن‬‫بال‬ ُ‫صنيف‬َ‫ت‬‫ال‬ ِ‫واألضالع‬ . 2 َ‫ب‬ْ‫س‬ِِّ‫ن‬‫بال‬ َ‫ث‬َ‫ل‬َّ‫ث‬ُ‫م‬‫ال‬ ِ‫ف‬ِِّ‫ن‬‫ص‬ ُ‫ه‬‫ا‬َ‫ي‬‫وا‬َ‫ز‬ِ‫ل‬ ِ‫ة‬ ‫ه؟‬ِ‫ع‬َ‫ال‬ْ‫ض‬‫وأل‬ َ‫أ‬ ‫منها‬ ٍِّ‫ل‬‫ك‬ َ ‫قياس‬ ‫ا‬َ‫ي‬‫وا‬َ‫ز‬ ُ‫الث‬َ‫ث‬ ُ‫ه‬َ‫ل‬ ُ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬‫ال‬ ‫ا‬ِ‫ذ‬‫ه‬ ْ‫ن‬ِ‫م‬ ُّ‫ل‬َ‫ق‬ 90 ‫ا‬ً‫ذ‬‫إ‬ ، ‫وايا‬َّ‫الز‬ ُّ‫د‬‫حا‬ ُ‫ث‬َّ‫ل‬‫فالمث‬ . َ‫أل‬‫ا‬ ُ‫ق‬‫تطاب‬ُ‫م‬ َ‫و‬ُ‫ه‬‫ف‬ ‫ا‬ً‫ذ‬‫إ‬ ،ٌ‫طابقة‬َ‫ت‬ُ‫م‬ ِ‫ه‬ِ‫ع‬‫ال‬ْ‫ض‬َ‫أ‬ ُّ‫ل‬ُ‫ك‬‫و‬ ‫ا‬ً‫ض‬ْ‫ي‬َ‫أ‬ ِ‫الع‬ْ‫ض‬ .
  • 9.
  • 10. ‫إلى‬ ‫أتي‬َ‫ي‬ ‫ا‬ِّ‫م‬ِ‫م‬ ٍ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬ َّ‫ل‬ُ‫ك‬ ْ ‫ف‬ِِّ‫ن‬َ‫ص‬ : َ‫أ‬ ،‫وايا‬َّ‫الز‬ ِِّ‫د‬‫حا‬ ِ‫ج‬ ِ ‫ر‬َ‫ف‬ْ‫ن‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ِ‫م‬ِ‫ئ‬‫قا‬ ْ‫و‬ ‫ا‬ ِ‫تطابق‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ن‬ْ‫ي‬َ‫ع‬ْ‫ل‬‫ض‬َّ‫ل‬‫ا‬ ِ‫ابق‬َ‫ط‬‫ت‬ُ‫م‬ ‫إلى‬َ‫و‬ ،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ِ‫ف‬ِ‫ل‬َ‫ت‬ْ‫خ‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫الع‬ْ‫ض‬َ‫أل‬ ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬ . 1 ‫المثلث‬ ‫هذا‬ ‫به‬ ‫من‬ ‫أكبر‬ ‫قياسها‬ ‫زاوية‬ 90 ‫منفرج‬ ‫فهو‬ ً‫ا‬‫ْإذ‬‫؛‬ ‫الزاوية‬ . ‫وبه‬ ‫متطاب‬ ‫فهو‬ ً‫ا‬‫متساويين؛إذ‬ ‫ضلعين‬ ً‫ا‬‫أيض‬ ‫ق‬ ‫الضلعين‬ .
  • 11. ‫إلى‬ ‫أتي‬َ‫ي‬ ‫ا‬ِّ‫م‬ِ‫م‬ ٍ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬ َّ‫ل‬ُ‫ك‬ ْ ‫ف‬ِِّ‫ن‬َ‫ص‬ : َ‫أ‬ ،‫وايا‬َّ‫الز‬ ِِّ‫د‬‫حا‬ ِ‫ج‬ ِ ‫ر‬َ‫ف‬ْ‫ن‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ِ‫م‬ِ‫ئ‬‫قا‬ ْ‫و‬ ‫ا‬ ِ‫تطابق‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ن‬ْ‫ي‬َ‫ع‬ْ‫ل‬‫ض‬َّ‫ل‬‫ا‬ ِ‫ابق‬َ‫ط‬‫ت‬ُ‫م‬ ‫إلى‬َ‫و‬ ،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ِ‫ف‬ِ‫ل‬َ‫ت‬ْ‫خ‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫الع‬ْ‫ض‬َ‫أل‬ ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬ . 2 ‫من‬ ‫أقل‬ ‫منها‬ ‫كل‬ ‫قياس‬ ‫زوايا‬ ‫ثالث‬ ‫له‬ ‫المثلث‬ ‫هذا‬ 90 ̊ ، ‫الزوايا‬ ‫حاد‬ ‫فالمثلث‬ ً‫ا‬‫إذ‬ . ،‫متطابقة‬ ‫أضالعه‬ ‫وكل‬ ‫فه‬ ً‫ا‬‫إذ‬ ‫و‬ ‫األضالع‬ ‫متطابق‬ .
  • 12. ‫إلى‬ ‫أتي‬َ‫ي‬ ‫ا‬ِّ‫م‬ِ‫م‬ ٍ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬ َّ‫ل‬ُ‫ك‬ ْ ‫ف‬ِِّ‫ن‬َ‫ص‬ : َ‫أ‬ ،‫وايا‬َّ‫الز‬ ِِّ‫د‬‫حا‬ ِ‫ج‬ ِ ‫ر‬َ‫ف‬ْ‫ن‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ِ‫م‬ِ‫ئ‬‫قا‬ ْ‫و‬ ‫ا‬ ِ‫تطابق‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ن‬ْ‫ي‬َ‫ع‬ْ‫ل‬‫ض‬َّ‫ل‬‫ا‬ ِ‫ابق‬َ‫ط‬‫ت‬ُ‫م‬ ‫إلى‬َ‫و‬ ،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ِ‫ف‬ِ‫ل‬َ‫ت‬ْ‫خ‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫الع‬ْ‫ض‬َ‫أل‬ ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬ . 3 ‫المثلث‬ ‫هذا‬ ‫به‬ ‫يساوي‬ ‫قياسها‬ ‫زاوية‬ 90 ̊ ‫قائم‬ ‫فالمثلث‬ ً‫ا‬‫إذ‬، ‫الزاوية‬ . ‫األضالع‬ ‫مختلف‬ ‫فهو‬ ً‫ا‬‫إذ‬،‫مختلفة‬ ‫أضالعه‬ ‫وأطوال‬ .
  • 13. 4 ‫تحدث‬ ٍ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬ ْ‫ي‬َ‫ع‬ْ‫ل‬ ِ ‫ض‬ ْ‫ي‬َ‫ل‬‫و‬ُ‫ط‬ ُ‫ع‬‫مجمو‬ َ‫كان‬ ‫إذا‬ ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬ ِ‫متطابق‬ 6 ،‫سم‬ َ‫ب‬‫إجا‬ ْ‫ح‬َ‫ْر‬‫ش‬‫ا‬ ‫؟‬ِ‫ث‬ِ‫ل‬‫ا‬ِّ‫ث‬‫ال‬ ِ‫ع‬ْ‫ل‬ِّ ِ ‫الض‬ ُ‫ل‬‫طو‬ ‫ما‬َ‫ف‬ ‫ك‬َ‫ت‬ ‫أضالعه‬ ‫فجميع‬ ً‫ا‬‫إذ‬ ،‫األضالع‬ ‫متطابق‬ ‫المثلث‬ ‫أن‬ ‫بما‬ ‫يساوي‬ ‫ضلعيه‬ ‫مجموع‬ ‫أن‬ ‫بما‬ ‫متساوية‬ 6 ‫بقسمة‬ ً‫ا‬‫إذ‬ ،‫سم‬ 6 ‫على‬ 2 ‫الضلع‬ ‫طول‬ ً‫ا‬‫إذ‬ ‫الضلع‬ ‫طول‬ ‫على‬ ‫نحصل‬ ‫يساوي‬ 3 ‫الضلع‬ ‫طول‬ ‫إذا‬، ‫األضالع‬ ‫متطابق‬ ‫أنه‬ ‫بما‬ ‫سم‬ ‫يساوي‬ ‫الثالث‬ 3 ‫سم‬ .
  • 14.
  • 15. ‫إلى‬ ‫أتي‬َ‫ي‬ ‫ا‬ِّ‫م‬ِ‫م‬ ٍ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬ َّ‫ل‬ُ‫ك‬ ْ ‫ف‬ِِّ‫ن‬َ‫ص‬ : ِ‫م‬ِ‫ئ‬‫قا‬ ْ‫و‬َ‫أ‬ ،‫وايا‬َّ‫الز‬ ِِّ‫د‬‫حا‬ ،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ِ‫ج‬ ِ ‫ر‬َ‫ف‬ْ‫ن‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ْ‫خ‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬ ِ‫تطابق‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ن‬ْ‫ي‬َ‫ع‬ْ‫ل‬‫ض‬َّ‫ل‬‫ا‬ ِ‫ابق‬َ‫ط‬‫ت‬ُ‫م‬ ‫إلى‬َ‫و‬ ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬ ِ‫ف‬ِ‫ل‬َ‫ت‬ . 5 ‫من‬ ‫أقل‬ ‫منها‬ ‫كل‬ ‫قياس‬ ‫زوايا‬ ‫ثالث‬ ‫له‬ ‫المثلث‬ ‫هذا‬ 90 ̊ ً‫ا‬‫إذ‬، ‫الزوايا‬ ‫حاد‬ ‫فالمثلث‬ . ‫أضالعه‬ ‫كل‬ ‫أن‬ ‫وبما‬ ‫متساوية‬ ‫فهو‬ ‫إذا‬، ‫األضالع‬ ‫متطابق‬ .
  • 16. ‫إلى‬ ‫أتي‬َ‫ي‬ ‫ا‬ِّ‫م‬ِ‫م‬ ٍ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬ َّ‫ل‬ُ‫ك‬ ْ ‫ف‬ِِّ‫ن‬َ‫ص‬ : ِ‫م‬ِ‫ئ‬‫قا‬ ْ‫و‬َ‫أ‬ ،‫وايا‬َّ‫الز‬ ِِّ‫د‬‫حا‬ ،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ِ‫ج‬ ِ ‫ر‬َ‫ف‬ْ‫ن‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ْ‫خ‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬ ِ‫تطابق‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ن‬ْ‫ي‬َ‫ع‬ْ‫ل‬‫ض‬َّ‫ل‬‫ا‬ ِ‫ابق‬َ‫ط‬‫ت‬ُ‫م‬ ‫إلى‬َ‫و‬ ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬ ِ‫ف‬ِ‫ل‬َ‫ت‬ . 6 ‫المثلث‬ ‫هذا‬ ‫به‬ ‫من‬ ‫أقل‬ ‫منها‬ ‫كل‬ ‫قياس‬ ‫زوايا‬ ‫ثالث‬ 90 ̊ ، ‫الزوايا‬ ‫حاد‬ ‫مثلث‬ ‫فهو‬ ً‫ا‬‫إذ‬ . ‫مختلفة‬ ‫أضالعه‬ ‫جميع‬ ‫أن‬ ‫وبما‬ ‫في‬ ‫األضالع‬ ‫مختلف‬ ‫مثلث‬ ‫فهو‬ ً‫ا‬‫إذ‬،‫الطول‬ .
  • 17. ‫إلى‬ ‫أتي‬َ‫ي‬ ‫ا‬ِّ‫م‬ِ‫م‬ ٍ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬ َّ‫ل‬ُ‫ك‬ ْ ‫ف‬ِِّ‫ن‬َ‫ص‬ : ِ‫م‬ِ‫ئ‬‫قا‬ ْ‫و‬َ‫أ‬ ،‫وايا‬َّ‫الز‬ ِِّ‫د‬‫حا‬ ،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ِ‫ج‬ ِ ‫ر‬َ‫ف‬ْ‫ن‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ْ‫خ‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬ ِ‫تطابق‬ُ‫م‬ ْ‫و‬َ‫أ‬ ،ِ‫ن‬ْ‫ي‬َ‫ع‬ْ‫ل‬‫ض‬َّ‫ل‬‫ا‬ ِ‫ابق‬َ‫ط‬‫ت‬ُ‫م‬ ‫إلى‬َ‫و‬ ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬ ِ‫ف‬ِ‫ل‬َ‫ت‬ . ‫المثلث‬ ‫هذا‬ ‫به‬ ‫يساوي‬ ‫قياسها‬ ‫زاوية‬ 90 ̊ ‫قائم‬ ‫مثلث‬ ‫فهو‬ ً‫ا‬‫إذ‬، ‫الزاوية‬ . ،‫الطول‬ ‫في‬ ‫مختلفة‬ ‫أضالعه‬ ‫جميع‬ ‫أن‬ ‫وبما‬ ‫األضالع‬ ‫مختلف‬ ‫مثلث‬ ‫فهو‬ ً‫ا‬‫إذ‬ . 7
  • 18. 8 ُ‫م‬َّ‫ل‬ُّ‫س‬‫ال‬ ُ‫ه‬ُ‫ع‬َ‫ن‬ْ‫ص‬َ‫ي‬ ‫ذي‬َّ‫ل‬‫ا‬ َ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬‫ال‬ ِ‫ف‬ ِ ‫ص‬ ‫؟‬ُ‫ط‬ِ‫ئ‬‫الحا‬َ‫و‬ ‫والحائط‬ ‫السلم‬ ‫يصنعه‬ ‫الذي‬ ‫المثلث‬ ‫أن‬ ‫بما‬ ‫به‬ ‫تس‬ ‫زاوية‬ ‫اوي‬ 90 ̊ ‫الزاوية‬ ‫قائم‬ ‫مثلث‬ ‫فهو‬ ً‫ا‬‫إذ‬، . ‫أضالعه‬ ‫جميع‬ ‫أن‬ ‫وبما‬ ‫األضالع‬ ‫مختلف‬ ‫مثلث‬ ‫فهو‬ ً‫ا‬‫إذ‬،‫الطول‬ ‫في‬ ‫مختلفة‬ .
  • 19. 9 ‫ذ‬َّ‫ل‬‫ا‬ َ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬‫ال‬ ِ‫ف‬ِِّ‫ن‬َ‫ص‬ ُ‫ه‬ُ‫ع‬َ‫ن‬ْ‫ص‬َ‫ت‬ ‫ي‬ ِ‫ن‬ْ‫ب‬ُ‫ج‬‫ال‬ َِِ‫ل‬‫قا‬ ُّ‫واف‬َ‫ح‬ ‫الجبن‬ َ‫قال‬ ‫واف‬ ‫تصنعه‬ ‫الذي‬ ‫المثلث‬ ‫أن‬ ‫بما‬ ‫به‬ ‫قي‬ ‫زاوية‬ ‫اسها‬ ‫أكبر‬ ‫من‬ 90 ‫الزاوية‬ ‫منفرج‬ ‫مثلث‬ ‫فهو‬ ً‫ا‬‫إذ‬ ،‫درجة‬ . ‫أن‬ ‫وبما‬ ‫به‬ ‫مثلث‬ ‫فهو‬ ً‫ا‬‫إذ‬،‫الطول‬ ‫في‬ ‫متساويين‬ ‫ضلعين‬ ‫متطابق‬ ‫الضلعين‬ .
  • 20. 10 ُ‫ياس‬ِ‫ق‬‫ال‬ : َ‫تطابق‬ُ‫م‬ ‫ا‬ً‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬ ُ‫ح‬‫با‬َ‫ص‬ ْ‫ت‬َ‫م‬َ‫س‬َ‫ر‬ ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬ . ُ‫غ‬ُ‫ل‬ْ‫ب‬َ‫ي‬ ِ‫ه‬‫في‬ ِ‫ن‬ْ‫ي‬َ‫ع‬ْ‫ل‬ ِ ‫ض‬ ِ‫ل‬‫طو‬ ُ‫ع‬‫مو‬ْ‫ج‬َ‫م‬ َ‫كان‬ ‫إذا‬ 12 ،‫سم‬ ‫؟‬ِ‫ث‬ِ‫ل‬‫ا‬ِّ‫ث‬‫ال‬ ِ‫ع‬ْ‫ل‬ِّ ِ ‫الض‬ ُ‫ل‬‫طو‬ ‫ما‬َ‫ف‬ ً‫ا‬‫إذ‬،‫األضالع‬ ‫متطابق‬ ‫صباح‬ ‫رسمته‬ ‫الذي‬ ‫المثلث‬ ‫أن‬ ‫بما‬ ‫ضلعين‬ ‫مجموع‬ ‫أن‬ ‫بما‬،‫الطول‬ ‫في‬ ‫متساوية‬ ‫أضالعه‬ ‫جميع‬ ‫يساوي‬ 12 ‫سم‬ ‫بقسمة‬ ‫الضلع‬ ‫طول‬ ‫ايجاد‬ ‫يمكن‬ 12 ‫على‬ 2 ً‫ا‬‫إذ‬ ‫يساوي‬ ‫الثالث‬ ‫الضلع‬ ‫طول‬ 6 ‫سم‬ .
  • 21. ُ‫ياس‬ِ‫ق‬‫ال‬ : َّ‫ض‬‫ال‬ َ‫طابق‬َ‫ت‬ُ‫م‬ ‫ا‬ً‫ث‬َّ‫ل‬‫ث‬ُ‫م‬ ُ‫طان‬ْ‫ل‬ُ‫س‬ َ‫م‬َ‫س‬َ‫ر‬ ِ‫ن‬ْ‫ي‬َ‫ع‬ْ‫ل‬ . ‫ساوي‬ُ‫ي‬ ِ‫ه‬ِ‫ع‬‫ال‬ْ‫ض‬َ‫أ‬ ِ‫د‬َ‫ح‬َ‫أ‬ ُ‫ل‬‫طو‬ َ‫كان‬ ‫إذا‬ 5 ُ‫ل‬‫طو‬َ‫و‬ ،‫سم‬ ‫ر‬َ‫خ‬‫اآل‬ 3 ‫؟‬ِ‫ث‬ِ‫ل‬‫ا‬ِّ‫ث‬‫ال‬ ِ‫ع‬ْ‫ل‬ِّ ِ ‫الض‬ ُ‫ل‬‫طو‬ ‫ا‬َ‫م‬َ‫ف‬ ،‫سم‬ 11 ‫فالمثل‬ ً‫ا‬‫إذ‬،‫الضلعين‬ ‫متطابق‬ ‫سلطان‬ ‫رسمه‬ ‫الذي‬ ‫المثلث‬ ‫أن‬ ‫بما‬ ‫ث‬ ‫به‬ ‫في‬ ‫متساويين‬ ‫ضلعين‬ ‫الطولوبما‬ ‫ضلع‬ ‫هناك‬ ‫أن‬ 5 ‫يكون‬ ‫أن‬ ‫يمكن‬ ‫سم‬ ‫يساويه‬ ‫الثالث‬ ‫الضلع‬ ‫الثالث‬ ‫الضلع‬ ‫طول‬ ‫يصبح‬ ‫وبالتالي‬ 5 ‫سم‬ . ‫أويمكن‬ ‫طوله‬ ‫الذي‬ ‫الضلع‬ ‫الثالث‬ ‫الضلع‬ ‫يساوي‬ ‫أن‬ 3 ‫سم‬ ‫وبالتالي‬، ‫يساوي‬ ‫الثالث‬ ‫الضلع‬ ‫طول‬ ‫يكون‬ 3 ‫سم‬ .
  • 22.
  • 23. ْ‫م‬َ‫م‬‫ال‬ ُ‫مة‬ ِ ‫عاص‬ ُ‫ياض‬ِِّ ‫الر‬ ِ‫ة‬َ‫ك‬َ‫ل‬ ِ‫د‬‫و‬ُ‫ع‬ُّ‫س‬‫ال‬ ِ‫ة‬َّ‫ي‬‫ب‬َ‫ر‬َ‫ع‬‫ال‬ ِ‫ة‬َّ‫ي‬ . ْ‫ن‬ِ‫وم‬ ِ‫ة‬‫المملك‬ ‫دن‬ُ‫م‬ ِ‫ل‬‫اح‬َّ‫س‬‫ال‬ ‫على‬ ُ‫م‬‫َّما‬‫د‬‫ال‬ ِ‫ل‬‫َّما‬‫ش‬‫ال‬ ‫في‬ ‫وتبوك‬ ،‫قي‬ ِ ‫َّر‬‫ش‬‫ال‬ ِِّ‫ي‬‫الغرب‬ .
  • 24. 12 ِ‫ة‬‫الخريط‬ ‫على‬ ْ ‫ف‬ِِّ‫ن‬‫ص‬ ِ ‫ص‬َ‫ي‬ ‫الذي‬ َ‫ث‬َّ‫ل‬‫ث‬ُ‫م‬‫ال‬ َ‫ن‬ْ‫ي‬َ‫ب‬ ُ‫ل‬ ‫و‬ ِ ‫ِض‬‫ا‬‫ي‬ ِِّ‫والر‬ ِ‫َّمام‬‫د‬‫ال‬ ‫تبوك‬ ‫اه‬َ‫ي‬‫لزوا‬ ِ‫ة‬َ‫ب‬‫س‬ِِّ‫ن‬‫بال‬ ِ‫ه‬ِ‫ع‬‫ضال‬َ‫أل‬‫و‬ ‫بين‬ ‫يصل‬ ‫الذي‬ ‫المثلث‬ ‫أن‬ ‫بما‬ ‫الدمام‬ ‫يح‬ ‫وتبوك‬ ‫والرياض‬ ‫توي‬ ‫من‬ ‫أكبر‬ ‫قياسها‬ ‫زاوية‬ ‫علي‬ 90 ̊ ‫الزاوية‬ ‫منفرج‬ ‫المثلث‬ ً‫ا‬‫إذ‬، . ‫األض‬ ‫مختلف‬ ‫مثلث‬ ‫الطول؛فهو‬ ‫في‬ ‫مختلفة‬ ‫أضالعه‬ ‫أن‬ ‫بما‬ ‫الع‬ .
  • 25.
  • 26. ٌ‫ة‬َ‫ح‬‫تو‬ْ‫ف‬َ‫م‬ ٌ‫ة‬َ‫ل‬‫أ‬َ‫س‬َ‫م‬ : ِ‫م‬ ‫ا‬ً‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬ ْ‫م‬ُ‫س‬ْ‫ار‬ ‫أتي‬َ‫ي‬ ‫ا‬ِّ‫م‬ِ‫م‬ ٍ‫ع‬ ْ‫و‬َ‫ن‬ ِِّ‫ل‬ُ‫ك‬ ْ‫ن‬ : 13 ْ‫ض‬َ‫أل‬‫ا‬ ‫ف‬ِ‫ل‬َ‫ت‬ْ‫خ‬ُ‫م‬َ‫و‬ ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ‫م‬ِ‫ئ‬‫قا‬ ِ‫الع‬ .
  • 27. ٌ‫ة‬َ‫ح‬‫تو‬ْ‫ف‬َ‫م‬ ٌ‫ة‬َ‫ل‬‫أ‬َ‫س‬َ‫م‬ : ِ‫م‬ ‫ا‬ً‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬ ْ‫م‬ُ‫س‬ْ‫ار‬ ‫أتي‬َ‫ي‬ ‫ا‬ِّ‫م‬ِ‫م‬ ٍ‫ع‬ ْ‫و‬َ‫ن‬ ِِّ‫ل‬ُ‫ك‬ ْ‫ن‬ : 14 ِ ِّ‫ض‬‫ال‬ ِ‫تطابق‬ُ‫م‬َ‫و‬ ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ ‫ج‬ ِ ‫ر‬َ‫ف‬ْ‫ن‬ُ‫م‬ ‫لعين‬ .
  • 28. ‫تط‬ُ‫م‬‫ال‬ ِ‫ث‬َّ‫ل‬َ‫ث‬ُ‫م‬ْ‫ل‬ِ‫ل‬ ُ‫ن‬ِ‫ك‬ْ‫م‬ُ‫ي‬ ْ‫ل‬َ‫ه‬ ْ‫ن‬َ‫أ‬ ِ‫الع‬ْ‫ض‬َ‫أل‬‫ا‬ ِ‫ابق‬ ِّ ِ ‫ض‬َ‫و‬ ‫؟‬ِ‫ة‬َ‫ي‬ِ‫و‬‫ا‬ِّ‫الز‬ َ‫ج‬ ِ ‫ر‬َ‫ف‬ْ‫ن‬ُ‫م‬ َ‫كون‬َ‫ي‬ َ‫ك‬َ‫ت‬َ‫ب‬‫إجا‬ ْ‫ح‬ . 15 ‫منفرج‬ ‫يكون‬ ‫أن‬ ‫األضالع‬ ‫المتطابق‬ ‫للمثلث‬ ‫يمكن‬ ‫ال‬ ‫طول‬ ‫زاد‬ ‫المثلث‬ ‫في‬ ‫الزاوية‬ ‫قياس‬ ‫زاد‬ ‫كلما‬ ‫ألن‬،‫الزارية‬ ‫أك‬ ‫تكون‬ ‫المنفرجة‬ ‫الزاوية‬ ‫أن‬ ‫وبما‬،‫لها‬ ‫المقابل‬ ‫الضلع‬ ‫بر‬ ‫يكون‬ ‫لها‬ ‫المقابل‬ ‫الضلع‬ ‫فإن‬،‫اآلخرين‬ ‫الزاويتين‬ ‫من‬ ‫أكبر‬ ‫اآلخرين‬ ‫الضلعين‬ ‫من‬ . ‫المتط‬ ‫للمثلث‬ ‫زاوية‬ ‫أي‬ ‫وقياس‬ ‫ابق‬ ‫تساوي‬ ‫األضالع‬ 60 ̊ .