دليل معلم مقرر رياضيات 1

2 2         1  ‫ﺍﺳﺘﻌﻤﻞ ﻧﺘﺎﺋﺞ ﺍﻻﺧﺘﺒﺎﺭ ﺍﻟﺴﺮﻳﻊ ﻭﻣﺨﻄﻂ‬   ￯‫ﺍﻟﻤﻌﺎﻟﺠﺔ ﻟﻤﺴﺎﻋﺪﺗﻚ ﻋﻠﻰ ﺗﺤﺪﻳﺪ ﻣﺴﺘﻮ‬ F G 1  A (2 -1 ‫ﺴﺘﻌﻤﻞ ﻣﻊ ﺍﻟﺪﺭﺱ‬.‫ﺍﺳﺘﻌﻤﻞ ﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ)ﺗ‬ ‫ ﻓﻘﻢ“ ﻓﻲ‬،‫ ﻭﺍﻟﻌﺒﺎﺭﺓ ”ﺇﺫﺍ‬،‫ﺍﻟﻤﻌﺎﻟﺠﺔ ﺍﻟﻤﻨﺎﺳﺐ‬ KL . ‫ﺍﺳﺘﻌﻤﻞ ﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ‬ F ￯‫ﺍﻟﺠﺪﻭﻝ ﺳﺘﺴﺎﻋﺪﻙ ﻋﻠﻰ ﺗﺤﺪﻳﺪ ﺍﻟﻤﺴﺘﻮ‬ J H C ‫ﺍﻟﻤﻨﺎﺳﺐ ﻟﻠﻤﻌﺎﻟﺠﺔ ﻭﺍﻗﺘﺮﺍﺡ ﻣﺼﺎﺩﺭ ﻟﻜﻞ‬ Q BD .￯‫ﻣﺴﺘﻮ‬ PM ‫ ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‬.‫( ﻛﻢ ﻣﺴﺘﻮ￯ ﻳﻈﻬﺮ ﻓﻲ ﺍﻟﺸﻜﻞ؟ ﺍﺫﻛﺮﻫﺎ‬1 A, F, D .‫( ﺳﻢﹼ ﺛﻼﺙ ﻧﻘﺎﻁ ﺗﻘﻊ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ‬2 .‫( ﻛﻢ ﻣﺴﺘﻮ￯ ﻳﻈﻬﺮ ﻓﻲ ﺍﻟﺸﻜﻞ؟ ﺍﺫﻛﺮﻫﺎ‬a ‫ ﻓﻲ ﺍﻟﻤﺴﺘﻮ￯ ﻧﻔﺴﻪ؟ ﻭ ﹼﺿﺢ‬B, C, D ‫( ﻫﻞ ﺗﻘﻊ ﺍﻟﻨﻘﺎﻁ‬3 :‫ﺳﺘﺔ ﻣﺴﺘﻮﻳﺎﺕ ﻫﻲ‬ ‫ ﺍﻧﻈﺮ ﺍﻟﻬﺎﻣﺶ‬.‫ﺇﺟﺎﺑﺘﻚ‬ . FGL, JHM, FKP, GLM, FGH, KLM ‫ﻳﻮﺿﻊ ﺟﻬﺎﺯ ﻣﺴﺎﺣﺔ ﺍﻷﺭﺍﺿﻲ ﻋﻠﻰ ﺣﺎﻣﻞ‬  (4  .‫( ﺳ ﹼﻢ ﺛﻼﺙ ﻧﻘﺎﻁ ﺗﻘﻊ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ‬b .‫ ﺗﻘﻊ ﻋﻠﻰ ﺍﺳﺘﻘﺎﻣﺔ ﻭﺍﺣﺪﺓ‬M, Q, H ‫ﺍﻟﻨﻘﺎﻁ‬ ‫ ﻫﻞ ﺗﻘﻊ ﺍﻟﺮﺅﻭﺱ ﺍﻟﺴﻔﻠﻴﺔ ﻟﻠﻘﻮﺍﺋﻢ ﺍﻟﺜﻼﺛﺔ‬.‫ﺛﻼﺛﻲ ﺍﻟﻘﻮﺍﺋﻢ‬ ‫ﻧﻌﻢ‬ ‫ﻓﻲ ﺍﻟﻤﺴﺘﻮ￯ ﻧﻔﺴﻪ؟‬   .‫ ﻓﻲ ﺍﻟﻤﺴﺘﻮﻱ ﻧﻔﺴﻪ؟ ﻭ ﹼﺿﺢ ﺇﺟﺎﺑﺘﻚ‬F, K, J ‫( ﻫﻞ ﺗﻘﻊ ﺍﻟﻨﻘﺎﻁ‬c .FKPJ ￯‫ ﺗﻘﻊ ﺟﻤﻴﻌﻬﺎ ﻓﻲ ﺍﻟﻤﺴﺘﻮ‬F, K, J ‫ ﺍﻟﻨﻘﺎﻁ‬.‫ﻧﻌﻢ‬ 1   2  (2 -2, 2 -3 ‫)ﺗﺴﺘﻌﻤﻞ ﻣﻊ ﺍﻟﺪﺭﺳﻴﻦ‬   25%   :‫ﺃﻭﺟﺪ ﻗﻴﺎﺱ ﻛ ﱟﻞ ﻣﻦ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻵﺗﻴﺔ‬   .m∠1 ‫ﺃﻭﺟﺪ‬ 1 37° 2 23° 113˚ ∠1 (5  2  m∠1 + 37°+ 90°= 180° 4 157˚ ∠2 (6   m∠1 = 53° 1 67° 90˚ ∠3 (7 3  23˚ ∠4 (8   3  (2 -6 ‫ ﺇﻟﻰ‬2 -4 ‫)ﺗﺴﺘﻌﻤﻞ ﻣﻊ ﺍﻟﺪﺭﻭﺱ‬ :‫ ﺍﻟﻤﻌﻄﺎﺓ ﻓﻲ ﻛﻞ ﻣﻌﺎﺩﻟﺔ ﻣﻤﺎ ﻳﺄﺗﻲ‬a , b ‫ ﻟﻘﻴﻢ‬x ‫ﺃﻭﺟﺪ ﻗﻴﻤﺔ‬ www.obeikaneducation.com ، a + 8 = b ( x - 7 ) ‫ ﻓﻲ ﺍﻟﻤﻌﺎﺩﻟﺔ‬x ‫ﺃﻭﺟﺪ ﻗﻴﻤﺔ‬ -1 a + 8 = -4(x - b), a = 8 , b = 3 (9 . a = 12 , b = 10 ‫ﺇﺫﺍ ﻛﺎﻥ‬   _16 b = 3 x + 4a, a = -9 , b = 12 (10  a + 8 = b(x - 7) 1 _a + 2 = 5x, a = 18 ,b = -1 (11   2 3  50%  a = 12 , b = 10 12 + 8 = 10(x - 7) b + 13     20 = 10x - 70 ‫ﻳﻘﺪﻡ ﻣﻌﺮﺽ ﻫﺪﻳﺔ ﺑﺴﻌﺮ ﺗﺸﺠﻴﻌﻲ ﻗﺪﺭﻩ‬  (12   70 90 = 10x  ‫ ﺇﺫﺍ ﺩﻓﻊ ﺃﺣﻤﺪ ﻭﺃﺧﻮﻩ‬.‫ ﺭﻳﺎ ﹰﻻ ﻋﻨﺪ ﺷﺮﺍﺀ ﺑﻄﺎﻗ ﹶﺘﻲ ﺩﺧﻮﻝ‬15 www.obeikaneducation.com ‫ ﺛﻢ‬،‫ ﻓﺎﻛﺘﺐ ﻣﻌﺎﺩﻟﺔ ﺗﻤﺜﻞ ﻣﺎ ﺩﻓﻌﻪ ﺃﺣﻤﺪ ﻭﺃﺧﻮﻩ‬،‫ ﺭﻳﺎ ﹰﻻ‬95 10 x=9 .‫ﹸﺣ ﱠﻠﻬﺎ ﻹﻳﺠﺎﺩ ﺛﻤﻦ ﺑﻄﺎﻗﺔ ﺍﻟﺪﺧﻮﻝ ﺍﻟﻮﺍﺣﺪﺓ‬ ‫ ﺭﻳﺎ ﹰﻻ‬40 ‫ ؛‬15 + 2x = 95 www.obeikaneducation.com 2  85 2 2  B C D , C D A , A B C , A B D ،4 (1 ‫ ﺍﻟﻨﻘﺎﻁ ﺍﻟﺜﻼﺙ ﺗﻘﻊ ﻓﻲ ﺍﻟﻘﺎﻋﺪﺓ ﻟﻠﺸﻜﻞ ﺍﻟﺘﻲ‬،‫( ﻧﻌﻢ‬3 . B C D ￯‫ﺗﻤﺜﻞ ﺍﻟﻤﺴﺘﻮ‬ 85 2 2

 2- 1  Lines and Transeversal  1     .￯‫ﹸﺗﻈﻬﺮ ﹸﻏﺮﻓﺔ ﺍﻟﺨﺪﺍﻉ ﺍﻟﺒﺼﺮﻱ ﺃﻥ ﺍﻟﺸﺨﺺ ﺍﻟﻮﺍﻗﻒ ﻓﻲ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻴﻤﻨﻰ ﺃﻛﺒﺮ ﻣﻦ ﺍﻟﺸﺨﺺ ﺍﻟﻮﺍﻗﻒ ﻓﻲ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻴﺴﺮ‬ 2-1 ‫ﺍﺳﺘﻌﻤﺎﻝ ﻋﻼﻗﺎﺕ ﺍﻟﺰﻭﺍﻳﺎ ﻭﺍﻟﻘﻄﻊ‬ .‫ ﻳﺒﺪﻭ ﺍﻟﺤﺎﺋﻄﺎﻥ ﺍﻷﻣﺎﻣﻲ ﻭﺍﻟﺨﻠﻔﻲ ﻣﺘﻮﺍﺯﻳﻴﻦ ﻓﻲ ﺣﻴﻦ ﺃﻧﻬﻤﺎ ﻟﻴﺴﺎ ﻛﺬﻟﻚ‬،‫ﻭﻓﻲ ﺍﻟﻤﻨﻈﺮ ﺍﻷﻣﺎﻣﻲ‬     .‫ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﻹﺛﺒﺎﺕ ﻧﻈﺮﻳﺎﺕ‬ 2-1   ‫ﺗﻌ ﹼﺮﻑ ﺍﻟﻌﻼﻗﺎﺕ ﺑﻴﻦ ﻣﺴﺘﻘﻴﻤﻴﻦ ﺃﻭ‬ A 1 - 8 1 - 5  .‫ﻣﺴﺘﻮﻳﻴﻦ‬ P F  ‫ﺗﺴﻤﻴﺔ ﺃﺯﻭﺍﺝ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﻨﺎﺗﺠﺔ ﻋﻦ‬    .‫ﻣﺴﺘﻘﻴﻤﻴﻦ ﻭﻗﺎﻃﻊ ﻟﻬﻤﺎ‬ CG   2-1 D    ‫ﺍﺳﺘﻌﻤﺎﻝ ﻣﻴﻞ ﺍﻟﻤﺴﺘﻘﻴﻢ؛ ﻻﺳﺘﻘﺼﺎﺀ‬  ‫ﺍﻟﻌﻼﻗﺎﺕ ﺍﻟﻬﻨﺪﺳﻴﺔ ﺍﻟﺘﻲ ﺗﺘﻀﻤﻦ‬ .‫ ﻭﻟﻜﻨﻬﻤﺎ ﻓﻲ ﺍﻟﺤﻘﻴﻘﺔ ﻟﻴﺴﺎ ﺃﻓﻘﻴﻴﻦ‬،‫ﻭﻳﺒﺪﻭ ﺍﻟﺴﻘﻒ ﻭﺍﻷﺭﺿﻴﺔ ﺃﻓﻘﻴﻴﻦ‬  ‫ﺍﻟﻤﺴﺘﻘﻴﻤﺎﺕ ﺍﻟﻤﺘﻮﺍﺯﻳﺔ ﻭﺍﻟﻤﺴﺘﻘﻴﻤﺎﺕ‬ ‫ ﺍﺳ ﹸﺘﻌ ﹺﻤﻠ ﹾﺖ ﻣﺴﺘﻘﻴﻤﺎﺕ ﻣﺘﻮﺍﺯﻳﺔ ﻭﻣﺘﻘﺎﻃﻌﺔ ﻭﻣﺘﺨﺎﻟﻔﺔ ﺑﺎﻹﺿﺎﻓﺔ ﺇﻟﻰ‬ .‫ﺍﻟﻤﺘﻌﺎﻣﺪﺓ‬  .‫ﻣﺴﺘﻮﻳﺎﺕ ﻣﺘﻘﺎﻃﻌﺔ ﻭﺃﺧﺮ￯ ﻣﺘﻮﺍﺯﻳﺔ؛ ﻟﺘﺼﻤﻴﻢ ﻏﺮﻓﺔ ﺍﻟﺨﺪﺍﻉ ﻛﻤﺎ ﻳﺘﻀﺢ ﻓﻲ ﺍﻟﺮﺳﻢ ﺍﻟﺴﺎﺑﻖ‬  2       parallel lines   JK  “‫ﺍﻃﻠﺐ ﺇﻟﻰ ﺍﻟﻄﻼﺏ ﻗﺮﺍﺀﺓ ﻓﻘﺮﺓ ”ﻟﻤﺎﺫﺍ ؟‬     L M JK LM     skew lines ‫• ﺃﻱ ﺍﻟﺸﺨﺼﻴﻦ ﺩﺍﺧﻞ ﺍﻟﻐﺮﻓﺔ ﻳﺒﺪﻭ ﺃﻛﺒﺮ‬  ‫ﺣﺠ ﹰﻤﺎ ﻟﻠﻨﺎﻇﺮ؛ ﺍﻟﺬﻱ ﻓﻲ ﺍﻷﻣﺎﻡ ﺃﻡ ﺍﻟﺬﻱ ﻓﻲ‬ parallel planes .‫ﺍﻟﺨﻠﻒ؟ ﺍﻟﺬﻱ ﻓﻲ ﺍﻷﻣﺎﻡ‬ A   ‫• ﻟﻤﺎﺫﺍ ﺗﺒﺪﻭ ﺍﻷﺟﺴﺎﻡ ﻛﺄﻧﻬﺎ ﺃﻛﺒﺮ ﺃﻭ ﺃﺻﻐﺮ‬ Bm  ‫ﻣﻤﺎ ﻫﻲ ﻋﻠﻴﻪ ﻓﻲ ﺍﻟﺤﻘﻴﻘﺔ ﺑﺎﺳﺘﻌﻤﺎﻝ‬ transversal ‫ﺍﻟﻤﺴﺘﻘﻴﻤﺎﺕ ﻏﻴﺮ ﺍﻟﻤﺘﻮﺍﺯﻳﺔ؟ ﺇﺟﺎﺑﺔ‬ ℓ, m   ‫ ﻷﻥ ﺍﻟﻤﺴﺎﻓﺔ ﺑﻴﻦ ﺍﻟﻤﺴﺘﻘﻴﻤﻴﻦ‬:‫ﻣﻤﻜﻨﺔ‬   ‫ﺗﺘﻐﻴﺮ؛ ﻓﻌﻨﺪﻣﺎ ﻳﻘﺘﺮﺏ ﺍﻟﻤﺴﺘﻘﻴﻤﺎﻥ ﺃﺣﺪﻫﻤﺎ‬ A, B  interior angles ،‫ﻣﻦ ﺍﻵﺧﺮ ﻳﻈﻬﺮ ﺍﻟﺠﺴﻢ ﺑﻴﻨﻬﻤﺎ ﺃﻛﺒﺮ‬ ‫ﻭﻋﻨﺪﻣﺎ ﻳﺘﺒﺎﻋﺪﺍﻥ ﻳﻈﻬﺮ ﺍﻟﺠﺴﻢ ﺑﻴﻨﻬﻤﺎ‬  .‫ﺃﺻﻐﺮ‬ exterior angles  consecutive angles LMJKJK LM     ‫ ﻓﺈﻧﻬﺎ ﺗﻜﻮﻥ ﻣﺘﻮﺍﺯﻳﺔ‬،‫ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺃﻭ ﺃﻧﺼﺎﻑ ﺍﻟﻤﺴﺘﻘﻴﻤﺎﺕ ﺃﺟﺰﺍ ﹰﺀ ﻣﻦ ﻣﺴﺘﻘﻴﻤﺎﺕ ﻣﺘﻮﺍﺯﻳﺔ ﺃﻭ ﻣﺘﺨﺎﻟﻔﺔ‬ .‫ﺃﻭ ﻣﺘﺨﺎﻟﻔﺔ ﺃﻳ ﹰﻀﺎ‬ alternate interior angles  1    K : ‫ﺣ ﱢﺪﺩ ﻛ ﹼﹰﻼ ﻣﻤﺎ ﻳﺄﺗﻲ ﻣﺴﺘﻌﻤ ﹰﻼ ﻗﻄﻌﺔ ﺍﻟﺠﺒﻦ ﻓﻲ ﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ‬ L ___ alternate exterior angles .JP ‫( ﺟﻤﻴﻊ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺍﻟﺘﻲ ﺗﻮﺍﺯﻱ‬a J  ___ ___ Q KQ , LR corresponding angles R ___ www.obeikaneducation.com .KL ‫( _ﺟ_ﻤ_ﻴﻊ ﺍ_ﻟﻘ__ﻄﻊ ﺍ_ﻟ_ﻤﺴﺘﻘﻴﻤﺔ ﺍﻟﺘﻲ ﺗﺨﺎﻟﻒ‬b JP , PQ , PR P . PQR ￯‫( ﻣﺴﺘﻮ￯ ﻳﻮﺍﺯﻱ ﺍﻟﻤﺴﺘﻮ‬c .PQR ￯‫ ﻫﻮ ﺍﻟﻤﺴﺘﻮ￯ ﺍﻟﻮﺣﻴﺪ ﺍﻟﻤﻮﺍﺯﻱ ﻟﻠﻤﺴﺘﻮ‬JKL ￯‫ﺍﻟﻤﺴﺘﻮ‬  2 86 2-1     (88) •  (88, 91) • (88, 91) • (14) •  (14) • (14) •  (6) •   (9)  • (6) • (8) • (10)  • (8) • (9) • (9)  • (10)  •  2 86

 BC  ✓  AD   ‫ ﻳﺒ ﹼﻴﻦ ﻛﻴﻔﻴﺔ ﺗﺤﺪﻳﺪ ﺍﻟﻌﻼﻗﺎﺕ ﺑﻴﻦ‬1 : ‫ﺣﺪﺩ ﻛ ﹼﹰﻼ ﻣﻤﺎ ﻳﺄﺗﻲ ﻣﺴﺘﻌﻤ ﹰﻼ ﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ‬  GH ___ ___ ___ ‫ ﻭﻋﻠﻰ ﺍﻟﻄﻼﺏ ﺃﻥ‬،‫ﺍﻟﻤﺴﺘﻮﻳﺎﺕ ﺍﻟﻤﺘﻮﺍﺯﻳﺔ‬ EH , GF , ED , FA . BC ‫( ﺟﻤﻴﻊ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺍﻟﺘﻲ ﺗﺨﺎﻟﻒ‬1A FE  1A FE BC  ‫ﻳﻜﻮﻧﻮﺍ ﻗﺎﺩﺭﻳﻦ ﻋﻠﻰ ﺗﺤﺪﻳﺪ ﻣﺴﺘﻮ￯ ﻓﻲ‬ ___ ___  AB :‫ ﺇﺟﺎﺑﺔ ﻣﻤﻜﻨﺔ‬. EH ‫( ﻗﻄﻌﺔ ﻣﺴﺘﻘﻴﻤﺔ ﺗﻮﺍﺯﻱ‬1B ‫ ﻭﺗﺤﺪﻳﺪ ﺟﻤﻴﻊ ﺍﻟﻤﺴﺘﻮﻳﺎﺕ ﺍﻟﺘﻲ‬،‫ﺍﻟﺸﻜﻞ‬  ABG ￯‫ ﺍﻟﻤﺴﺘﻮ‬. DCH ￯‫( ﺟﻤﻴﻊ ﺍﻟﻤﺴﺘﻮﻳﺎﺕ ﺍﻟﺘﻲ ﺗﻮﺍﺯﻱ ﺍﻟﻤﺴﺘﻮ‬1C BCF .‫ﺗﻮﺍﺯﻳﻪ‬ ￯‫ ﺍﻟﻘﺎﻃﻊ ﻫﻮ ﺍﻟﻤﺴﺘﻘﻴﻢ ﺍﻟﺬﻱ ﻳﻘﻄﻊ ﻣﺴﺘﻘﻴﻤﻴﻦ ﺃﻭ ﺃﻛﺜﺮ ﻓﻲ ﺍﻟﻤﺴﺘﻮ‬ BC ‫ ﻳﺸ ﹼﻜﻞ ﺛﻤﺎﻧﻲ‬t ‫ ﻻﺣﻆ ﺃﻥ ﺍﻟﻤﺴﺘﻘﻴﻢ‬.q , r ‫ ﻗﺎﻃﻊ ﻟﻠﻤﺴﺘﻘﻴﻤﻴﻦ‬t ‫ ﺍﻟﻤﺴﺘﻘﻴﻢ‬،‫ ﻓﻔﻲ ﺍﻟﺸﻜﻞ ﺃﺩﻧﺎﻩ‬.‫ﻧﻔﺴﻪ ﻭﻓﻲ ﻧﻘﺎﻁ ﻣﺨﺘﻠﻔﺔ‬ FE .‫ ﻭﺃﺯﻭﺍﺝ ﻣﺤﺪﺩﺓ ﻣﻦ ﻫﺬﻩ ﺍﻟﺰﻭﺍﻳﺎ ﻟﻬﺎ ﺃﺳﻤﺎﺀ ﺧﺎﺻﺔ‬. q , r ‫ﺯﻭﺍﻳﺎ ﻣﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﻴﻦ‬       ∠3, ∠4, ∠5, ∠6   q, r  ✓ ∠1, ∠2, ∠7, ∠8   q, r ‫ﺍﺳﺘﻌﻤﻞ ﺗﻤﺎﺭﻳﻦ ”ﺗﺤﻘﻖ ﻣﻦ ﻓﻬﻤﻚ“ ﺑﻌﺪ ﻛﻞ‬  t  .‫ﻣﺜﺎﻝ؛ ﻟﻠﺘﺤﻘﻖ ﻣﻦ ﻣﺪ￯ ﻓﻬﻢ ﺍﻟﻄﻠﺒﺔ ﺍﻟﻤﻔﺎﻫﻴﻢ‬ 12 ∠6∠3∠5∠4 t 43 q    ∠6∠4∠5∠3    56 r t 87   ‫ ﺣ ﹼﺪﺩ ﻛ ﹰﹼﻼ ﻣﻤﺎ ﻳﺄﺗﻲ ﺑﺎﺳﺘﻌﻤﺎﻝ‬1  ∠8∠2∠7∠1  :‫ﺍﻟﺼﻨﺪﻭﻕ ﺃﺩﻧﺎﻩ‬ t E F ∠6∠2∠5∠1  D C ∠8∠4∠7∠3 t q, r  2  H G ‫ ﺃﻭ ﻣﺘﺒﺎﺩﻟﺘﻴﻦ‬،‫ ﺻ ﱢﻨﻒ ﻛﻞ ﺯﻭﺝ ﻣﻦ ﺍﻟﺰﻭﺍﻳﺎ ﻓﻴﻤﺎ ﻳﺄﺗﻲ ﺇﻟﻰ ﺯﺍﻭﻳﺘﻴﻦ ﻣﺘﺒﺎﺩﻟﺘﻴﻦ ﺩﺍﺧﻠ ﹰﹼﻴﺎ‬،‫ﻣﺴﺘﻌﻤ ﹰﻼ ﺍﻟﺸﻜﻞ ﺍﻟﻤﺠﺎﻭﺭ‬ A B :‫ ﺃﻭ ﻣﺘﺤﺎﻟﻔﺘﻴﻦ‬،‫ ﺃﻭ ﻣﺘﻨﺎﻇﺮﺗﻴﻦ‬،‫ﺧﺎﺭﺟ ﹼﹰﻴﺎ‬ 8 1 2 ∠6 ‫∠ ﻭ‬7 (b ∠1 ‫∠ ﻭ‬5 (a 7 ‫ﻣﺘﺤﺎﻟﻔﺘﺎﻥ‬ ‫ﻣﺘﺒﺎﺩﻟﺘﺎﻥ ﺧﺎﺭﺟ ﹰﹼﻴﺎ‬ 6 34 ‫( ﺟﻤﻴﻊ ﺍﻟﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﺍﻟﺘﻲ‬a 5 ∠2 ‫∠ ﻭ‬6 (d ∠2 ‫∠ ﻭ‬4 (c ‫ﻣﺘﺒﺎﺩﻟﺘﺎﻥ ﺩﺍﺧﻠ ﹰﹼﻴﺎ‬ ‫ﻣﺘﻨﺎﻇﺮﺗﺎﻥ‬ ___ ___ ___ ___ ‫( ﻣﺘﺒﺎﺩﻟﺘﺎﻥ ﺩﺍﺧﻠ ﹼﹰﻴﺎ‬2A ‫( ﻣﺘﻨﺎﻇﺮﺗﺎﻥ‬2B AD, EH, FG . BC ‫ﺗﻮﺍﺯﻱ‬ ‫( ﻣﺘﺒﺎﺩﻟﺘﺎﻥ ﺧﺎﺭﺟ ﹼﹰﻴﺎ‬2C ___ ‫ﺗﺨﺎﻟﻒ‬ ‫ﻣﺴﺘﻘﻴﻤﺔ‬ ‫ﻗﻄﻌﺔ‬ (b  ✓ ‫( ﻣﺘﺤﺎﻟﻔﺘﺎﻥ‬2D .EH ___ ___ ___ ___ ∠3 ‫∠ ﻭ‬2 (2D ∠8 ‫∠ ﻭ‬4 (2C ∠7 ‫∠ ﻭ‬5 (2B ∠7 ‫∠ ﻭ‬3 (2A CF ‫ ﺃﻭ‬BG ‫ ﺃﻭ‬CD ‫ ﺃﻭ‬AB .ABG ￯‫( ﻣﺴﺘﻮ￯ ﻳﻮﺍﺯﻱ ﺍﻟﻤﺴﺘﻮ‬c 87  2- 1 CDE ￯‫ﺍﻟﻤﺴﺘﻮ‬ ‫ ﺍﺭﺳﻢ ﺷﻜ ﹰﻼ‬ ! ‫ﺛﻼﺛﻲ ﺍﻷﺑﻌﺎﺩ ﻋﻠﻰ ﺍﻟﺴﺒﻮﺭﺓ )ﻣﻨﺸﻮﺭ‬ ‫ ﻋﻨﺪﻣﺎ ﺗﺤﺘﺎﺝ‬ ‫ ﻭﺍﺧﺘﺮ ﻃﻼ ﹰﺑﺎ‬،(‫ﺭﺑﺎﻋﻲ ﺃﻭ ﻫﺮﻡ ﺭﺑﺎﻋﻲ‬ ‫ﺇﻟﻰ ﺃﻥ ﺗﻘﺮﺭ ﻣﺎ ﺇﺫﺍ ﻛﺎﻧﺖ ﺍﻟﻘﻄﻊ‬ ‫ﻟﻴﺤ ﱢﺪﺩﻭﺍ ﺍﻷﺣﺮﻑ ﺍﻟﺘﻲ ﻳﻮﺍﺯﻱ ﺑﻌﻀﻬﺎ‬ ￯‫ ﻓﻼ ﺗﻤ ﹼﺪﻫﺎ ﻟﺘﺮ‬،‫ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﻣﺘﻘﺎﻃﻌﺔ‬ .‫ﺑﻌ ﹰﻀﺎ‬ ‫ﺇﻥ ﻛﺎﻧﺖ ﺗﺘﻘﺎﻃﻊ ﺇﺫﺍ ﹸﻣﺪ ﹾﺕ ﺃﻛﺜﺮ؛ ﻷﻥ‬ .‫ﻟﻠﻘﻄﻊ ﺍﻟﻤﺴﺘﻘﻴﻤﺔ ﻃﻮ ﹰﻻ ﻣﺤﺪ ﹰﺩﺍ‬ 87  2-1