1. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 

2. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 

‫ﻣ‬‫ﻫﻨﺪﺳﻴﺔ‬ ‫ﻔﺎﻫﻴﻢ‬ 
 
‫ﺍﻟﺘﻄﺎﺑﻖ‬ 
 
‫ﺍﻟﻤﺜﻠﺜﺎﺕ‬ ‫ﺗﻄﺎﺑﻖ‬
 
‫ﺍﻟﺘﻮﺍﺯﻱ‬ 
 
‫ﺍﻟﻬﻨﺪﺳﻴﺔ‬ ‫ﺍﻷﻧﺸﺎﺀﺍﺕ‬ 
 

3. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 
‫ﺍﻷﻭﻝ‬ ‫ﺍﻟﺪﺭﺱ‬
 
 
 
)(:- 
 
 
 
 
 
 
 
 
)(:- 
 
 
 
 
 
 
)(:- 
 
 
 
 
 
 
)(:- 
 
 
 
 
 
–
‫ﻫﻨﺪﺳﻴﺔ‬ ‫ﻣﻔﺎﻫﻴﻢ‬

hf.hf=hf=٤‫ﺳﻢ‬
hf،hf
hfhfhf
):hf≠hf(
h‫ب‬ ٤‫ﺳﻢ‬
 
 
hf 
:-hf‫ﺥ‬hfhf

)(
hf.hf≠fh
)hf‫ﺥ‬hf‫ﺥ‬hf(
 
‫ﺲ‬‫ﺳ‬،‫ﺹ‬،.......
):-hf‫ﺲ‬‫ﺳ‬ ‫ﺥ‬(
)hf‫ﺥ‬hf‫ﺥ‬hf(
‫أ‬
‫ب‬
‫ﺳ‬ h
‫ب‬
hf

4. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 
)(:- 
 
 
 
 
 
 
 
 
‫ﺍﻟﺰﺍﻭﻳﺔ‬ ‫ﻗﻴﺎﺱ‬:-
 
)٥
(،)/
(،)//
(
١٥
=٦٠/
،١/
=٦٠//
)hfO(≠R)hfO(
)hfO(fh،fO.
R)hfO(.
‫ﺍﻟﺰﻭﺍﻳﺎ‬ ‫ﺃﻧﻮﺍﻉ‬:-‫ﻫﻲ‬ ‫ﺃﻧﻮﺍﻉ‬ ‫ﻋﺪﺓ‬ ‫ﺇﱃ‬ ‫ﺎ‬‫ﻗﻴﺎﺳﺎ‬ ‫ﲝﺴﺐ‬ ‫ﺍﻟﺰﻭﺍﻳﺎ‬ ‫ﺗﻨﻘﺴﻢ‬:
‫ﺍﻟﺰﺍﻭﻳﺔ‬‫ﻗﻴﺎﺳﻬﺎ‬‫ﺭﲰﻬﺎ‬
‫ﺯﺍﻭﻳﺔ‬‫ﺻﻔﺮﻳﺔ‬
=٥

‫ﺯﺍﻭﻳﺔ‬‫ﺣﺎﺩﺓ‬
 
‫ﺻﻔﺮ‬٥
٩٠٥
‫ﺯﺍﻭﻳﺔ‬‫ﻗﺎﺋﻤﺔ‬
=٥
٩٠=٦٠/
٨٩٥
=٦٠//
٥٩/
٨٩٥
‫ﺯﺍﻭﻳﺔ‬‫ﻣﻨﻔﺮﺟﺔ‬
٩٠٥

١٨٠٥
‫ﺯﺍﻭﻳﺔ‬‫ﻣﺴﺘﻘﻴﻤﺔ‬
=٥
١٨٠=٦٠/
١٧٩٥
=٦٠//
٥٩/
١٧٩٥
‫ﺯﺍﻭﻳﺔ‬‫ﻣﻨﻌﻜﺴ‬‫ﺔ‬
١٨٠٥

٣٦٠٥
.
f 
 
 
)‫ب‬hO(،)Ohf(،)h(
fhhO
‫ﺯ‬ ‫ﺯ‬ ‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
O
f hO
f h
f
h
f
O
f hO
h
f
O
O
h
■
h
‫ﺿﻠﻊ‬
‫ﺿﻠﻊ‬
‫رأس‬
f
O

5. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 

‫ﻣﻼﺣﻈﺔ‬:-
٣٦٠٥
‫ﻣﺜﺎﻝ‬:R)hfO(=١٢٠٥
:R)hfO(=٣٦٠٥
‫ـ‬‫ــ‬١٢٠٥
=٢٤٠٥
)١(‫اﻻﺗﻰ‬ ‫اﻟﺠﺪول‬ ‫اﻛﻤﻞ‬:-
R)hfO(٤٥٥
١٨٠٥
٢٠٠٥
١٥٠٥
٩٠٥
٦٠/
١٧٩٥
‫ﻧﻮﻋﻬﺎ‬
)٢(‫اﻻﺗﻰ‬ ‫اﻟﺠﺪول‬ ‫اﻛﻤﻞ‬:-
R)hfO(١٣٥٥
٥٨٥
٨٠٥
١٠٠٥
١١٠٥
٩٠٥
R)hfO(‫ﺍﳌﻨﻌﻜﺴﺔ‬

)(‫ﺍﳌﺘﺠﺎﻭﺭﺗﺎﻥ‬ ‫ﺍﻟﺰﺍﻭﻳﺘﺎﻥ‬:- 
‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﻓﻰ‬:-
)hfx(،)xfO(:-
ffx
fh،fOfx
 
‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﻓﻰ‬:
)١()hfx(،)xfO(
fO،fh‫ب‬x
‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﰲ‬:
)٢()hfO(،)Oxi(
 
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺍﻟﺰﻭﺍﻳﺎ‬ ‫ﺑﲔ‬ ‫ﺍﻟﻌﻼﻗﺎﺕ‬ ‫ﺑﻌﺾ‬
 
h
xO
f
‫ﺯ‬‫ﺯ‬
‫ﻫﺎﻣﺔ‬ ‫ﻣﻼﺣﻈﺎﺕ‬
‫ﺯ‬ ‫ﺯ‬
h
f
O
x
‫ﺯ‬‫ﺯ‬
hf
i
O
x

6. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 
 
)(‫ﺍﳌﺘ‬ ‫ﺍﻟﺰﺍﻭﻳﺘﺎﻥ‬‫ﺘﺎﻣﺘﺎﻥ‬:- 
َ‫ﻼ‬‫ﻓﻤﺜ‬:٣٠°،٦٠°:٣٠°+٦٠°=٩٠°
)(‫ﺍﳌﺘ‬ ‫ﺍﻟﺰﺍﻭﻳﺘﺎﻥ‬‫ﻜﺎﻣﻠﺘﺎﻥ‬:- 
َ‫ﻼ‬‫ﻓﻤﺜ‬:١٢٥°،٥٥°:١٢٥°+٥٥°=١٨٠°
)(‫ﺍﻟﺰﺍﻭﻳﺘﺎﻥ‬‫ﺍﳌﺘﺠﺎﻭﺭﺗﺎﻥ‬‫ﺍﳌﺘ‬‫ﻜﺎﻣﻠﺘﺎﻥ‬:- 
 
 
=° 
 
Y                                                       
Y                                                          
Y                   )                        (                   
َ‫ﻼ‬‫ﻓﻤﺜ‬:‫ﻛﺎن‬ ‫إذا‬:
)f(
)h(‫ﻓﺈن‬:R)‫ب‬(=R)O(
)O(
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺗﺘﻤﻢ‬
‫ﺗﺘﻤﻢ‬
=° 
 
Y                                              
Y                                                               
Y                  )                        (                   
َ‫ﻼ‬‫ﻓﻤﺜ‬:‫ﻛﺎن‬ ‫إذا‬:
)f(
)h(‫ﻓﺈن‬:R)‫ب‬(=R)O(
)O(
‫ﺗﻜﻤﻞ‬
‫ﺗﻜﻤﻞ‬
‫ﺯ‬‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬

=° 

7. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 
)(‫ﺍﻟﺰﺍﻭﻳﺘﺎﻧ‬‫ﺑﺎﻟﺮﺃﺱ‬ ‫ﺎﳌﺘﻘﺎﺑﻠﺘﺎﻥ‬:- 
 
h‫ب‬∩Ox=}‫م‬{
:R)h‫م‬x(=R)O‫ب‬ ‫م‬(.
،R)h‫م‬O(=R)x‫م‬f(.
)(‫ﻧﻘﻄﺔ‬ ‫ﺣﻮﻝ‬ ‫ﺍﳌﺘﺠﻤﻌﺔ‬ ‫ﺍﻟﺰﺍﻭﻳﺎ‬:- 
)١(‫ﻓﻤﺜﻼ‬َ:‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﻓﻰ‬:
:R)‫ب‬hx(=٣٥°،R)xhi(=١٠٠°
،R)ihO(=٤٥°
‫ﻓﺈﻥ‬:R)‫ب‬hx(+R)xhi(+R)Ohi(
=٣٥°+١٠٠°+٤٥°=١٨٠°
Eh‫ب‬،hO.
 
‫ﺯ‬‫ﺯ‬
‫ﺯ‬ ‫ﺯ‬‫م‬
●
hx
f O
●
XX
=° 
‫ﻣﺘﺠﺎﻭﺭﺗﲔ‬ ‫ﻟﺰﺍﻭﻳﺘﲔ‬ ‫ﺍﳌﺘﻄﺮﻓﺎﻥ‬ ‫ﺍﻟﻀﻠﻌﺎﻥ‬
)=°(
 
 
)≠°(
 
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬ ‫ﺯ‬
٣٥
٥
fhO
x
٤٥
٥
i
١٠٠
٥

8. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 
)٢(‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﻓﻰ‬:
‫ﻫﻞ‬Oh،O‫ب‬‫ﻻ‬ ‫ﺃﻡ‬ ‫ﻭﺍﺣﺪﺓ‬ ‫ﺇﺳﺘﻘﺎﻣﺔ‬ ‫ﻋﻠﻲ‬،‫؟‬ ‫ﻭﳌﺎﺫﺍ‬
eR)h‫ب‬x+ (R)x‫ب‬O= (٧٠+١٠٠
=١٧٠°≠°.
E‫ب‬h،‫ب‬O.
‫ﹶ‬‫ﻼ‬‫ﻓﻤﺜ‬:‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﻓﻰ‬:
:R)h‫ب‬ ‫م‬(=٣٠°،R)‫ب‬‫م‬O(=٦٠°
‫ﻓﺈﻥ‬:R)h‫ب‬ ‫م‬(+R)‫ب‬‫م‬O(
=٣٠°+٦٠°=٩٠°
E)h‫م‬O(‫ﻗﺎﺋﻤﺔ‬ ‫ﺯﺍﻭﻳﺔ‬.
E‫م‬hn‫م‬O.
‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﻓﻔﻲ‬:
fx)hfO(
R)hfx(=R)xfO= (R)hfO(
fx)hfO(
‫ﺯ‬‫ﺯ‬١٠٠٥
h
٧٠٥
‫ب‬
O
x
‫ﺯ‬‫ﺯ‬
‫ﺯ‬‫ﺯ‬
‫ﺯ‬‫م‬
O
h
f
٦٠٥
٣٠٥
١
٢
‫ﺯ‬ ‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺢ‬‫ﳒ‬
‫ﺢ‬‫ﳒ‬
h‫ب‬
O
x
‫ﻣﻼﺣﻈــــــــﺔ‬
)=°( 
‫ﺍﻟﺰﺍﻭﻳﺔ‬ ‫ﻣﻨﺼﻒ‬
 

9. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 
)١(‫ﺍﻻﺗﻰ‬ ‫ﺍﳉﺪﻭﻝ‬ ‫ﺍﻛﻤﻞ‬:-
‫ﺍﻟﺰﺍﻭﻳﺔ‬ ‫ﻗﻴﺎﺱ‬٧٥٥
٩٠٥
٢٠٥
٤٥٥
‫ﻣﺘﻤﻤﺘﻬﺎ‬ ‫ﻗﻴﺎﺱ‬٦٧٥
٣٠٥
‫ﻣﻜﻤﻠﺘﻬﺎ‬ ‫ﻗﻴﺎﺱ‬١٥٤٥
٣٠/
١١٥٥
)٢(‫ﺍﻟﺘﺎﻟﻴﺔ‬ ‫ﺍﻷﺷﻜﺎﻝ‬ ‫ﻣﻦ‬ ‫ﻛﻞ‬ ‫ﰲ‬:
‫ﻛﺎﻧﺖ‬ ‫ﺇﺫﺍ‬:‫م‬Jhf‫ﺑﺎﻟﻌﻼﻣﺔ‬ ‫ﺇﻟﻴﻬﺎ‬ ‫ﺍﳌﺸﺎﺭ‬ ‫ﺍﻟﺰﺍﻭﻳﺔ‬ ‫ﻗﻴﺎﺱ‬ ‫ﻓﺄﻭﺟﺪ‬)‫؟‬: (
R)x‫ب‬ ‫م‬(=...........R)x‫ب‬ ‫م‬(=...........R)x‫ب‬ ‫م‬(=...........
)٣(‫ﰲ‬‫ﺍﻟﺘﺎﻟﻴﺔ‬ ‫ﺍﻷﺷﻜﺎﻝ‬ ‫ﻣﻦ‬ ‫ﻛﻞ‬:
OhOf 
 
 
 
 
)٤(‫ﺍﻟﺘﺎﻟﻴﺔ‬ ‫ﺍﻷﺷﻜﺎﻝ‬ ‫ﻣﻦ‬ ‫ﻛﻞ‬ ‫ﰲ‬:
fhnfO 
 
 
 
 
 
R)h‫ب‬x(=...........R)x‫ب‬O(=...........‫س‬=........... 
–
‫ﺯ‬
h
x
‫م‬‫ب‬
٥٠°‫؟‬
٣٥°
١٠٠°
h
x
‫م‬‫ب‬
‫؟‬
O
h
x
‫م‬‫ب‬
٤٥°
‫؟‬
■
O
‫ﺯ‬‫ﺯ‬
* ٦٠°*
h
x
O‫ب‬
i
١٠٨° ٦٢°
x
O ‫ب‬ h
٣٥°
١٠٠°
h
x
‫م‬‫ب‬
‫؟‬
O
‫ب‬
O
h
x
٦٥٥
‫ب‬
O
h
x
*
* *
i
x
‫ب‬
O
h
‫س‬
٤٠°
*
i
‫ﺯ‬‫ﺯ‬

10. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 
 
)٥(‫ﺍﻟﺰﺍﻭﻳﺔ‬ ‫ﻗﻴﺎﺱ‬ ‫ﺃﻭﺟﺪ‬ ‫ﺍﻟﺘﺎﻟﻴﺔ‬ ‫ﺍﻷﺷﻜﺎﻝ‬ ‫ﻣﻦ‬ ‫ﻛﻞ‬ ‫ﰲ‬‫ﺷﻜﻞ‬ ‫ﻛﻞ‬ ‫ﺃﺳﻔﻞ‬ ‫ﺍﻟﻄﻠﻮﺑﺔ‬:
 
 
 
 
 
R)‫ب‬‫م‬O(=...........R)O‫م‬x(=...........R)h‫م‬‫ب‬(=........... 
 
 
 
 
 
 
 
R)h‫م‬x(=...........R)h‫م‬x(=...........R)‫ب‬‫م‬x(=........... 
 
)٦(‫ﺍﻟﺘﺎﻟﻴﺔ‬ ‫ﺍﻷﺷﻜﺎﻝ‬ ‫ﻣﻦ‬ ‫ﻛﻞ‬ ‫ﰲ‬:
‫م‬O)h‫م‬x( 
 
 
 
 
 
 
 
R)x‫م‬O(=...........R)‫ب‬‫م‬i(=...........R)i‫م‬‫ب‬(=...........
 
 
 
  
 
 
O
f
‫م‬ ١٥٠°
١٠٠°
h
h
f
O
x
i
‫م‬ ١١٠°
٤٥°
٩٠°
‫م‬
h
f
O
x
٥٠°
١٠٠°
١٤٠°
••
‫م‬
■
h
f
O
x
٥٤°
٨٦°‫م‬
f
h
O • •
x
١٢٠°
•
• ‫م‬
١١٠°
٧٠° • h
f
O
x
‫ﺯ‬‫ﺯ‬‫ﺯ‬
‫ﺯ‬‫ﺯ‬‫ﺯ‬
‫ﺯ‬
‫م‬
x
١١٠°
٩٠° ‫ب‬
h
O
i
٥٠°
٤٥°
h
O
‫م‬‫ب‬
x
■
h
‫م‬ i
f
O
x
‫ﺯ‬‫ﺯ‬ ‫ﺯ‬

11. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 
‫ﺍ‬ ‫ﺍﻟﺪﺭﺱ‬‫ﻟﺜﺎﻟﺚ‬
 
 
:h‫ب‬=Oxh‫ب‬≡Ox
‫ﺻﺤﻴﺢ‬ ‫ﻭﺍﻟﻌﻜﺲ‬ 
 
.
:h‫ب‬≡Ox:h‫ب‬=Ox
 
 
 
 
 
:R)h‫ب‬O= (R)‫ع‬ ‫ص‬ ‫س‬(
:)h‫ب‬O(≡)‫ع‬ ‫ص‬ ‫س‬(
‫ﺻﺤﻴﺢ‬ ‫ﻭﺍﻟﻌﻜﺲ‬ 
 
 
 
:)h‫ب‬O(≡)‫ع‬ ‫ص‬ ‫س‬(
:R)h‫ب‬O= (R)‫ع‬ ‫ص‬ ‫س‬(
َ ‫ﺃﻭﻻ‬:‫ﻣﺴﺘﻘﻴﻤﺘﲔ‬ ‫ﻗﻄﻌﺘﲔ‬ ‫ﺗﻄﺎﺑﻖ‬:-

f h
Ox
َ‫ﺛﺎﻧﻴﺎ‬:‫ﺯﺍﻭﻳﺘﲔ‬ ‫ﺗﻄﺎﺑﻖ‬:-
‫ﺯ‬
‫ﺯ‬‫ﺯ‬
‫ﺯ‬
‫ﺯ‬‫ﺯ‬
‫ﺯ‬
h‫ﺢ‬‫ﳒ‬
f
O
‫ع‬
‫ﺢ‬‫ﳒ‬
‫س‬‫ص‬
‫ﺯ‬
 
.

12. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 
‫ﻓﻤﺜﻼ‬::
)١(.
‫أن‬ ‫أي‬:
h‫ب‬=‫ل‬ ‫س‬،‫ب‬O=‫ع‬ ‫ل‬
Ox=‫ص‬ ‫ع‬،hx=‫ص‬ ‫س‬
)٢(.
‫أن‬ ‫أي‬:
R)h= (R)‫س‬(،R)‫ب‬= (R)‫ل‬(
R)O= (R)‫ع‬(،R)x= (R)‫ص‬(
h‫ب‬Ox≡‫ل‬ ‫س‬‫ص‬ ‫ع‬
 
َ‫ﺛﺎﻟﺜﺎ‬:‫ﻣﻀﻠﻌﲔ‬ ‫ﺗﻄﺎﺑﻖ‬)‫ﻫﻨﺪﺳﻴﲔ‬ ‫ﺷﻜﻠﲔ‬ ‫ﺃﻯ‬(:-

 
‫ﺯ‬‫ﺯ‬‫ﺯ‬‫ﺯ‬
‫ﺯ‬ ‫ﺯ‬‫ﺯ‬ ‫ﺯ‬
 
‫ﲢﻘﻖ‬ ‫ﺇﺫﺍ‬ ‫ﺍﳌﻀﻠﻌﺎﻥ‬ ‫ﻳﺘﻄﺎﺑﻖ‬‫ﺍﻟﺸﺮﻃﺎﻥ‬ 
)١(
)٢(
‫ﺻﺤﻴﺢ‬ ‫ﻭﺍﻟﻌﻜﺲ‬:‫ﻓﺈﻥ‬ ‫ﻣﻀﻠﻌﺎﻥ‬ ‫ﺗﻄﺎﺑﻖ‬ ‫ﺇﺫﺍ‬ ‫ﺃﻧﻪ‬ ‫ﺃﻱ‬ 
)١( 
)٢( 
O
x‫ص‬
‫ع‬
‫ل‬ ‫س‬ h ‫ب‬

13. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 
‫ﻓﻤﺜﻼ‬::h‫ب‬Ox≡h‫ص‬ ‫س‬ ‫ب‬
‫ﻓﺈ‬‫ن‬:
)١(‫ب‬O=‫ب‬‫س‬
،Ox=‫ص‬ ‫س‬
،h x=h، ‫ص‬h‫ب‬
)٢(R)x= (R)‫ص‬(
،R)O= (R)‫س‬(
،R)xh‫ب‬= (R)‫ص‬h‫ب‬(
،R)h‫ب‬O= (R)h‫ب‬‫س‬(
‫ﺑﻨﻔﺴﻚ‬ ‫ﺣﺎﻭﻝ‬:‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﻓﻰ‬:- 
‫ص‬ ‫س‬&‫م‬≡h‫ب‬O‫م‬
 
١(‫ص‬ ‫س‬≡.............٢(‫ص‬&................... =
٣(‫م‬O≡...............٤(h‫م‬............. =‫ﺳﻢ‬
٥(h‫س‬............ =‫ﺳﻢ‬٦(R)‫ص‬= (R(......... )
٧(R)O= (R(........ )٨(R)‫س‬= (R(......... )
٩(R)O‫م‬h...... = (٥
١٠(R)h.............. = (٥
١١(R)&‫م‬O....... = (٥
١٢(R)O‫س‬ ‫م‬....... = (٥
‫ﺯ‬
‫ﺯ‬ ‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬‫ﺯ‬
x
h
‫ص‬
‫س‬ ‫ب‬ O
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ص‬
&O
‫ب‬
‫س‬h ‫م‬
٦٥٥
٦‫ﺳﻢ‬
■ ■

14. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 

)(‫ﻳﺄﺗﻲ‬ ‫ﻣﺎ‬ ‫ﺃﻛﻤﻞ‬:-
١(‫ﻛﺎﻧﺘﺎ‬ ‫اذا‬ ‫اﻟﻤﺴﺘﻘﯿﻤﺘﺎن‬ ‫اﻟﻘﻄﻌﺘﺎن‬ ‫ﺗﺘﻄﺎﺑﻖ‬.............................
٢(‫ﻛﺎﻧﺘﺎ‬ ‫اذا‬ ‫اﻟﺰاوﯾﺘﺎن‬ ‫ﺗﺘﻄﺎﺑﻖ‬............................................
٣(‫اذ‬ ‫اﻟﻤﻀﻠﻌﺎن‬ ‫ﯾﺘﻄﺎﺑﻖ‬‫ﻛﻞ‬ ‫ﯾﻄﺎﺑﻖ‬ ‫ﺑﺤﯿﺚ‬ ‫رءوﺳﯿﮭﻤﺎ‬ ‫ﺑﯿﻦ‬ ‫ﺗﻘﺎﺑﻞ‬ ‫وﺟﺪ‬.........‫وﻛﻞ‬...........‫اﻟﻤﻀﻠﻊ‬ ‫ﻓﻰ‬
‫ﻓﻰ‬ ‫ﻧﻈﯿﺮه‬ ‫اﻻول‬..............................
٥(‫ﺷﻜﻠﯿﻦ‬ ‫اﻟﻰ‬ ‫ﯾﻘﺴﻤﮫ‬ ‫اﻟﺸﻜﻞ‬ ‫ﺗﻤﺎﺛﻞ‬ ‫ﻣﺤﻮر‬.............................
٦(‫ﻛﺎن‬ ‫إذا‬h‫ب‬≡Ox‫ﻓﺎن‬h‫ب‬=........................
٧(‫ﻛﺎن‬ ‫إذا‬)h(≡)‫ب‬(‫وﻛﺎن‬R)h= (٥٠٥
‫ﻓﺈ‬‫ن‬R)‫ب‬........ = (٥
٨(‫ص‬ ‫س‬ ‫ﻛﺎن‬ ‫اذا‬≡h‫ب‬‫ﻓﺈ‬‫ن‬:‫ص‬ ‫س‬–h‫ب‬................... =
٩(‫اﻟﻤﻀﻠﻊ‬ ‫ﻛﺎن‬ ‫إذا‬h‫ب‬Ox≡‫ل‬ ‫ع‬ ‫ص‬ ‫س‬ ‫اﻟﻤﻀﻠﻊ‬‫ﻓﺈ‬‫ن‬:R)hOx................. = (
١٠(‫ﻛﺎﻧﺖ‬ ‫اذا‬O‫ﻣﻨﺘﺼﻒ‬h‫ب‬‫ﻓﺈ‬‫ن‬hO..........‫ب‬O
)(‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﰲ‬:-:-
١(‫اﻟﺮأس‬‫ب‬‫ﺗﻨﺎﻇﺮ‬‫ا‬‫ﻟﺮأس‬............
٢(‫اﻟﻤﻀﻠﻊ‬h‫ب‬O x≡‫اﻟﻤﻀﻠﻊ‬.......
٣(h‫ب‬≡....................
٤(‫ل‬&............. =‫ﺳﻢ‬
٥(R)h= (R(....... )
٦(‫ص‬ ‫س‬.............. =
٧(R)‫ص‬..................... = (
٨(Ox.......................... =
٩(xi‫ـــ‬‫ص‬ ‫ع‬=..................
)(‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﰲ‬:-h‫ب‬O‫و‬≡‫اﻟﻤﻀﻠﻊ‬ixO‫و‬:-
١(h‫ب‬.......................... =
٢(h‫و‬.......................... =
٣(Ox...................... =
٤(O‫ﺿﻠﻊ‬ ‫و‬.................
٥(R)i= (R(........ )
٦(R)‫ب‬= (R(........... )
٧(R)‫و‬Ox= (R(.......... )٨(R)i‫و‬O...... = (٥
٩(‫ب‬x........ =‫ﺳﻢ‬١٠(R)‫و‬Ox...... = (٥
١١(R)h‫و‬i............................ = (٥
١٢(‫اﻟﻤﻘﺎﺑﻞ‬ ‫اﻟﺸﻜﻞ‬ ‫ﺗﻤﺎﺛﻞ‬ ‫ﻣﺤﻮر‬.................
‫ﺢ‬‫ﳒ‬ ×
‫ﺽ‬‫ﻆ‬‫ﻇ‬
‫ل‬
‫س‬‫ص‬
‫ع‬
‫ﺢ‬‫ﳒ‬ ×
‫ﺽ‬‫ﻆ‬‫ﻇ‬ ‫أ‬
‫ب‬‫ﺟـ‬
‫ء‬
‫ﺯ‬‫ﺯ‬ ‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬

h
١٢٠٥
٥‫ﺳﻢ‬
‫و‬
i
x O ‫ب‬

15. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 
‫ﺍ‬ ‫ﺍﻟﺪﺭﺱ‬‫ﻟﺮﺍﺑﻊ‬
 
 
:h‫ب‬،‫ب‬O،hO
:)h(،)‫ب‬(،)O(
 
 
 
 
 
h‫ب‬O‫ع‬ ‫ص‬ ‫س‬ ،:
)١(h‫ب‬=‫ص‬ ‫س‬،hO=، ‫ع‬ ‫س‬‫ب‬O=‫ع‬ ‫ص‬
)٢(R)h= (R)‫س‬(،R)‫ب‬= (R)‫ص‬(،R)O= (R)‫ع‬(
:∆h‫ب‬O≡´‫ع‬ ‫ص‬ ‫س‬
‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﰲ‬:-‫إ‬‫ﺛﺒﺖ‬‫أ‬‫ن‬:h∆‫ب‬O≡∆xi‫و‬
h∆∆‫ب‬O،xi‫و‬
h‫ب‬=xi
‫ﻓﯿﮭﻤﺎ‬R)‫ب‬(=R)i(
‫ب‬‫ﺟـ‬=i‫و‬
Eh∆‫ب‬O≡∆xi‫و‬
hO=x‫و‬
R)h= (R)x(
R)O= (R)‫و‬(

‫ﺯ‬‫ﺯ‬‫ﺯ‬
‫ﺯ‬‫ﺯ‬‫ﺯ‬ ‫ﺯ‬‫ﺯ‬ ‫ﺯ‬
 




 

 


 
‫ﺯ‬ ‫ﺯ‬
‫ﺯ‬
‫ﺯ‬‫ﺯ‬
‫ﺯ‬
h
fO
h
f‫ع‬ ‫ص‬
O
‫س‬
* *
h
‫ب‬O
x
i‫و‬

16. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 
‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﰲ‬:-‫أ‬‫ﺛﺒﺖ‬‫أ‬‫ن‬:∆h‫ب‬O≡∆xi‫و‬
h∆∆‫ب‬O،xi‫و‬
R)O= (R)‫و‬(
‫ﻓﯿﮭﻤﺎ‬R)‫ب‬= (R)i(
‫ب‬‫ﺟـ‬=i‫و‬
Eh∆‫ب‬O≡∆xi‫و‬
h‫ب‬=xi
hO=x‫و‬
R)h= (R)x(
‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﰲ‬:-‫أ‬‫ﺛﺒﺖ‬‫أ‬‫ن‬:h∆‫ب‬O≡∆xi‫و‬
∆∆h‫ب‬O،xi‫و‬
h‫ب‬=xi
‫ﻓﯿﮭﻤﺎ‬hO=x‫و‬
‫ب‬O=i‫و‬
E∆h‫ب‬O≡∆xi‫و‬‫ان‬ ‫وﻳﻨﺘﺞ‬:
R)‫ب‬= (R)i(
R)O= (R)‫و‬(
R)h= (R)x(


 
‫ﺯ‬ ‫ﺯ‬
‫ﺯ‬
‫ﺯ‬‫ﺯ‬
‫ﺯ‬

 
i
•• *
h
fO
*
x
‫و‬
‫ﺯ‬‫ﺯ‬
‫ﺯ‬ ‫ﺯ‬
‫ﺯ‬‫ﺯ‬
i
h
fO
x
‫و‬



17. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 
‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﰲ‬:-‫أ‬‫ﺛﺒﺖ‬‫أ‬‫ن‬:h∆‫ب‬O≡∆xi‫و‬
∆∆h‫ب‬O،xi‫و‬
‫ب‬O=i‫و‬
‫ﻓﯿﮭﻤﺎ‬hO=x‫و‬
R)‫ب‬(=R)i= (٩٠٥
E∆h‫ب‬O≡∆xi‫و‬‫ان‬ ‫وﻳﻨﺘﺞ‬:
h‫ب‬=xi
R)O(=R)‫و‬(
R)h= (R)x(
‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﰲ‬:-
 
 
∆∆h‫ب‬x،hOx
h‫ب‬=hO
‫ﻓﯿﮭﻤﺎ‬‫ب‬x=Ox
hx‫ﻣﺸﺘﺮك‬ ‫ﺿﻠﻊ‬
E∆h‫ب‬x≡h∆Ox‫وﻳﻨﺘﺞ‬‫ان‬
R)‫ب‬(=R)O(
R)‫ب‬hx(=R)Ohx(
R)‫ب‬xh(=R)Oxh(
‫اﻟﺘﻄﺎﺑﻖ‬ ‫ﺣﺎﻟﺔ‬: 


 
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
hx
fOi‫و‬
■■
‫ﺯ‬‫ﺯ‬
‫ﺯ‬‫ﺯ‬
‫ﺯ‬‫ﺯ‬

f
x
O
h

18. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 
‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﰲ‬:-R)O(‫؟‬

∆∆h‫م‬O،‫ب‬‫م‬x
‫م‬h=‫م‬‫ب‬
R)h= (R)‫ب‬(
R)h‫م‬O= (R)‫ب‬‫م‬x= (٨٠٥
.
E∆h‫م‬O≡∆‫ب‬‫م‬x‫وﻳﻨﺘﺞ‬‫أ‬‫ن‬R)O= (R)x= (٦٠٥
‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﰲ‬:-‫ﻣﺎﻳﺄﺗﻰ‬ ‫أﻛﻤﻞ‬:-
∆∆............،.............
١.......................... (
‫ﻓﯿﮭﻤﺎ‬٢........................ (
٣........................ (
E∆............≡∆.............‫وﻳﻨﺘﺞ‬‫ا‬‫ن‬:-
١........................... (
٢........................... (
٣.......................... (
‫اﻟﺘﻄﺎﺑﻖ‬ ‫ﺣﺎﻟﺔ‬.............................................................. :
‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﰲ‬:-‫ﻣﺎﻳﺄﺗﻰ‬ ‫أﻛﻤﻞ‬:-
١(‫ل‬ ‫س‬................. =
٢(R)‫س‬....... = (٥
٣(‫ﺿﻠﻊ‬ ‫ل‬ ‫ص‬..............
٤(‫ع‬ ‫ص‬................. =
٥(R)‫ع‬ ‫ص‬ ‫س‬= (R....... = (............)٥
٦(R)‫ص‬ ‫ل‬ ‫س‬= (R...... )........... = (....٥
‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﰲ‬:-
R)hfO(‫؟‬
‫ﺯ‬
‫ﺯ‬‫ﺯ‬
‫ﺯ‬
‫ﺯ‬‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
h
fxO
•
•
٨٠٥
٦٠٥
‫ص‬
‫ع‬
‫ل‬
‫س‬
٥٠٥
‫ﺯ‬f
O
x
h
٥٠٥
١١٠٥
h
‫ب‬
٥
O
x
‫م‬

19. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 
‫ﺍ‬ ‫ﺍﻟﺪﺭﺱ‬‫ﳋﺎﻣﺲ‬
 
)٣،٥(،)٤،٦(
)١،٥(،)٢،٦(،)٤،٧(،)٣،٨(
)٤،٥(،)٣،٦(
//،:-
R)٣= (R)٥(
R)٤= (R)٦(
R)١= (R)٥(
R)٢= (R)٦(
R)٤= (R)٨(
R)٣= (R)٧(
R)٤+ (R)٥= (١٨٠٥

R)٣+ (R)٦= (١٨٠٥


‫ﳌﺴﺘﻘﻴﻤﲔ‬ ‫ﻣﺴﺘﻘﻴﻢ‬ ‫ﻗﻄﻊ‬ ‫ﻣﻦ‬ ‫ﺍﻟﻨﺎﲡﺔ‬ ‫ﺍﻟﺰﻭﺍﻳﺎ‬:-
)(:-
)(:-
)(:-
٦
١

٢
٤
٥
٧٨

٣
‫ﻣﺘﻮﺍﺯﻳﲔ‬ ‫ﳌﺴﺘﻘﻴﻤﲔ‬ ‫ﻣﺴﺘﻘﻴﻢ‬ ‫ﻗﻄﻊ‬ ‫ﻣﻦ‬ ‫ﺍﻟﻨﺎﲡﺔ‬ ‫ﺍﻟﺰﻭﺍﻳﺎ‬ ‫ﺃﺯﻭﺍﺝ‬ ‫ﺑﲔ‬ ‫ﺍﻟﻌﻼﻗﺔ‬:-
‫ﻓ‬ ‫ﻣﺘﻮﺍﺯﻳﻴﻦ‬ ‫ﻣﺴﺘﻘﻴﻤﻴﻦ‬ ‫ﻣﺴﺘﻘﻴﻢ‬ ‫ﻗﻄﻊ‬ ‫ﺍﺫﺍ‬‫ﺈ‬‫ﻥ‬:-
 
 
 
١٢
٣ ٤
٥٦
٧ ٨




20. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 
‫ﺑﺎﻟﻌﻼﻣﺔ‬ ‫ﺇﻟﻴﻬﺎ‬ ‫ﺍﳌﺸﺎﺭ‬ ‫ﺍﻟﺰﺍﻭﻳﺔ‬ ‫ﻗﻴﺎﺱ‬ ‫ﺃﻭﺟﺪ‬ ‫ﺍﻵﺗﻴﺔ‬ ‫ﺍﻷﺷﻜﺎﻝ‬ ‫ﻣﻦ‬ ‫ﻛﻞ‬ ‫ﻓﻰ‬)‫؟‬(‫ﺍﻟﺴﺒﺐ‬ ‫ﺫﻛﺮ‬ ‫ﻣﻊ‬:-

‫ﺷﻜﻞ‬)١(:R)‫ن‬ ‫م‬x(=°R)‫ن‬ ‫م‬x(=R)h‫ن‬ ‫م‬()(
‫ﺷﻜﻞ‬)٢(:R)‫ن‬ ‫و‬ ‫ع‬(=°R)‫و‬ ‫ع‬i(=R)‫س‬i‫م‬(=°)(
:R)‫ن‬ ‫و‬ ‫ع‬(=°-°=°
‫ﺷﻜﻞ‬)٣(:R)fOx(=°:R)f(=R)h(=°)( 
)f()O( 
R)fOx(=°‫ـــ‬°=° 
 
‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﰲ‬:-
‫اوﺟﺪ‬:R)f(،R)O(،R)fhO(‫؟‬   
 
 
xi........... //،hf
R)f= (R........... = (......... )٥
‫ﺑـ‬..............)١(
xi........... //،hO
R)O= (R........... = (.............. )٥
‫ﺑـ‬.........)٢(
R)xhi................ = (٥

R)fhO= (١٨٠٥
–......... )٥
............ +٥
(
R)hfO= (١٨٠٥
–............٥
........... =٥
)٣(
 
‫م‬
x
‫؟‬
i h
f O
°
hf
Ox
i
‫ن‬
‫و‬
°
‫؟‬›
›
i
‫س‬‫ص‬
‫و‬
‫م‬
‫ل‬ ‫ع‬
‫ن‬
‫؟‬
°
›
›
‫ﺷﻜﻞ‬)٢( ‫ﺷﻜﻞ‬)١(‫ﺷﻜﻞ‬)٣(
‫ﺯ‬‫ﺯ‬‫ﺯ‬
‫ﺯ‬‫ﺯ‬‫ﺯ‬
‫ﺯ‬
‫ﺯ‬‫ﺯ‬‫ﺯ‬
‫ﺯ‬‫ﺯ‬
‫ﺯ‬
›
›
‫ﺯ‬ ‫ﺯ‬‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
h
O
x
٧٠٥
f
٥٠٥
i


21. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 
‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﰲ‬:-R)hOi(‫؟‬
 
 
hf//Ox،............
R)hOx= (١٨٠٥
–...........
.............. =
Ox//‫و‬i،...........
R)xOi.............. = (–..................=................
R)hOi(.................... =+.....................=...............٥
R)hOi= (٣٦٠٥
–..........=...............٥
‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﰲ‬:-R)xOi(‫؟‬

hf//Ox،fO
R)xOf= (١٨٠٥
–٦٠٥
=١٢٠٥

=٣٦٠٥
 
R)xOi(=٣٦٠٥
–)١٢٠٥
+١٢٠٥
(
=٣٦٠٥
–٢٤٠٥
=١٢٠٥
‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﰲ‬:-R)ihf(،R)O(،R)ihO(‫؟‬

hi//fO،hf
R)ihf(=R)f= (٥٠٥
)١(
hi//fO،Oh
R)O(=R)xhi= (٧٠٥
)٢(
R)xhO= (١٨٠٥
)(
R)fhO= (١٨٠٥
–)٧٠٥
+٥٠٥
(
=١٨٠٥
–٥
١٢٠=٦٠٥
R)ihO= (٥٠٥
+٦٠٥
=١١٠٥
)٣(
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
O
‫و‬i
h
f
x
٤٠٥
٣٠٥
f
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬ ‫ﺯ‬‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬
‫ﺯ‬h
٥٠٥
i
fO
x
٧٠٥
١٢٠
٥
O
hx
i
٦٠
٥
‫ﺯ‬
‫ﺯ‬

22. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 
‫ﻳﻜﻮﻥ‬ ‫ﳌﺎﺫﺍ‬ ‫ﺑﲔ‬ ‫ﺍﻵﺗﻴﺔ‬ ‫ﺍﻷﺷﻜﺎﻝ‬ ‫ﻣﻦ‬ ‫ﻛﻞ‬ ‫ﻓﻰ‬hf//Ox‫ﺍﻟﺴﺒﺐ‬ ‫ﺫﻛﺮ‬ ‫ﻣﻊ‬:-
fh//Ox‫ﻷﻥ‬:fh//Ox‫ﻷﻥ‬:fh//Ox‫ﻷﻥ‬:
R)ihO= (R)ihO= (°R)ihO= (R)ihO= (°R)ihO= (R)ihO= (°
)) () (
(
 
‫ﻣﺴﺘﻘﻴﻤﲔ‬ ‫ﺗﻮﺍﺯﻱ‬ ‫ﺷﺮﻁ‬:-
‫ﺍﻵﺗﻴﺔ‬ ‫ﺍﻟﺤﺎﻻﺕ‬ ‫ﺇﺣﺪﻯ‬ ‫ﻭﺣﺪﺛﺖ‬ ‫ﺛﺎﻟﺚ‬ ‫ﻣﺴﺘﻘﻴﻢ‬ ‫ﻗﻄﻌﻬﻤﺎ‬ ‫ﺇﺫﺍ‬ ‫ﺍﻟﻤﺴﺘﻘﻴﻤﺎﻥ‬ ‫ﻳﺘﻮﺍﺯﻯ‬:-
)١(
)٢(
)٣(
°
‫م‬
›
›
hf
Ox
i
‫و‬
°
‫ن‬›
›
i
hf
‫و‬
‫م‬
x O
‫ن‬
‫؟‬
°
°
‫م‬
O
x
h
f
i‫و‬
‫ن‬ °°
‫ﻫﻨﺪﺳﻴﺔ‬ ‫ﺣﻘﺎﺋﻖ‬:-
)(
.
)( 
)(


‫ﺯ‬‫ﺯ‬‫ﺯ‬‫ﺯ‬‫ﺯ‬‫ﺯ‬

23. ‫اﻟﺨﻮا‬‫ر‬‫اﻟ‬ ‫ﻓﻰ‬ ‫زﻣﻰ‬‫ﻬﻨﺪﺳﺔ‬‫اﻹﻋﺪادى‬ ‫اﻷول‬ ‫اﻟﺼﻒ‬
‫أ‬/‫ﻣﺤﻤﻮد‬ ‫ﻣﺤﻤﺪ‬‫ت‬/٠١٢١١١٦٥٠١٢ 
‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﰲ‬:-x‫؟‬ ‫ص‬
 
 
:hx//‫س‬‫ص‬//fO،hf
،h‫س‬=‫س‬f
:x‫ص‬=‫ص‬O=٢‫ﺳﻢ‬
‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﰲ‬:-h‫؟‬ ‫ص‬
 
 
:fO//x‫و‬//‫س‬i//‫ص‬‫ع‬،h‫ع‬
،h‫و‬=‫و‬i=i‫ع‬
:hx=x‫س‬=‫س‬‫ص‬=٥‫ﺳﻢ‬
:h‫ص‬=١٥‫ﺳﻢ‬
‫ﺍﳌﻘﺎﺑﻞ‬ ‫ﺍﻟﺸﻜﻞ‬ ‫ﰲ‬:-‫ص‬O‫؟‬

‫أن‬ ‫ﺣﯿﺚ‬:hx//‫س‬‫ص‬//‫ع‬‫ل‬//fO،h‫ب‬،xO
،h‫س‬=‫س‬‫ع‬=‫ع‬f
x‫ص‬=‫ص‬‫ل‬=‫ل‬O==٤‫ﺳﻢ‬
:‫ص‬O=٤+٤=٨‫ﺳﻢ‬
h
‫ص‬
x
O f
‫س‬
٢‫ﺳﻢ‬
١٢
٣
١٢‫ﺳﻢ‬
h
‫ب‬
‫س‬
‫ع‬
x
‫ص‬
‫ل‬
O
x
h fO
‫س‬
‫ص‬‫ع‬
i
‫و‬
٥‫ﺳﻢ‬