Înţelegere:
1. Notez: x= m (∡ AOB) ! fig. 33
y = m( ∡BOC)
z = m( ∡COA)
x= y= z x+ y+ z = 360°
x+ y+ z = 360°
____________ x+ x+ x= 360°
x= ?°
y= ?° 3x = 360° I :3
z= ?°
x = 120°
x= y=z= 120°
Probă: x+ y+ z= 360°
120°+ 120°+ 120°= 360°
360°=360°
2. Notez: x= m (∡ AOB) Observaţie! Nr. consecutive cresc din 1
y = m( ∡BOC) în 1!
z = m( ∡COD)
v = m(∡ DOE
w = m( ∡ EOA)
x+ y+ z+ v+ w = 360°
x< z< y< v< w
______________________
x= ? , y= ?, z= ?, v=? , w=?
x+ y+ z+ v+ w = 360°
x+ ( x+ 1) +( x+ 1+ 1)+ ( x+ 1+ 1+ 1)+ (x+ 1+ 1+ 1+ 1)=360°
5x+ 10= 360
5x = 360- 10
5x = 350 I : 5
x = 70°
y = 70+ 1 z= 71+ 1 v= 72+ 1 w= 73+ 1
y =71° z = 72° v= 73° w= 74°
probă: x+y+z+v+w= 360°
70°+ 71°+72°+ 73°+74=360°
360° = 360°
3. Notez: x= m (∡ AOB)
y = m( ∡BOC)
z = m( ∡COD)
v = m(∡ DOE
w = m( ∡ EOF)
t = m( ∡ FOA)
x = 31° x+ y+ z+ v+ w+ t= 180°
y =94°
z= 41° 31°+ 94°+ 41°+ w+ 16°+w+ 80°=360°
v = w+ 16°
w= ?° 2w+ 262°= 360°
t = 80°
___________ 2w = 360° - 262°
w = ?°
v+ w= ?° 2w = 98° I:2
w = 49°
v= 49°+ 16° v+ w =
v =65° 65°+ 49°=
114°
probă: 31°+ 94°+ 41°+ 65°+ 49°+ 80°= 360°
360°= 360°
Aprofundare şi exersare
5. Notez: x= m (∡ AOB)
y = m( ∡BOC)
z = m( ∡COD)
v = m(∡DOA)
x= z x+ y+ z+ t= 360°
y= t
x+ y+ z+ t= 360° x=( y+ z+ t):3 ⇒ 3x = y+ z+ t
x=( y+ z+ t):3
_______________ 3x = y+ x+ y
x=?, y= ? , z= ?, t=? 3x - x = 2y
2x = 2y I : 2
x = y
x+ y+ z+ t = 360°
x+ x+ x+ x = 360°
4x = 360° I : 4
x = 90° ⇒ x= y= z= t= 90°
6.a) Notez: x= m (∡ AOB)
y = m( ∡BOC)
z = m( ∡COA)
x = 80° x+ y+ z = 360°
y = 130°
__________ 80°+ 130°+ z= 360°
z = ?°
210°+ z= 360°
z= 360° - 210°
z = 150°
b) Notez: x= m (∡ AOB)
y = m( ∡BOC)
z = m( ∡COD)
v = m(∡ DOE
w = m( ∡ EOF)
x = ?° x+ y+ z+ v+ w= 360°
y = 50°
z = 2?°+ 20° ?°+ 50° + 2?° + 20° + 90°+ 4?°- 10°=360°
v = 90°
w = 4?° - 10° 7?°+150°= 360°
______________
? fiecare 7?° = 360°- 210°
?° = 210° I : 7
?° = 30°= x°
z= 2·30°+ 20° w= 4·30° - 10°
z = 60° + 20° w = 120° - 10°
z = 80° w = 110°
probă : x+ y+ z+ v+ w = 360°
30°+ 50°+ 80°+ 90°+ 110°= 360°
360° = 360°
ex. 7 Notez: x= m(∡AOC) = m(∡ B'OC') = m(∡ BOC)
t = m(∡AOB') = m(∡BOA')
t = 104° x+ x+ t+ x+ x+t = 360°
x= x
______________________ 4x+ 2t = 360°
a) x= ?°
b) x= x ? 4x +2·104°= 360°
c) [OC = bisect.∡AOB
[OC' = bisect. ∡ A'OB' 4x + 208°= 360°
4x = 360°- 208°
4x = 152° I : 4
x = 38°
m(∡AOC) = 38° = m(∡B'OC') = m(∡A'OC') = m(∡BOC)
! m(∡BOC) =m(∡B'OC') = ∡ opuse la vf.
m(∡A'OB) = m(∡AOB') =∡opuse la vf.
m(∡AOC) = m(∡A'OC') = ∡ opuse la vf.
m(∡AOC') = m(∡AOB') + m(∡ B'OC')
= x + t
= 38° + 104°
= 142°
b) m(∡A'OC') = 38° = m(∡COB)
c) din m ( ∡ AOC) = m(∡ COB) = 38° ⇒ [ OC = bisect. ∡ AOB
din m (∡ A'OC') = m( ∡B'OC') = 38° ⇒ [OC' = bisect. ∡A'OB"
