Ψηφιακό Σχολείο - Γεωμετρία Α΄ Λυκείου - 2.16 Απλές σχέσεις γωνιών - Θεώρημα Ι

K

ω

ω

φ

t

ω

φ

t

K'

Οι γωνίες φ,t απο εφεξής ---> Μη εφεξής

t

φ

Οι γωνίες ω,t απο εφεξής ---> Μη εφεξής

t

ω

t

ω

φ

t

Να δικαιολογήσετε ότι οι κατακορυφήν γωνίες φ,ω είναι ίσες.

Να μετακινήσετε τους δρομείς.

Βοήθεια

t

φ

t

φ

t

ω

t

ω

Ποια είναι η σχέση των γωνιων φ,ω με τη γωνία t;

Κατακορυφήν 


   γωνίες

B = (11533.766256465924, 6430.311853594785)
A = (1.099894436320091E7, -9470819.897380667)
C = (6.1834896414302465, 4.38493567796899)
D = (13.89777951864031, 8.035111863770824)
a = Line(C, D)
E = (10.4733874268056, 1.5250038210520882)
b = Line(C, E)
F = Point(b)
G = Point(a)
r = 2.5 (slider from 0 to 5, step 0.1)
c = Circle(C, r)
H, I = Intersect(c, a)
J, K = Intersect(c, b)
d = CircleSector(C, K, I)
f = CircleSector(C, J, H)
e = CircleSector(C, H, K)
α = Angle(E, C, D)
g = CircleSector(C, I, K)
h = CircleSector(C, H, J)
k_1 = CircleSector(C, J, I)
L = (-5.143345739253797, -1.5607121298319369)
k = 0 (slider from 0 to 2, step 0.0010)
L'_1 = Dilate(L, k, C)
i = Line(L, b)
j = Line(L, a)
p = Circle(L, r)
M, N = Intersect(p, j)
O, P = Intersect(p, i)
P' = Dilate(P, k, K)
M'_1 = Dilate(M, k, H)
O'_1 = Dilate(O, k, J)
q = CircleSector(L'_1, M'_1, P')
l = Line(O'_1, P')
s = CircleSector(L'_1, O'_1, M'_1)
Q_1 = (13.145165872083231, -0.9209905302584195)
m = Line(Q_1, b)
n = Line(Q_1, a)
t = Circle(Q_1, r)
R, S = Intersect(t, m)
T, Q = Intersect(t, n)
w = 0 (slider from 0 to 2, step 0.0010)
Q'_1 = Dilate(Q_1, w, C)
T' = Dilate(T, w, H)
S'_1 = Dilate(S, w, K)
U'_1 = Dilate(Q, w, I)
c_1 = CircleSector(Q'_1, T', S'_1)
d_1 = CircleSector(Q'_1, S'_1, U'_1)
a_1 = Line(T', U'_1)
Q'_2 = Dilate(Q_1, w, C)
R' = Dilate(R, w, J)
Q' = Dilate(Q, w, I)
S' = Dilate(S, w, K)
e_1 = CircleSector(Q'_2, R', Q')
f_1 = CircleSector(Q'_2, Q', S')
b_1 = Line(R', S')
M' = Dilate(M, k, H)
O' = Dilate(O, k, J)
L'_2 = Dilate(L, k, C)
N' = Dilate(N, k, I)
g_1 = CircleSector(L'_2, M', O')
h_1 = CircleSector(L'_2, O', N')
i_1 = Line(M', N')
o = false
j_1 = Ray(L'_2, M'_1)
U = Midpoint(O, M)
l_1 = Line(U, l)
m_1 = Line(U, j_1)
p_1 = Circle(U, r)
V, W = Intersect(p_1, l_1)
Z, A_1 = Intersect(p_1, m_1)
V' = Dilate(V, k - 1, O)
U' = Dilate(U, k - 1, L)
B_1 = Dilate(A_1, k - 1, M)
q_1 = CircleSector(L, M, P)
r_1 = CircleSector(U', V', B_1)
s_1 = Ray(L, M)
C_1 = Midpoint(M, O)
t_1 = Line(C_1, i_1)
a_2 = Line(C_1, b)
c_2 = Circle(C_1, r)
D_1, E_1 = Intersect(c_2, t_1)
F_1, G_1 = Intersect(c_2, a_2)
d_2 = CircleSector(L, O, N)
H_1 = Dilate(C_1, k - 1, L)
I_1 = Dilate(D_1, k - 1, M)
J_1 = Dilate(F_1, k - 1, O)
e_2 = CircleSector(H_1, I_1, J_1)
b_2 = Ray(L'_2, O')
f_2 = Ray(L, O)
L_1 = Intersect(c_1, d_1, 1)
i_2 = Ray(Q'_2, L_1)
p_2 = CircleSector(Q_1, T, S)
K_1 = Midpoint(Q, S)
h_2 = Line(K_1, a_1)
j_2 = Line(K_1, i_2)
k_2 = Circle(K_1, r)
M_1, N_1 = Intersect(k_2, h_2)
O_1, P_1 = Intersect(k_2, j_2)
R_1 = Dilate(K_1, w - 1, Q_1)
S_1 = Dilate(N_1, w - 1, Q)
T_1 = Dilate(P_1, w - 1, S)
q_2 = CircleSector(R_1, T_1, S_1)
l_2 = Ray(Q_1, S)
n_2 = Ray(R_1, S_1)
r_2 = Ray(R_1, T_1)
s_2 = Ray(Q_1, T)
t_2 = CircleSector(Q_1, R, Q)
U_1 = Midpoint(S, Q)
V_1 = (14.311717024246704, 4.19678226632972)
m_2 = Line(U_1, b_1)
a_3 = Ray(Q'_2, Q')
b_3 = Line(U_1, a_3)
c_3 = Circle(U_1, r)
W_1, Z_1 = Intersect(c_3, m_2)
A_2, B_2 = Intersect(c_3, b_3)
C_2 = Dilate(U_1, w - 1, Q_1)
D_2 = Dilate(Z_1, w - 1, S)
E_2 = Dilate(B_2, w - 1, Q)
d_3 = CircleSector(C_2, E_2, D_2)
e_3 = Ray(C_2, D_2)
f_3 = Ray(C_2, E_2)
g_3 = Ray(Q_1, Q)
h_3 = Ray(Q_1, R)
g_2 = Ray(H_1, I_1)
i_3 = Ray(H_1, J_1)
j_3 = Ray(L, N)
n_1 = Ray(U', V')
k_3 = Ray(U', B_1)
l_3 = Ray(L, P)
Α = (31.73472294204308, -8.08587244548182)
Β = (37.003018467942624, -8.08587244548182)