Parallel Lines

Demonstration of the "F" (Corresponding) and "Z" (Alternate) angles

F

F Type 1

F Type 2

F - Tracing Paper

Z

Z - Tracing Paper

Z Type 1

Z Type 2

A = (-4.46, 0.906)
B = (4.208, 1.104)
a = Segment(A, B)
C = (-3.404, 3.414)
u = Vector(A, B)
D = Translate(C, u)
v = Vector(C, D)
b = Segment(C, D)
D_1 = (-3.47, -1.25)
E = (3.702, 5.13)
c = Segment(D_1, E)
d = Circle(D, 2)
e = Circle(B, 2)
F = Intersect(d, b, 1)
f = Segment(F, D)
G = Intersect(e, a, 1)
g = Segment(G, B)
α = Angle(c, b)
β = Angle(a, c)
H = Intersect(c, b)
I = Intersect(c, a)
J = Midpoint(H, D)
K = Midpoint(I, B)
L = Midpoint(D_1, I)
h = PolyLine(L, I, K, I, H, J, H, L)
M = (5.704, 2.028)
k = Circle(M, 7)
N = Point(k)
N' = Mirror(N, M)
N'_1 = Rotate(N, 90*°, M)
O = Mirror(N'_1, M)
P = Point(h)
i = Segment(P, J)
Q = Midpoint(i)
w = Vector(Q, M)
L' = Translate(L, w)
I' = Translate(I, w)
K' = Translate(K, w)
I'_1 = Translate(I, w)
H' = Translate(H, w)
J' = Translate(J, w)
H'_1 = Translate(H, w)
L'_1 = Translate(L, w)
h' = PolyLine(L', I', K', I'_1, H', J', H'_1, L'_1)
j = Line(M, xAxis)
l = Ray(M, N)
R = Intersect(k, j, 2)
S = Intersect(k, j, 2)
γ = Angle(S, M, N)
L'' = Rotate(L', γ - 45°, M)
I'' = Rotate(I', γ - 45°, M)
K'' = Rotate(K', γ - 45°, M)
T = Rotate(I'_1, γ - 45°, M)
H'' = Rotate(H', γ - 45°, M)
J'' = Rotate(J', γ - 45°, M)
U = Rotate(H'_1, γ - 45°, M)
V = Rotate(L'_1, γ - 45°, M)
h'' = PolyLine(L'', I'', K'', T, H'', J'', U, V)
m = false
δ = Angle(T, H'', J'')
ε = Angle(L'', T, K'')
n = Ray(T, H'')
p = Circle(H'', J'')
Z = Intersect(p, n, 2)
B_1 = Intersect(p, n, 1)
q = Ray(L'', H'')
r = Circle(H'', J'')
W, A_1 = Intersect(r, q)
s = Segment(H'', A_1)
ζ = Angle(J'', H'', A_1)
η = Angle(K'', T, H'')
o = false
t = false
z = false
C_1 = Midpoint(H, C)
E_1 = Midpoint(I, H)
F_1 = Midpoint(I, B)
a_1 = PolyLine(C_1, H, E_1, I, F_1)
G_1 = (4.054, 1.786)
c_1 = Circle(G_1, 7)
H_1 = Point(c_1)
I_1 = Mirror(H_1, G_1)
J_1 = Rotate(H_1, 90*°, G_1)
N_1 = Rotate(I_1, 90*°, G_1)
b_1 = false
u_1 = Vector(E_1, G_1)
K_1 = Translate(C_1, u_1)
H'_2 = Translate(H, u_1)
L_1 = Translate(E_1, u_1)
I'_2 = Translate(I, u_1)
M_1 = Translate(F_1, u_1)
d_1 = PolyLine(K_1, H'_2, L_1, I'_2, M_1)
f_1 = Line(G_1, xAxis)
g_1 = Ray(G_1, H_1)
O_1 = Point(f_1)
θ = Angle(O_1, G_1, H_1)
P_1 = Rotate(K_1, θ - 45°, G_1)
Q_1 = Rotate(H'_2, θ - 45°, G_1)
R_1 = Rotate(L_1, θ - 45°, G_1)
S_1 = Rotate(I'_2, θ - 45°, G_1)
T_1 = Rotate(M_1, θ - 45°, G_1)
h_1 = PolyLine(P_1, Q_1, R_1, S_1, T_1)
e_1 = false
ι = Angle(P_1, Q_1, L_1)
κ = Angle(T_1, S_1, L_1)
k_1 = Circle(Q_1, P_1)
p_1 = Circle(S_1, T_1)
i_1 = Ray(P_1, Q_1)
j_1 = Ray(T_1, S_1)
U_1 = Intersect(p_1, j_1, 2)
V_1 = Intersect(p_1, j_1, 2)
W_1, Z_1 = Intersect(p_1, j_1)
A_2, B_2 = Intersect(k_1, i_1)
C_2 = Rotate(T_1, 180*°, S_1)
l_1 = Segment(S_1, C_2)
m_1 = Segment(Q_1, B_2)
λ = Angle(L_1, Q_1, B_2)
μ = Angle(L_1, S_1, C_2)
n_1 = false
o_1 = false