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a+b+c=360⁰

Class 9: Triangles 
Sum of the exterior angles of a triangle

benny varghese
pgt mathematics
jnv pathanamthitta
kerala-686511

Plz drag A,B,C to get various cases of triangle

You may drag vertices A,B or C to change the triangle and then 
click the button “StartAnimation”

To verify the 
exterior angle  sum 
property in a triangle

t = 0.5 (slider from 0.5 to 6.5, step 0.01)
C = (-0.9, -0.54)
B = (3.5, 3.42)
P3: (x(B) + (Min[Max[t - 4, 0], 1] * (x(C) - x(B))), y(B) + (Min[Max[t - 4, 0], 1] * (y(C) - y(B))))
A = (8.22, -1.04)
P3' = Rotate((x(P3) + (x(C) - x(B)) / Distance[B, C], y(P3) + (y(C) - y(B)) / Distance[B, C]), (Min[Max[t - 5, 0], 1] * (180° - Angle[A, C, B])), P3)
P2: (x(A) + (Min[Max[t - 2, 0], 1] * (x(B) - x(A))), y(A) + (Min[Max[t - 2, 0], 1] * (y(B) - y(A))))
P2' = Rotate((x(P2) + (x(B) - x(A)) / Distance[A, B], y(P2) + (y(B) - y(A)) / Distance[A, B]), (Min[Max[t - 3, 0], 1] * (180° - Angle[C, B, A])), P2)
P1: (x(C) + (Min[Max[t - 0, 0], 1] * (x(A) - x(C))), y(C) + (Min[Max[t - 0, 0], 1] * (y(A) - y(C))))
P1' = Rotate((x(P1) + (x(A) - x(C)) / Distance[C, A], y(P1) + (y(A) - y(C)) / Distance[C, A]), (Min[Max[t - 1, 0], 1] * (180° - Angle[B, A, C])), P1)
poly1, c_1, a_1, b_1 = Polygon(A, B, C)
e = Segment(A, (x(A) + (x(A) - x(C)) / Distance[C, A], y(A) + (y(A) - y(C)) / Distance[C, A]))
f_1 = Segment(B, (x(B) + (x(B) - x(A)) / Distance[A, B], y(B) + (y(B) - y(A)) / Distance[A, B]))
g = Segment(C, (x(C) + (x(C) - x(B)) / Distance[B, C], y(C) + (y(C) - y(B)) / Distance[B, C]))
P4: (x(C) + (Min[Max[t - 6, 0], 1] * (x(A) - x(C))), y(C) + (Min[Max[t - 6, 0], 1] * (y(A) - y(C))))
P4': (x(P4) + (x(A) - x(C)) / Distance[C, A], y(P4) + (y(A) - y(C)) / Distance[C, A])
P = If(t ≤ 2, P1, If[t ≤ 4, P2, If[t ≤ 6, P3, P4]])
Q = If(t ≤ 2, P1', If[t ≤ 4, P2', If[t ≤ 6, P3', P4']])
u = Vector(P, Q)
O = Midpoint(b_1)
a = Angle((x(A) + (x(A) - x(C)) / Distance[C, A], y(A) + (y(A) - y(C)) / Distance[C, A]), A, B)
b = Angle((x(B) + (x(B) - x(A)) / Distance[A, B], y(B) + (y(B) - y(A)) / Distance[A, B]), B, C)
α = Angle(P1', P1, (x(P1) + (x(A) - x(C)) / Distance[C, A], y(P1) + (y(A) - y(C)) / Distance[C, A]))
β = Angle(P2', P2, (x(P2) + (x(B) - x(A)) / Distance[A, B], y(P2) + (y(B) - y(A)) / Distance[A, B]))
c = Angle((x(C) + (x(C) - x(B)) / Distance[B, C], y(C) + (y(C) - y(B)) / Distance[B, C]), C, A)
γ = Angle(P3', P3, (x(P3) + (x(C) - x(B)) / Distance[B, C], y(P3) + (y(C) - y(B)) / Distance[B, C]))
aα: (a * α)
d = OrthogonalLine(P, u)
f = Circle(P, 0.5)
G = Point(f)
k = Circle(Q, 0.6)
I, K = Intersect(k, f)
h = Line(I, u)
F, H = Intersect(f, h)
δ: round(α)
ε: round(β)
ζ: 360° - δ - ε
text1: "a=" + (LaTeX[δ]) + ""
text2: "b=" + (LaTeX[ε]) + ""
text3: "c=" + (LaTeX[ζ]) + ""
stroke1 = PolyLine((-3.5, 2.68), (NaN, NaN), (-3.5, 2.68), (-3.54, 2.74), (-3.58, 2.8000000000000003), (-3.62, 2.8600000000000003), (-3.62, 2.9400000000000004), (-3.62, 3.02), (-3.56, 3.08), (-3.54, 3.16), (-3.48, 3.2), (-3.44, 3.2600000000000002), (-3.38, 3.2800000000000002), (-3.3200000000000003, 3.32), (-3.24, 3.38), (-3.22, 3.44), (-3.22, 3.52), (-3.22, 3.6), (-3.22, 3.6799999999999997), (-3.22, 3.78), (-3.2, 3.84), (-3.14, 3.92), (-3.1, 3.98), (-3.04, 4), (-2.98, 4.04), (-2.9, 4.04), (-2.82, 4.04), (-2.74, 4.04), (-2.66, 4.04), (-2.58, 4.04), (-2.52, 4.06), (-2.46, 4.1), (-2.42, 4.16), (-2.38, 4.239999999999999), (-2.3000000000000003, 4.279999999999999), (-2.24, 4.3), (-2.18, 4.34), (-2.12, 4.359999999999999), (-2.04, 4.38), (-1.96, 4.38), (-1.86, 4.38), (-1.78, 4.38), (-1.72, 4.359999999999999), (-1.6600000000000001, 4.34), (-1.58, 4.319999999999999), (-1.52, 4.3), (-1.44, 4.3), (-1.36, 4.279999999999999), (-1.3, 4.26), (-1.22, 4.26), (-1.1400000000000001, 4.26), (-1.04, 4.26), (-0.98, 4.279999999999999), (-0.92, 4.3), (-0.84, 4.3), (-0.76, 4.319999999999999), (-0.68, 4.34), (-0.58, 4.34), (-0.48, 4.34), (-0.4, 4.34), (-0.32, 4.34), (-0.22, 4.34), (-0.16, 4.319999999999999), (-0.1, 4.279999999999999), (-0.02, 4.26), (0.06, 4.26), (0.14, 4.239999999999999), (0.22, 4.239999999999999), (0.3, 4.239999999999999), (0.42, 4.239999999999999), (0.5, 4.239999999999999), (0.64, 4.239999999999999), (0.76, 4.239999999999999), (0.84, 4.239999999999999), (0.92, 4.239999999999999), (0.98, 4.2), (1.04, 4.14), (1.1, 4.1), (1.16, 4.04), (1.18, 3.96), (1.2, 3.88), (1.24, 3.78), (1.28, 3.6999999999999997), (1.3, 3.64), (1.3, 3.54), (1.3, 3.46), (1.3, 3.38), (1.3, 3.2800000000000002), (1.3, 3.18), (1.3, 3.1), (1.3, 3), (1.3, 2.9200000000000004), (1.28, 2.8600000000000003), (1.2, 2.8400000000000003), (1.1400000000000001, 2.8000000000000003), (1.06, 2.7800000000000002), (1, 2.7600000000000002), (0.9400000000000001, 2.74), (0.88, 2.7), (0.8200000000000001, 2.64), (0.76, 2.5400000000000005), (0.76, 2.4600000000000004), (0.74, 2.3400000000000003), (0.72, 2.2800000000000002), (0.66, 2.24), (0.6, 2.2000000000000006), (0.54, 2.1800000000000006), (0.46, 2.1400000000000006), (0.42, 2.0800000000000005), (0.36, 2.0200000000000005), (0.3, 1.9800000000000004), (0.26, 1.9200000000000006), (0.2, 1.8600000000000005), (0.16, 1.8000000000000005), (0.08, 1.8000000000000005), (-0.02, 1.8000000000000005), (-0.1, 1.8000000000000005), (-0.18, 1.8000000000000005), (-0.32, 1.8200000000000005), (-0.4, 1.8200000000000005), (-0.48, 1.8200000000000005), (-0.56, 1.8200000000000005), (-0.66, 1.8000000000000005), (-0.72, 1.7600000000000005), (-0.8, 1.7000000000000006), (-0.8200000000000001, 1.6400000000000006), (-0.88, 1.5600000000000007), (-0.92, 1.5000000000000007), (-0.98, 1.4600000000000006), (-1.06, 1.4200000000000006), (-1.12, 1.4000000000000006), (-1.18, 1.3800000000000008), (-1.24, 1.3600000000000008), (-1.32, 1.3600000000000008), (-1.42, 1.4000000000000006), (-1.5, 1.4800000000000006), (-1.56, 1.5400000000000007), (-1.62, 1.5800000000000005), (-1.68, 1.6000000000000005), (-1.76, 1.6400000000000006), (-1.82, 1.6600000000000006), (-1.8800000000000001, 1.6800000000000006), (-1.94, 1.7200000000000006), (-2.02, 1.7200000000000006), (-2.12, 1.7200000000000006), (-2.2, 1.7200000000000006), (-2.2600000000000002, 1.7000000000000006), (-2.34, 1.7000000000000006), (-2.4, 1.6800000000000006), (-2.48, 1.6800000000000006), (-2.58, 1.6800000000000006), (-2.68, 1.6800000000000006), (-2.7800000000000002, 1.6800000000000006), (-2.84, 1.7000000000000006), (-2.9, 1.7600000000000005), (-2.98, 1.8000000000000005), (-3.04, 1.8600000000000005), (-3.12, 1.9200000000000006), (-3.18, 1.9600000000000004), (-3.2, 2.0200000000000005), (-3.22, 2.0800000000000005), (-3.22, 2.1800000000000006), (-3.22, 2.2600000000000002), (-3.22, 2.3400000000000003), (-3.24, 2.4000000000000004), (-3.3000000000000003, 2.4600000000000004), (-3.38, 2.5000000000000004), (-3.42, 2.5600000000000005), (-3.48, 2.5800000000000005), (-3.48, 2.66), (-3.48, 2.74), (-3.48, 2.8200000000000003), true)