The measures of the angle can be found by calculation using sine rule and
cosine rule.
The measures of the required angle are;
∠ABC = ∠A'B'C' = ∠A'₁B'₁C'₁ ≈ 119.74°
∠BCA = ∠B'C'A' = ∠B'₁C'₁A'₁ ≈ 37.88°
∠BAC = ∠B'A'C' = ∠B'₁A'₁C'₁ ≈ 22.38°
By calculation, we have;
∠A'₂B'₂C'₂ ≈ 105.26
∠B'₂C'₂A'₂ ≈ 53.97°
∠B'₂A'₂C'₂ ≈ 20.77°
Reasons:
Length of AB = Distance between points B(-3, -3), A(-6, -1)
AB = √((-3 - (-6))² + (-3 - (-1))²) = √(13)
Length AC = Distance between points A(-6, -1), and C(-1, -2)
AC = √((-6 - (-1))² + (-1 - (-2))²) = √(26)
Length BC = Distance between points B(-3, -3), and C(-1, -2)
BC = √((-3 - (-1))² + (-3 - (-2))²) = √(5)
By cosine rule, we have;
AC² = BC² + AB² - 2×BC×AB ×cos∠ABC
cos∠ABC = (AC² - (BC² + AB²))/(2×BC×AB) = (26 - (13 + 5))/(-2×√(5)×√(13))
∠ABC = 119.74
AC/(sin(119.74) = AB/(sin(∠BCA))
(√(26))/(sin(172.97) = √(13)/(sin(∠BCA))
sin(∠BCA) = √(13)/((√(26))/(sin(119.74))
∠BCA = 37.88°
∠BAC = 180 - (119.74 + 37.88) = 22.38°
Length of segment B'₁C'₁ = √5
Length of segment B'₁A'₁ = √(13)
Length of segment A'₁C'₁ = √(26)
By calculation, we have;
Length A'₂B'₂ = √(26)
Length A'₂C'₂ = √(37)
Length B'₂C'₂ = √(5)
cos∠A'₂B'₂C'₂ = (37 - (26 + 5))/(-2 × √(26) × √(5))
∠A'₂B'₂C'₂ = 105.26
(√(37))/(sin(105.26) = √(26)/(sin(∠BCA))
sin(∠BCA) = √(26)/((√(37))/(sin(105.26))
∠B'₂C'₂A'₂ = 53.97°
∠B'₂A'₂C'₂ = 180 - (105.26 + 53.97) = 20.77
Therefore, by similar triangles, we have;
∠ABC = ∠A'B'C' = ∠A'₁B'₁C'₁ ≈ 119.74°
∠BCA = ∠B'C'A' = ∠B'₁C'₁A'₁ ≈ 37.88°
∠BAC = ∠B'A'C' = ∠B'₁A'₁C'₁ ≈ 22.38°
By calculation, we have;
∠A'₂B'₂C'₂ ≈ 105.26
∠B'₂C'₂A'₂ ≈ 53.97°
∠B'₂A'₂C'₂ ≈ 20.77°
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