Measure the angles of ABC and it’s three images and record the measurements in the table. Round each answer to the nearest degree.

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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Answer

(∠ABC) 108 (for all)

(∠ACB) 27 (for all)

(∠CAB) 45 (for all)

Step-by-step explanation:

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