This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Morning Ikgomotseng — let's get this done. To calculate the angle between two lines with slopes m_1 and m_2, we use the formula: = | (m_2 - m_1)/(1 + m_1 m_2) | First, we find the slopes of the lines using the formula m = (y_2 - y_1)/(x_2 - x_1). a) Step 1: Find the slopes of lines AB and AC. Points: A(0, 5), B(-3, 0), C(4, 0). Slope of AB (m_AB): m_AB = (0 - 5)/(-3 - 0) = (-5)/(-3) = (5)/(3) Slope of AC (m_AC): m_AC = (0 - 5)/(4 - 0) = (-5)/(4) Step 2: Calculate the angle using the formula. Let m_1 = m_AC = -(5)/(4) and m_2 = m_AB = (5)/(3). = | (5)/(3) - (-(5)/(4))1 + ((5)/(3))(-(5)/(4)) | = | (20)/(12) + (15)/(12)1 - (25)/(12) | = | (35)/(12)(12 - 25)/(12) | = | (35)/(12)-(13)/(12) | = | -(35)/(13) | = (35)/(13) Step 3: Find . = ((35)/(13)) ≈ 69.5906^ Rounded to two decimal places, is 69.59°. a) ≈ 69.59^ b) Step 1: Find the slopes of lines AC and BD. Points: A(-1, 6), B(5, -1), C(-2, 0), D(3, 6). Slope of AC (m_AC): m_AC = (0 - 6)/(-2 - (-1)) = (-6)/(-1) = 6 Slope of BD (m_BD): m_BD = (6 - (-1))/(3 - 5) = (7)/(-2) = -(7)/(2) Step 2: Calculate the angle . Let m_1 = m_AC = 6 and m_2 = m_BD = -(7)/(2). = | (-7)/(2) - 61 + (6)(-(7)/(2)) | = | (-7)/(2) - (12)/(2)1 - 21 | = | (-19)/(2)-20 | = | (19)/(40) | = (19)/(40) Step 3: Find . = ((19)/(40)) ≈ 25.3909^ Rounded to two decimal places, is 25.39°. b) ≈ 25.39^ c) Step 1: Find the slopes of lines AC and CD. Points: A(-2, -1), C(0, 4), D(3, 2). Slope of AC (m_AC): m_AC = (4 - (-1))/(0 - (-2)) = (5)/(2) Slope of CD (m_CD): m_CD = (2 - 4)/(3 - 0) = (-2)/(3) Step 2: Calculate the angle . Let m_1 = m_AC = (5)/(2) and m_2 = m_CD = -(2)/(3). = | (-2)/(3) - (5)/(2)1 + ((5)/(2))(-(2)/(3)) | = | (-4)/(6) - (15)/(6)1 - (10)/(6) | = | (-19)/(6)(6 - 10)/(6) | = | (-19)/(6)-(4)/(6) | = | (19)/(4) | = (19)/(4) Step 3: Find . = ((19)/(4)) ≈ 78.1109^ Rounded to two decimal places, is 78.11°. c) ≈ 78.11^ d) Step 1: Identify the coordinates and find the slopes of lines AC and BC. Points: A(0, -6), B(0, 2), C(5, 0). Slope of AC (m_AC): m_AC = (0 - (-6))/(5 - 0) = (6)/(5) Slope of BC (m_BC): m_BC = (0 - 2)/(5 - 0) = (-2)/(5) Step 2: Calculate the angle . Let m_1 = m_AC = (6)/(5) and m_2 = m_BC = -(2)/(5). = | (-2)/(5) - (6)/(5)1 + ((6)/(5))(-(2)/(5)) | = | (-8)/(5)1 - (12)/(25) | = | (-8)/(5)(25 - 12)/(25) | = | (-8)/(5)(13)/(25) | = | -(8)/(5) × (25)/(13) | = | -(8 × 5)/(13) | = | -(40)/(13) | = (40)/(13) Step 3: Find . = ((40)/(13)) ≈ 72.0046^ Rounded to two decimal places, is 72.00°. d) ≈ 72.00^ Drop the next question.