The answer: m∡BCD = 130° . _____________________________________ Explanation: ______________________________ m∡BCD = 9x - 5 = our answer. _____________________________ Note: (9x - 5) + (m∡C IN Δ ACB)= 180 ; ____________________________ Reason: all angles on straight line add up to 180. ___________________________ Note: In Δ ACB; m∡A + m∡B + m∡c = 180. _________________________________________ Reason: All three angles in any triangle add up to 180. __________________________________________ Given Δ ACB, we are given: _____________ m∡C= ? m∡B = (4x + 5) m∡A = 65 _____________________ So, given Δ ACB; m∡A + m∡B + m∡c = 180; →Plug in our known values and rewrite: ___________________________________ Given Δ ACB; 65 + 4x + 5 + (m∡c) = 180; →Simplify, and rewrite: ___________________________________ Given Δ ACB; 4x + 70 + (m∡c) = 180; →Subtract "70" from each side of the equation; and rewrite: ___________________________________ Given Δ ACB; 4x + (m∡C) = 110; →Subtract "4x" from EACH SIDE of the equation; to isolate: "(m∡c)" on one side of the equation; and "solve in terms of "(m∡C)" ; ______________________________________________ Given Δ ACB' m∡C = 110 - 4x ; __________________________________________ So, we know that: (110 - 4x) + (9x - 5) = 180; (since all angles on a straight line add up to 180. ____________________ We can solve for "x". ____________________ (110 - 4x) + (9x - 5) = 180; ________________________ Rewrite as: ___________ (110 - 4x) + 1(9x - 5) = 180 ; (Note: there is an implied coefficient of "1"; since anything multiplied by "1" equals that same value). _______________________ Note the "distributive property of multiplication": _________________ a*(b+c) = ab + ac ; AND: a*(b - c) = ab - ac . _______________________ So, +1(9x - 5) = (+1*9x) - (+1*5) = 9x - 5 ; __________________________ So we can rewrite: ___________________ (110 - 4x) + (9x - 5) = 180 ; as: ________________________ 110 - 4x + 9x - 5 = 180 ; We can simplify this by combining "like terms" on the "left-hand side" of the equation: _________________________________________________ 110 - 5 = 105 ; -4x + 9x = 5x; ______________ So, rewrite as: 5x + 105 = 180; Subtract "105" from EACH side; to get: _____________________________________ 5x = 75 ; Now, divide each side of the equation by "5"; to get: x = 15. _____________________________________________ Now, we want to know: m∡BCD; which equals: _____________________________________________ 9x - 5 ; let us substitute "15" for "x"; and solve: ______________________ 9x - 5 = 9*(15) - 5 = 135 - 5 = 130. _____________________________ The answer: m∡BCD = 130° ________________________
