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°
________________________
_____________________________________
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°
________________________
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