The answer is:
" m∠BAC = 45° ; m∠ABC, = 64° ; and: m∠ACB = 70° ."
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Step-by-step explanation:
To find the measure of ∠BCA {"m∠BCA"} :
<BCA is supplementary to " 110 " ; that is, both, or all, of all the measurements that form a "line or line segment"; or any portions thereof,
that can be written in: "slope-intercept format"
that is, on Cartesian plane: " y = mx + b " ;
in which: "y" is isolated on the "left-hand side" of the equation; as a single, isolated variable;
and in which: "m" is the "slope" (if applicable); and the coefficient to the "x" value'
and in which "b" is the "y-intercept"; or more precisely"
the value of the "y-coordinate"—in: "(x, y)" format —at which:
" x = 0" ; that is, in the format: " (0, b)."
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So, to find
m∠BCA : Subtract our given value, "110" ; from "180" ; as follows:
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" (180 − 110) = 70" ;
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Now: Note a triangle, by definition, has 3 (three) sides and 3 (three) angles—and all three (3) angles of a triangle add up to 180°.
So: We just solved for m∠ACB ; which is: 70° .
Now shall solve for the value of "x" ; & then find the value of all 3 (three) sides of this triangle
Since all angles within a triangle add up to 180: Let's add them up:
" 70 + 3x + 20 + 3x = 180 " ;
Combine the "like terms" on the "left-hand side" of the equation:
" +3x + 3x = 6x " ; "+ 70 + 20 = 90 ";
And rewrite the equation:
" 6x + 90 = 180 " ;
Now, subtract "90" from Each Side of the equation:
" 6x + 90 − 90 = 180 − 90 " ;
to get: " 6x = 90 " ;
Now, divide Each side of the equation by "6" ;
to isolate "x" on one side of the equation ;
and to solve for "x" :
" 6x / 6 = 90 / 6 ;
to get: " x = 15 " ;
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Now, we are asked to find:
1) " m∠BAC = 3x " ; Plug in "15" for "x" ; and solve:
"3x = 3(15) = 45 " ;
2) " m∠ABC = "(3x + 20)" ; Plug in "15" for "x"; and solve:
"3(15) + 20 = 45 + 20 = 65 " ;
3) " m∠ACB = 70 " ; (as per our calculated value);
To check our work—do these 3 (three) numbers add up to 180?
"45 + 65 + 70 =? 180 ? " ;
" 110 + 70 = " 180 "? Yes!
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So: The answer is as follows:
" m∠BAC = 45° ; m∠ABC, = 64° ; and: m∠ACB = 70° ."
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Hope this is helpful to you! Best wishes!