Question 651767
if the measure of angle {{{A }}}is equal to the measure of angle {{{B}}}, then we have 

{{{2(6x-5y)+4=54-3(2x-3y)}}}...we can solve this for {{{x}}}


{{{12x-10y+4=54-6x+9y}}}

{{{12x+6x=54+10y+9y-4}}}

{{{18x=19y+50}}}

{{{x=19y/18+50/18}}}

{{{x=1.06y+2.78}}}....plug it in measure of angle C: 


{{{4(5(1.06y+2.78)-6y)-2(4y-3(1.06y+2.78))+30}}}

{{{4(5.3y+13.9-6y)-2(4y-3.18y-8.34)+30}}}

{{{21.2y+55.6-24y-8y+6.36y+16.68+30}}}

measure of angle C: {{{-4.44y+102.28}}}

now we have: {{{x=1.06y+2.78}}}, then

the measure of angle {{{A=2(6(1.06y+2.78)-5y)+4=2(6.36y+16.68-5y)+4=2(1.36y+16.68)+4=2.72y+33.36+4=2.72y+37.36}}}
{{{A= 2.72y+37.36}}}

the measure of angle {{{B }}}={{{54-3(2.12y+5.56-3y)}}}={{{54-3(-0.88y+5.56)}}}={{{54-(-2.64y+16.68)}}}={{{54+2.64y-16.68}}}={{{2.64y+37.32}}}

{{{B= 2.64y+37.32}}} 

the measure of angle {{{C =-4.44y+102.28}}}

now in all measure of angles we have only one unknown variable, and we also know that  

{{{A+B+C=180}}}...plug in {{{A }}}, {{{B }}} and {{{C }}}

{{{(2.72y+37.36)+(2.64y+37.32)+(-4.44y+102.28)=180}}}...solve for {{{y}}}

{{{2.72y+37.36+2.64y+37.32-4.44y+102.28=180}}}

{{{0.92y+176.96=180}}}

{{{0.92y=180-176.96}}}

{{{0.92y=3.04}}}

{{{y=3.04/0.92}}}

{{{highlight(y=3.3)}}}

now find the measure of each angle:

{{{A= 2.72y+37.36}}}
{{{A= 2.72*3.3+37.36}}}
{{{A= 8.976+37.36}}}
{{{highlight(A= 46.34)}}}
 
{{{B= 2.64*3.3+37.32}}} 
{{{B=8.712+37.32}}}
{{{highlight(B=46.03)}}}
 
{{{C =-4.44*3.3+102.28}}}
{{{C =-14.652+102.28}}}
{{{highlight(C =87.63)}}}


check:


{{{A+B+C=180}}}

{{{46.34+46.03+87.63=180}}}

{{{180=180}}}