Question 214848
Using the fact that the sum of supplementary angles is 180 degrees, we can write two equations in x and y:
{{{A[1]+A[2] = 180}}} Substitute:
{{{A[1] = x^2+3y}}}
{{{A[2] = 20y+3}}} we get:
1). {{{(x^2+3y)+(20y+3) = 180}}} Similarly for angles 2 and 3.
{{{A[2] = 20y+3}}}
{{{A[3] = 3y+4x}}} Add the angles.
2). {{{(20y+3)+(3y+4x) = 180}}} Simplifying 1). and 2). we have:
1a). {{{x^2+23y+3 = 180}}}
2a). {{{4x+23y+3 = 180}}} Subtracting 2a) from 1a) we get:
3). {{{x^2-4x = 0}}} Factor an x.
3a). {{{x(x-4) = 0}}} so that:
{{{x = 0}}} or {{{x = 4}}} Substitute x = 0 into equation 1a) and solve for y.
{{{0^2+23y+3 = 180}}}
{{{23y = 177}}}
{{{y = 7.6957}}} or substitute x = 4 into equation 1a) and solve for y.
{{{(4)^2+23y+3 = 180}}}
{{{19+23y = 180}}} Subtract 19 from both sides.
{{{23y = 161}}} Divide both sides by 23.
{{{y = 7}}}
So angle 1 can have one of two measures:
{{{A[1] = x^2+3y}}} Substitute x = 0 and y = 7.6957
{{{A[1] = 0+3(7.6957)}}}
{{{highlight_green(A[1] = 23.0871)}}}degrees.
{{{A[2] = 20y+3}}}
{{{A[2] = 20(7.6957)+3}}}
{{{highlight_green(A[2] = 156.914)}}}degrees.
or...
{{{A[1] = x^2+3y}}} x=4 and y = 7.
{{{A[1] = 4^2+3(7)}}}
{{{A[1] = 16+21}}}
{{{highlight(A[1] = 37)}}}degrees.
{{{A[2] = 20y+3}}}
{{{A[2] = 20(7)+3}}}
{{{highlight(A[2] = 143)}}} degrees.
As you can see, in both cases, the sums of angle 1 and angle 2 are 180 degrees.