Exterior Angle Theorem basically says that if there's an angle that makes a 180° with one of the angles of a triangle, the other two angles of that triangle must be equal to the angle outside.
In the context of this problem, that means m∠SAB is equal to m∠B + m∠C. Putting that into an equation would look like this: [tex]19x + 8= 75 + 6x +11 \\ 19x + 8 = 86+ 6x[/tex].
From here, we can solve for x with the usual methods. Here's how it would look: [tex]19x +8-8 = 86+6x-8\\ 19x= 78 +6x \\ 19x -6x = 78 +6x -6x \\ 13x = 78 \\ \frac{13}{13}x = \frac{78}{3} \\ x = 6[/tex].
With x being six, we can now find the angle measures of all of the angles in the problem by plugging it into the angles that have x and using our rules along with our theorems from there. Let's start with ∠C, which is 6x + 11. It would look like this: [tex]C=6(6) +11 \\ C= 36 + 11 \\ C = 47[/tex]. Now we know that m∠C is 47°, we can find the angle outside through exterior angle theorem again. We can set it up and solve it like this:
[tex]A = 47 + 75 \\ A = 122[/tex]
So we know that the exterior angle is 122°. We also know that the interior angle (the one inside that we don't know yet) is a supplementary angle to our exterior angle (meaning that their angles add up to 180° and that they make a straight line). From this, we can find the angle by subtracting 122 from 180. This gets us 58°.
So, your angles measures are the exterior angle being 122°, the interior angle being 58°, and m∠C being 47°. Also, your x value is 6.
