We want to know if our friend is correct in their solution. This means that we need to find the measure of the exterior angle and compare it with the given answer. Let's now look at the given diagram and identify the exterior angle!
The exterior angle measures (3x-6)^(∘). This angle measure involves an algebraic expression, but we want to find its exact measure. To find the measure of the exterior angle, we can use the rule for the exterior angle measures of a triangle.
Exterior Angle Measures of a Triangle
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.
We will now identify the two nonadjacent interior angles.
The nonadjacent interior angles are 30^(∘) and x^(∘). By the rule for the exterior angle measures of a triangle, we can write an equation to relate the nonadjacent interior angles to the exterior angle.
30^(∘)+ x^(∘) = (3x-6)^(∘)
Now we can solve for x. When solving an equation we usually do not include the units so we will drop the degree symbols while we solve.
The measure of the exterior angle is 48^(∘), which is different from the 111^(∘) that our friend calculated. This means that our friend is not correct.
Extra
What Our Friend Did
Notice that our friend incorrectly wrote that the sum of the exterior angle and the nonadjacent interior angles of a triangle is 180^(∘). (3 x - 6)+ x+ 30 ≠ 180
By the rule for the exterior angle measures of a triangle, we know that the exterior angle is equal to the sum of the nonadjacent interior angles of a triangle. It is the rule for the interior angle measures of a triangle that the sums of the measures angles of a triangle is 180^(∘). In this case, our friend combined the two rules into a rule that does not exist.
