Final answer:
By applying the exterior angle theorem, the problem is solved to find that the measure of ∠hfg is 48 degrees, after determining the value of x to be 11.
Step-by-step explanation:
The problem presented involves solving for an angle in a triangle whose sides have been extended. The triangle is labeled fgh, with line segment fh being extended through point h to a new point, i. We're given that m∠hfg is represented by the expression (3x + 15)°, m∠ghi is (6x - 6)°, and m∠fgh is (x + 1)°. This scenario suggests we should employ the exterior angle theorem, which states that the measure of an exterior angle (m∠ghi in this case) of a triangle is equal to the sum of the measures of the two non-adjacent interior angles. Therefore, to find the value of x, we set m∠ghi equal to the sum of m∠hfg and m∠fgh:
(6x - 6)° = (3x + 15)° + (x + 1)°
Combining like terms yields:
6x - 6 = 4x + 16
Subtracting 4x from both sides:
2x - 6 = 16
Adding 6 to both sides:
2x = 22
Dividing by 2:
x = 11
Having found x, we can now substitute it back into the expression for m∠hfg to find the angle's measure:
m∠hfg = (3x + 15)° = (3(11) + 15)° = (33 + 15)° = 48°