To find the measures of the angles in triangle \( ABC \) using the Triangle Angle Sum Theorem, we start by setting up the equation based on the given expressions for the angles:
\[ \text{Angle } A = 4x + 14, \quad \text{Angle } B = 5x + 4, \quad \text{Angle } C = 6x - 3 \]
According to the Triangle Angle Sum Theorem, the sum of the angles in a triangle is 180 degrees:
\[ (4x + 14) + (5x + 4) + (6x - 3) = 180 \]
Now combining like terms:
\[ 4x + 5x + 6x + 14 + 4 - 3 = 180 \] \[ 15x + 15 = 180 \]
Next, we solve for \( x \):
\[ 15x = 180 - 15 \] \[ 15x = 165 \] \[ x = \frac{165}{15} = 11 \]
Now, substituting \( x = 11 \) back into the expressions for the angles:
\[ \text{Angle } A = 4(11) + 14 = 44 + 14 = 58 \] \[ \text{Angle } B = 5(11) + 4 = 55 + 4 = 59 \] \[ \text{Angle } C = 6(11) - 3 = 66 - 3 = 63 \]
Thus, the angles of triangle \( ABC \) are:
\[ 58^\circ, 59^\circ, \text{ and } 63^\circ \]
The correct option is:
59°, 58°, and 63°.
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