To find the measures of the angles of triangle \(a\), \(b\), and \(c\) using the Triangle Angle Sum Theorem, we know that the sum of the angles in a triangle is \(180^\circ\).
Given:
- Angle \(a\) = \(4x + 14\)
- Angle \(b\) = \(5x + 4\)
- Angle \(c\) = \(6x - 3\)
According to the Triangle Angle Sum Theorem: \[ (4x + 14) + (5x + 4) + (6x - 3) = 180 \]
Now, combine like terms: \[ 4x + 5x + 6x + 14 + 4 - 3 = 180 \] \[ 15x + 15 = 180 \]
Next, solve for \(x\): \[ 15x = 180 - 15 \] \[ 15x = 165 \] \[ x = \frac{165}{15} = 11 \]
Now, substitute \(x\) back into the angle expressions to find the measures of angles \(a\), \(b\), and \(c\).
For angle \(a\): \[ a = 4(11) + 14 = 44 + 14 = 58^\circ \]
For angle \(b\): \[ b = 5(11) + 4 = 55 + 4 = 59^\circ \]
For angle \(c\): \[ c = 6(11) - 3 = 66 - 3 = 63^\circ \]
Now, we have the angles:
- Angle \(a = 58^\circ\)
- Angle \(b = 59^\circ\)
- Angle \(c = 63^\circ\)
Thus, the measures of the angles for the triangle are \(59^\circ\), \(58^\circ\), and \(63^\circ\). Therefore, the correct response is:
59°, 58°, and 63°.
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