To find the measures of the angles in the triangle using the Triangle Angle Sum Theorem, which states that the sum of the angles in a triangle is always 180 degrees, we can start by setting up the equations based on the angles given.
Let:
- Angle A = \(4x + 14\)
- Angle B = \(5x + 4\)
- Angle C = \(6x - 3\)
According to the Triangle Angle Sum Theorem: \[ \text{Angle A} + \text{Angle B} + \text{Angle C} = 180^\circ \] Substituting the expressions for each angle: \[ (4x + 14) + (5x + 4) + (6x - 3) = 180 \]
Now, combine like terms: \[ 4x + 5x + 6x + 14 + 4 - 3 = 180 \] \[ 15x + 15 = 180 \]
Next, subtract 15 from both sides: \[ 15x = 165 \]
Now, divide by 15: \[ x = 11 \]
Now that we have the value of \(x\), we can substitute it back into the expressions for each angle to find their measures.
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Calculate Angle A: \[ A = 4(11) + 14 = 44 + 14 = 58^\circ \]
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Calculate Angle B: \[ B = 5(11) + 4 = 55 + 4 = 59^\circ \]
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Calculate Angle C: \[ C = 6(11) - 3 = 66 - 3 = 63^\circ \]
The angles of the triangle are:
- Angle A = \(58^\circ\)
- Angle B = \(59^\circ\)
- Angle C = \(63^\circ\)
Thus, the measures of the angles for the triangle are: 59°, 58°, and 63°.
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