The Triangle Angle Sum Theorem states that the sum of the angles in a triangle must equal \(180^\circ\). In this case, the angle measures of the triangle are given as \((x - 20)^\circ\), \((3x + 3)^\circ\), and \((2x - 7)^\circ\).
We can set up the equation:
\[ (x - 20) + (3x + 3) + (2x - 7) = 180 \]
Combining the terms:
\[ x - 20 + 3x + 3 + 2x - 7 = 180 \]
This simplifies to:
\[ (x + 3x + 2x) + (-20 + 3 - 7) = 180 \] \[ 6x - 24 = 180 \]
Now, we can solve for \(x\):
\[ 6x = 180 + 24 \] \[ 6x = 204 \] \[ x = 34 \]
Now that we have \(x = 34\), we can find the measures of the angles:
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First angle: \[ (x - 20) = 34 - 20 = 14^\circ \]
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Second angle: \[ (3x + 3) = 3(34) + 3 = 102 + 3 = 105^\circ \]
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Third angle: \[ (2x - 7) = 2(34) - 7 = 68 - 7 = 61^\circ \]
Now we have the angle measures:
- \(14^\circ\)
- \(105^\circ\)
- \(61^\circ\)
The largest angle in the triangle is \(\boxed{105^\circ}\).
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