To find the largest angle in a triangle with angle measures \((x−20)^\circ\), \((3x+3)^\circ\), and \((2x−7)^\circ\), we can start by using the Triangle Angle Sum Theorem, which states that the sum of the angles in a triangle is \(180^\circ\).
First, we set up the equation:
\[ (x - 20) + (3x + 3) + (2x - 7) = 180 \]
Now, simplify the left side:
\[ x - 20 + 3x + 3 + 2x - 7 = 180 \]
Combine like terms:
\[ (1x + 3x + 2x) + (-20 + 3 - 7) = 180 \]
\[ 6x - 24 = 180 \]
Now, solve for \(x\):
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Add \(24\) to both sides: \[ 6x = 204 \]
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Divide by \(6\): \[ x = 34 \]
Now we can find the measures of each angle:
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For \((x - 20)^\circ\): \[ x - 20 = 34 - 20 = 14^\circ \]
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For \((3x + 3)^\circ\): \[ 3x + 3 = 3(34) + 3 = 102 + 3 = 105^\circ \]
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For \((2x - 7)^\circ\): \[ 2x - 7 = 2(34) - 7 = 68 - 7 = 61^\circ \]
The angles are \(14^\circ\), \(105^\circ\), and \(61^\circ\). Thus, the largest angle is:
\[ \boxed{105^\circ} \]
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