Vertical angles are equal, so we can set the two angle measures equal to each other:
\[ m∠2 = m∠4 \]
This gives us the equation:
\[ 5x + 23 = 3x + 35 \]
Now, let's solve for \(x\):
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Subtract \(3x\) from both sides: \[ 5x - 3x + 23 = 35 \] \[ 2x + 23 = 35 \]
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Subtract 23 from both sides: \[ 2x = 35 - 23 \] \[ 2x = 12 \]
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Divide by 2: \[ x = 6 \]
Now that we have \(x\), we can substitute it back into either expression to find the measure of the angles. Let's use \(m∠2\):
\[ m∠2 = 5x + 23 = 5(6) + 23 = 30 + 23 = 53° \]
Now let's check \(m∠4\) to ensure they are equal:
\[ m∠4 = 3x + 35 = 3(6) + 35 = 18 + 35 = 53° \]
Thus, both angles measure \(53°\).
The measure of each angle is 53°.
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