Vertical angles are equal, which means we can set the two expressions for the angles equal to each other:
\[ 6x - 15 = 4x + 5 \]
To solve for \(x\), we can first get all the \(x\) terms on one side and the constant terms on the other:
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Subtract \(4x\) from both sides: \[ 6x - 4x - 15 = 5 \] \[ 2x - 15 = 5 \]
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Next, add 15 to both sides: \[ 2x = 5 + 15 \] \[ 2x = 20 \]
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Now, divide both sides by 2: \[ x = 10 \]
Now that we have the value of \(x\), we can find the measure of each angle by substituting \(x\) back into either of the angle expressions.
Let's substitute \(x = 10\) into the first angle: \[ 6x - 15 = 6(10) - 15 = 60 - 15 = 45° \]
Now, let's check \(x = 10\) in the second angle: \[ 4x + 5 = 4(10) + 5 = 40 + 5 = 45° \]
Both angles are indeed \(45°\).
Thus, the measure of each angle is: \[ \boxed{45°} \]