To find the measures of the angles, we need to set the expressions for the angles equal to each other since vertical angles are equal.
We have:
- m∠2 = (5x + 23)°
- m∠4 = (3x + 35)°
Setting them equal to each other: \[ 5x + 23 = 3x + 35 \]
Now, let's solve for \(x\):
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Subtract \(3x\) from both sides: \[ 5x - 3x + 23 = 35 \] This simplifies to: \[ 2x + 23 = 35 \]
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Next, subtract \(23\) from both sides: \[ 2x = 35 - 23 \] This simplifies to: \[ 2x = 12 \]
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Now, divide both sides by \(2\): \[ x = 6 \]
Now, substitute \(x\) back into either expression to find the measures of the angles. We'll use m∠2:
\[ m∠2 = 5(6) + 23 \] \[ m∠2 = 30 + 23 \] \[ m∠2 = 53° \]
Since the angles are equal, we also have: \[ m∠4 = 3(6) + 35 \] \[ m∠4 = 18 + 35 \] \[ m∠4 = 53° \]
Thus, the measure of each angle is 53°.