Question

given ( m parallel n ), find the value of x.

Explanation:

Step1: Identify the angle relationship

Since \( m \parallel n \) and the angles \( (3x + 6)^\circ \) and \( (4x - 29)^\circ \) are alternate interior angles, they are equal. So we set up the equation:
\( 3x + 6 = 4x - 29 \)

Step2: Solve for \( x \)

Subtract \( 3x \) from both sides:
\( 6 = x - 29 \)

Add 29 to both sides:
\( x = 6 + 29 \)
\( x = 35 \) Wait, wait, earlier miscalculation. Wait, let's do it again.

Wait, alternate interior angles: \( 3x + 6 = 4x - 29 \)

Subtract \( 3x \) from both sides: \( 6 = x - 29 \)

Then add 29 to both sides: \( x = 6 + 29 = 35 \)? Wait, but the initial answer was 41, that was wrong. Let's check again.

Wait, maybe the angles are corresponding angles? Wait, the diagram: the transversal t, so the angles \( (3x + 6) \) and \( (4x - 29) \) are alternate interior angles, so they should be equal. So:

\( 3x + 6 = 4x - 29 \)

Subtract \( 3x \): \( 6 = x - 29 \)

Add 29: \( x = 35 \). Wait, but the user's attempt was 41, which is wrong. Let's re-express.

Wait, maybe the angles are equal because they are alternate interior angles. Let's solve:

\( 3x + 6 = 4x - 29 \)

\( 6 + 29 = 4x - 3x \)

\( 35 = x \). So \( x = 35 \).

Wait, maybe I misread the diagram. Let me check again. The lines m and n are parallel, transversal t. The angle on m is \( (3x + 6) \), on n is \( (4x - 29) \). If they are alternate interior angles, then they are equal. So the equation is correct. So solving:

\( 3x + 6 = 4x - 29 \)

\( 4x - 3x = 6 + 29 \)

\( x = 35 \). So the correct answer is 35.

Answer:

\( x = 35 \)