Question

given $m \parallel n$, find the value of x. image of two parallel lines m and n cut by a transversal t, with angles $(6x - 7)^circ$ and $(3x + 16)^circ$ on line n

Explanation:

Step1: Identify angle relationship

Since \( m \parallel n \), the alternate interior angles are equal? Wait, no, wait. Wait, the two angles \( (6x - 7)^\circ \) and \( (3x + 16)^\circ \) are actually same - side? Wait, no, looking at the diagram, when two parallel lines are cut by a transversal, consecutive interior angles? Wait, no, wait, actually, the two angles \( (6x - 7)^\circ \) and \( (3x + 16)^\circ \) are supplementary? Wait, no, wait, no. Wait, no, when \( m \parallel n \), and the transversal \( t \) cuts them, the angle \( (6x - 7)^\circ \) and \( (3x + 16)^\circ \) are actually alternate interior angles? Wait, no, maybe I made a mistake. Wait, no, let's re - examine. If \( m \parallel n \), and the transversal \( t \) intersects them, then the angle \( (6x - 7)^\circ \) and \( (3x + 16)^\circ \) are equal? Wait, no, that can't be. Wait, no, maybe they are same - side interior angles? Wait, no, let's calculate. Wait, if we assume that \( (6x - 7) \) and \( (3x + 16) \) are equal (alternate interior angles), then:

Step2: Set up the equation

If alternate interior angles are equal (because \( m\parallel n \)), then \( 6x-7 = 3x + 16 \)

Step3: Solve for x

Subtract \( 3x \) from both sides: \( 6x-3x-7=3x - 3x+16 \), which gives \( 3x-7 = 16 \)

Add 7 to both sides: \( 3x-7 + 7=16 + 7 \), so \( 3x=23 \)? Wait, that can't be right. Wait, maybe I misidentified the angle relationship. Wait, no, maybe the two angles are supplementary? Wait, no, let's look again. Wait, the two angles \( (6x - 7)^\circ \) and \( (3x + 16)^\circ \) are adjacent and form a linear pair? Wait, no, on line \( n \), the two angles are on a straight line? Wait, no, the transversal cuts line \( n \), so the two angles \( (6x - 7) \) and \( (3x + 16) \) are adjacent and form a linear pair? Wait, no, that would mean they are supplementary. Wait, no, if \( m\parallel n \), then the angle above \( m \) (corresponding to \( (6x - 7) \)) and \( (3x + 16) \) are equal. Wait, maybe I made a mistake in the angle relationship. Let's start over.

Since \( m\parallel n \), the alternate interior angles are equal. The angle \( (6x - 7)^\circ \) and \( (3x + 16)^\circ \) are alternate interior angles? Wait, no, maybe the angle \( (6x - 7) \) and \( (3x + 16) \) are equal because they are alternate interior angles. So:

\( 6x-7=3x + 16 \)

Subtract \( 3x \) from both sides: \( 3x-7 = 16 \)

Add 7 to both sides: \( 3x=23 \), \( x=\frac{23}{3}\approx7.67 \). That doesn't seem right. Wait, maybe I messed up the angle relationship. Wait, maybe the two angles are same - side interior angles, so they are supplementary. So \( (6x - 7)+(3x + 16)=180 \)

Let's try that.

Step1 (corrected): Identify angle relationship (supplementary)

Since \( m\parallel n \), same - side interior angles are supplementary. So \( (6x - 7)+(3x + 16)=180 \)

Step2 (corrected): Simplify the equation

Combine like terms: \( 6x+3x-7 + 16=180 \), \( 9x + 9=180 \)

Step3 (corrected): Solve for x

Subtract 9 from both sides: \( 9x+9 - 9=180 - 9 \), \( 9x = 171 \)

Divide both sides by 9: \( x=\frac{171}{9}=19 \)

Answer:

\( x = 19 \)