Question

given ( m parallel n ), find the value of ( x ).
answer attempt 1 out of 2
( x = )

Explanation:

Step1: Identify Alternate Exterior Angles

Since \( m \parallel n \) and \( t \) is a transversal, the angles \( (x - 2)^\circ \) and \( (x - 14)^\circ \) are alternate exterior angles. Alternate exterior angles are equal when lines are parallel. So we set up the equation:
\( x - 2 = x - 14 \)? Wait, no, that can't be. Wait, maybe I misidentified. Wait, actually, looking at the diagram, maybe they are same - side or maybe corresponding? Wait, no, let's re - examine. Wait, maybe the angles are supplementary? Wait, no, alternate exterior angles: when two parallel lines are cut by a transversal, alternate exterior angles are congruent. Wait, but if we look at the positions, maybe the angle \( (x - 2)^\circ \) and the angle adjacent to \( (x - 14)^\circ \) are related. Wait, no, maybe the correct approach is that since \( m\parallel n \), the angle \( (x - 2)^\circ \) and \( (x - 14)^\circ \) are actually same - side exterior? No, that doesn't make sense. Wait, maybe I made a mistake. Wait, let's think again. If we have two parallel lines \( m \) and \( n \), and a transversal \( t \). The angle \( (x - 2)^\circ \) is on line \( m \), and \( (x - 14)^\circ \) is on line \( n \). If we consider the vertical angles or corresponding angles. Wait, maybe the angle \( (x - 2)^\circ \) and the angle that is supplementary to \( (x - 14)^\circ \) are equal? No, that's not right. Wait, perhaps the correct equation is \( x - 2=(180-(x - 14)) \)? No, that seems complicated. Wait, no, let's look at the diagram again. Wait, maybe the two angles \( (x - 2)^\circ \) and \( (x - 14)^\circ \) are actually equal because they are alternate exterior angles. Wait, but if we set \( x - 2=x - 14 \), we get \( - 2=-14 \), which is impossible. So I must have misidentified the angle relationship. Wait, maybe the angle \( (x - 2)^\circ \) and \( (x - 14)^\circ \) are same - side interior angles? No, same - side interior angles are supplementary. Wait, no, same - side exterior angles? No. Wait, maybe the angle \( (x - 2)^\circ \) and the angle equal to \( 180-(x - 14) \) are equal. Let's try that. So \( x - 2 = 180-(x - 14) \)

Step2: Solve the Equation

Expand the right - hand side: \( x - 2=180 - x+14 \)
Simplify the right - hand side: \( x - 2=194 - x \)
Add \( x \) to both sides: \( x+x - 2=194 - x+x \)
We get \( 2x - 2 = 194 \)
Add 2 to both sides: \( 2x-2 + 2=194 + 2 \)
\( 2x=196 \)
Divide both sides by 2: \( x=\frac{196}{2}=98 \)

Wait, but let's check again. Wait, maybe the correct angle relationship is that the two angles \( (x - 2)^\circ \) and \( (x - 14)^\circ \) are supplementary? No, that would be \( (x - 2)+(x - 14)=180 \)
Let's try that equation:
\( x - 2+x - 14 = 180 \)
Combine like terms: \( 2x-16 = 180 \)
Add 16 to both sides: \( 2x=180 + 16=196 \)
Divide by 2: \( x = 98 \)
Ah, that works. So the two angles are same - side interior angles? Wait, no, same - side interior angles are supplementary. Wait, maybe the diagram shows that the two angles are same - side interior angles. So when two parallel lines are cut by a transversal, same - side interior angles are supplementary. So we set up the equation \( (x - 2)+(x - 14)=180 \)

Step1 (Correct): Identify the Angle Relationship

Since \( m\parallel n \) and the two angles \( (x - 2)^\circ \) and \( (x - 14)^\circ \) are same - side interior angles (or same - side exterior? No, same - side interior angles are between the two lines), wait, actually, looking at the positions, the angle \( (x - 2)^\circ \) is above line \( m \), and \( (x - 14)^\circ \) is below line \( n \). Wait, maybe they are same - side exterior angles, and same - side exterior angles are supplementary. So we have the equation:
\( (x - 2)+(x - 14)=180 \)

Step2: Solve the Equation

Combine like terms: \( x+x-2 - 14=180 \)
\( 2x-16 = 180 \)
Add 16 to both sides: \( 2x=180 + 16 \)
\( 2x=196 \)
Divide both sides by 2: \( x=\frac{196}{2}=98 \)

Answer:

\( x = 98 \)