Question

given $m\parallel n$, find the value of x.

Explanation:

Step1: Identify angle relationship

Since \( m \parallel n \) and \( t \) is a transversal, the given angles are alternate exterior angles, so they are equal.
\[ 9x - 10 = 8x - 14 \]

Step2: Solve for \( x \)

Subtract \( 8x \) from both sides:
\[ 9x - 8x - 10 = 8x - 8x - 14 \]
\[ x - 10 = -14 \]

Add 10 to both sides:
\[ x - 10 + 10 = -14 + 10 \]
\[ x = -4 \] Wait, that can't be right. Wait, maybe I mixed up the angle types. Wait, no, maybe they are alternate interior? Wait, no, let's check the diagram again. Wait, maybe the angles are same - side? No, wait, maybe I made a mistake. Wait, no, let's re - examine. Wait, the angles: one is \( (9x - 10)^\circ \) on line \( m \), and \( (8x - 14)^\circ \) on line \( n \), with transversal \( t \). Wait, maybe they are corresponding angles? Wait, no, maybe I got the direction wrong. Wait, actually, if we consider that the angles are alternate exterior angles, but maybe they are supplementary? Wait, no, let's think again. Wait, maybe the angles are same - side exterior? No, same - side exterior angles are supplementary. Wait, maybe I made a mistake in the angle relationship. Let's start over.

Wait, when two parallel lines are cut by a transversal, alternate exterior angles are equal. But maybe in this case, the angles are actually alternate interior angles? Wait, no, let's look at the positions. The angle \( (9x - 10)^\circ \) is above line \( m \), and \( (8x - 14)^\circ \) is below line \( n \), on the opposite side of the transversal. Wait, maybe they are vertical angles? No, vertical angles are opposite each other when two lines intersect. Wait, maybe the angles are corresponding angles. Wait, no, let's check the sum. Wait, maybe I made a mistake in the initial assumption. Let's suppose that the angles are supplementary. Wait, no, let's re - derive.

Wait, if \( m \parallel n \), and the two angles are same - side exterior angles, then they are supplementary. So \( (9x - 10)+(8x - 14)=180 \)

Combine like terms:
\[ 9x+8x-10 - 14=180 \]
\[ 17x-24 = 180 \]

Add 24 to both sides:
\[ 17x=180 + 24=204 \]

Divide by 17:
\[ x=\frac{204}{17}=12 \]

Ah, I see, I misidentified the angle relationship. The angles are same - side exterior angles, so they are supplementary.

Answer:

\( x = 12 \)