Question

1. in the diagram below ( l parallel m ). if ( mangle 2 = (32x + 4)^circ ) and ( mangle 8 = (25x + 5)^circ ), find ( mangle 2 ). show all work.

(diagram of two parallel lines ( l ) and ( m ) cut by a transversal, with angles labeled 1 - 9)
( x = square )
( mangle 2 = square )

Explanation:

Step1: Identify angle relationship

Since \( l \parallel m \), \( \angle 2 \) and \( \angle 8 \) are alternate exterior angles, so they are supplementary (wait, no—alternate exterior angles are equal? Wait, no, looking at the diagram, \( \angle 2 \) and \( \angle 8 \): actually, \( \angle 2 \) and \( \angle 8 \) are same - side? Wait, no, let's correct. When two parallel lines are cut by a transversal, alternate exterior angles are equal? Wait, no, \( \angle 2 \) and \( \angle 7 \) are alternate exterior, but \( \angle 8 \) is vertical to \( \angle 6 \), and \( \angle 2 \) and \( \angle 8 \): actually, \( \angle 2 \) and \( \angle 8 \) are same - side exterior? No, wait, \( \angle 2 \) and \( \angle 8 \): let's see, \( \angle 2 \) and \( \angle 4 \) are vertical, \( \angle 8 \) and \( \angle 6 \) are vertical. Since \( l\parallel m \), \( \angle 4 \) and \( \angle 6 \) are same - side interior, so they are supplementary. But \( \angle 2=\angle 4 \) (vertical angles) and \( \angle 8 = \angle 6 \) (vertical angles), so \( \angle 2+\angle 8 = 180^{\circ}\)? Wait, no, maybe I made a mistake. Wait, the correct relationship: \( \angle 2 \) and \( \angle 8 \) are same - side exterior angles? Wait, no, let's look at the positions. \( \angle 2 \) is above line \( l \), to the right of the transversal. \( \angle 8 \) is below line \( m \), to the right of the transversal. Wait, actually, \( \angle 2 \) and \( \angle 8 \) are supplementary? Wait, no, let's use the fact that \( \angle 2 \) and \( \angle 8 \): since \( l\parallel m \), and the transversal, \( \angle 2 \) and \( \angle 8 \) are same - side exterior angles, so they are supplementary. Wait, no, alternate exterior angles: \( \angle 2 \) and \( \angle 7 \) are alternate exterior, \( \angle 3 \) and \( \angle 8 \) are alternate exterior? No, maybe I messed up. Wait, the problem says \( l\parallel m \), and we have \( \angle 2=(32x + 4)^{\circ}\) and \( \angle 8=(25x + 5)^{\circ}\). Wait, maybe \( \angle 2 \) and \( \angle 8 \) are supplementary? Wait, no, let's check the vertical angles. \( \angle 2 \) and \( \angle 4 \) are vertical, \( \angle 8 \) and \( \angle 6 \) are vertical. \( \angle 4 \) and \( \angle 6 \) are same - side interior angles, so \( \angle 4+\angle 6 = 180^{\circ}\). So \( \angle 2+\angle 8=180^{\circ}\) (because \( \angle 2 = \angle 4 \) and \( \angle 8=\angle 6 \)). So the equation is \( (32x + 4)+(25x + 5)=180 \).

Step2: Solve for x

Combine like terms: \( 32x+25x + 4 + 5=180 \)
\( 57x+9 = 180 \)
Subtract 9 from both sides: \( 57x=180 - 9=171 \)
Divide both sides by 57: \( x=\frac{171}{57}=3 \)

Step3: Find \( m\angle 2 \)

Substitute \( x = 3 \) into \( m\angle 2=(32x + 4)^{\circ}\)
\( m\angle 2=32\times3+4=96 + 4=100^{\circ}\)

Answer:

\( x = 3 \), \( m\angle 2=100^{\circ}\)