Question

two parallel lines are crossed by a transversal. what is the value of x? x = 21 x = 28 x = 35 x = 37 (3x + 4)° 115°

Explanation:

Step1: Identify angle relationship

Since \( y \) and \( z \) are parallel, the \( 115^\circ \) angle and \( (3x + 4)^\circ \) are same - side interior angles? Wait, no, actually, the \( 115^\circ \) angle and \( (3x + 4)^\circ \) should be supplementary? Wait, no, looking at the diagram, the \( 115^\circ \) angle and \( (3x + 4)^\circ \) are same - side interior angles? Wait, no, let's re - examine. If two parallel lines are cut by a transversal, same - side interior angles are supplementary. Wait, but also, the angle \( (3x + 4)^\circ \) and the angle adjacent to \( 115^\circ \) (vertical angles or corresponding angles)? Wait, actually, the \( 115^\circ \) angle and \( (3x + 4)^\circ \) are same - side interior angles? No, wait, the correct relationship: since the two lines are parallel, the angle \( (3x + 4)^\circ \) and \( 115^\circ \) are supplementary? Wait, no, let's think again. The angle \( (3x + 4)^\circ \) and the angle that is supplementary to \( 115^\circ \)? Wait, no, the correct equation: \( 3x+4 + 115=180 \)? Wait, no, that would be if they are same - side interior angles. Wait, let's check the answer options. Let's solve \( 3x + 4=115 - \) no, wait, maybe the angle \( (3x + 4)^\circ \) and \( 115^\circ \) are supplementary? Wait, no, let's do the math.

Wait, the correct approach: when two parallel lines are cut by a transversal, same - side interior angles are supplementary. So \( (3x + 4)+115 = 180 \)? Wait, no, that would give \( 3x=180 - 115 - 4=61 \), which is not one of the options. Wait, maybe the angle \( (3x + 4)^\circ \) is equal to \( 115^\circ \)? No, that would be corresponding angles. Wait, no, maybe I made a mistake. Wait, the other way: the angle \( (3x + 4)^\circ \) and \( 115^\circ \) are supplementary? Wait, no, let's check the answer options. Let's try \( 3x+4 = 115 - \) no, let's try the equation \( 3x + 4=180 - 115 \)? Wait, \( 180 - 115 = 65 \), no. Wait, maybe the angle \( (3x + 4)^\circ \) and \( 115^\circ \) are equal? No, that doesn't make sense. Wait, wait, maybe the angle \( (3x + 4)^\circ \) and \( 115^\circ \) are supplementary? Wait, no, let's solve for \( x \) using the options.

Let's test \( x = 37 \): \( 3\times37+4=111 + 4 = 115 \). Oh! Wait, so \( 3x + 4=115 \). So \( 3x=115 - 4=111 \), \( x = 37 \). So the angle \( (3x + 4)^\circ \) is equal to \( 115^\circ \), which means they are corresponding angles (since the lines are parallel). So the correct equation is \( 3x + 4=115 \).

Step2: Solve for \( x \)

Start with the equation \( 3x+4 = 115 \)
Subtract 4 from both sides: \( 3x=115 - 4=111 \)
Divide both sides by 3: \( x=\frac{111}{3}=37 \)

Answer:

\( x = 37 \) (the option \( x = 37 \))