Question

29
parallel lines p and q are intersected by transversal r, as shown.
what is the value of x?
30
use the relationships between the angles to find the value of x.
x = degrees

Explanation:

Problem 29

Step1: Identify angle relationship

Since \( p \parallel q \) and \( r \) is a transversal, the angle \( x^\circ \) and the \( 30^\circ \) angle are supplementary (they form a linear pair or use consecutive interior angles? Wait, actually, looking at the diagram, the angle adjacent to \( 30^\circ \) and \( x \) – wait, no, when two parallel lines are cut by a transversal, consecutive interior angles are supplementary, but also, the angle \( x \) and the angle that is supplementary to \( 30^\circ \)? Wait, no, actually, the angle \( x \) and the \( 30^\circ \) angle: wait, the angle on line \( p \) with \( 30^\circ \) – the vertical angle or the supplementary? Wait, no, let's think again. The angle \( x \) and the angle that is \( 180^\circ - 30^\circ \)? No, wait, the diagram: the transversal \( r \) intersects \( p \) and \( q \). The angle between \( p \) and \( r \) is \( 30^\circ \), and the angle \( x \) is on \( q \). Since \( p \parallel q \), the angle \( x \) and the angle supplementary to \( 30^\circ \)? Wait, no, actually, the angle \( x \) and the \( 30^\circ \) angle: wait, the angle \( x \) is a linear pair with the corresponding angle? Wait, no, let's see: the angle adjacent to \( 30^\circ \) on line \( p \) is \( 180^\circ - 30^\circ = 150^\circ \), but that's not right. Wait, no, the angle \( x \) and the \( 30^\circ \) angle: actually, the angle \( x \) is equal to \( 180^\circ - 30^\circ \)? No, wait, no. Wait, the angle \( x \) and the \( 30^\circ \) angle: when two parallel lines are cut by a transversal, the consecutive interior angles are supplementary. Wait, the angle \( 30^\circ \) and the angle \( x \): wait, no, the angle \( x \) is actually supplementary to \( 30^\circ \)? Wait, no, let's draw it mentally. Line \( p \) and \( q \) are parallel, transversal \( r \). The angle between \( p \) and \( r \) is \( 30^\circ \), so the angle on the other side of \( p \) (adjacent) is \( 180^\circ - 30^\circ = 150^\circ \), but that's not \( x \). Wait, no, the angle \( x \) is on line \( q \), and since \( p \parallel q \), the corresponding angle to the angle supplementary to \( 30^\circ \)? Wait, I think I made a mistake. Wait, the angle \( x \) and the \( 30^\circ \) angle: actually, the angle \( x \) is equal to \( 180^\circ - 30^\circ = 150^\circ \)? No, that can't be. Wait, no, the diagram: the angle \( x \) is a linear pair with the angle that is equal to \( 30^\circ \) (corresponding angle). Wait, no, corresponding angles are equal. Wait, the angle \( 30^\circ \) and the angle that is vertical to the angle adjacent to \( x \). Wait, maybe I should use the fact that consecutive interior angles are supplementary. Wait, the angle \( 30^\circ \) and the angle \( x \): if \( p \parallel q \), then the consecutive interior angles are supplementary. So the angle \( 30^\circ \) and the angle \( x \) are supplementary? No, \( 30 + x = 180 \)? Then \( x = 150 \). Wait, that makes sense. Because the angle between \( p \) and \( r \) is \( 30^\circ \), and the angle \( x \) is on \( q \), so they are consecutive interior angles, so they add up to \( 180^\circ \). So \( x = 180 - 30 = 150 \).

Step1: Determine angle relationship

Since \( p \parallel q \) and \( r \) is a transversal, \( x^\circ \) and \( 30^\circ \) are supplementary (consecutive interior angles).

Step2: Calculate \( x \)

\( x + 30 = 180 \)
\( x = 180 - 30 \)
\( x = 150 \)

Step1: Identify angle relationship

Since the two lines are parallel and cut by a transversal, the angle \( 102^\circ \) and the angle \( x + 15^\circ \) are corresponding angles? Wait, no, the angle \( 102^\circ \) and the angle adjacent to \( x + 15^\circ \) – wait, the angle \( 102^\circ \) and the angle \( x + 15^\circ \): are they supplementary? Wait, no, the diagram: the angle \( 102^\circ \) is on the top line, and \( x + 15^\circ \) is on the bottom line. Since the lines are parallel, the angle \( 102^\circ \) and the angle \( x + 15^\circ \) are same - side interior angles? No, wait, the angle \( 102^\circ \) and the angle \( x + 15^\circ \): let's see, the angle \( 102^\circ \) and the angle that is supplementary to \( x + 15^\circ \)? Wait, no, the angle \( 102^\circ \) and \( x + 15^\circ \): actually, the angle \( 102^\circ \) and \( x + 15^\circ \) are equal? No, wait, the angle \( 102^\circ \) and the angle adjacent to \( x + 15^\circ \) – wait, the angle \( 102^\circ \) and \( x + 15^\circ \): let's check the diagram. The top line has an angle of \( 102^\circ \) with the transversal, and the bottom line has an angle of \( x + 15^\circ \) with the transversal. Since the lines are parallel, the angle \( 102^\circ \) and \( x + 15^\circ \) are same - side interior angles? No, same - side interior angles are supplementary. Wait, yes! Same - side interior angles are supplementary when lines are parallel. So \( 102 + (x + 15)=180 \).

Step1: Set up the equation

Since the lines are parallel, same - side interior angles are supplementary:
\( 102 + (x + 15)=180 \)

Step2: Simplify the equation

\( x + 117 = 180 \)

Step3: Solve for \( x \)

\( x = 180 - 117 \)
\( x = 63 \)

Answer:

\( x = 150 \)

Problem 30