Question

6-8 additional practice

1. leveled practice if ( p parallel q ), what is the value of ( v )?

( angle u ) and ( angle v ) are (\boxed{}) angles.
so, ( angle u ) and ( angle v ) are (\boxed{})
( mangle v ) is (\boxed{}).
“measure of angle ( v ) is ?”

1. are ( angle 6 ) and ( angle 7 ) corresponding angles if ( a parallel b ) and ( c parallel d )? explain.
2. find ( mangle v ) given that ( p parallel q ), ( mangle u = 75.8^circ ), and ( mangle w = 104.2^circ ).
3. in the figure ( m parallel n ). what is the value of ( x )?

( x = \boxed{})

1. reasoning what value of ( x ) will show that line ( m ) is parallel to line ( n )? explain.

( x = \boxed{})
why?

Explanation:

Problem 1:

Step1: Identify angle relationship

Since \( p \parallel q \), \( \angle u \) and \( \angle v \) are same - side interior angles? Wait, no, looking at the diagram, \( \angle u = 104^{\circ} \), \( \angle w=76^{\circ} \), and \( \angle u \) and \( \angle v \): actually, \( \angle u \) and \( \angle v \) are same - side interior angles? Wait, no, \( \angle u \) and \( \angle v \) should be supplementary? Wait, no, \( \angle u \) and \( \angle v \): since \( p\parallel q \) and the transversal, \( \angle u \) and \( \angle v \) are same - side interior angles? Wait, no, \( \angle u = 104^{\circ} \), \( \angle w = 76^{\circ} \), and \( \angle v \) and \( \angle w \) are adjacent? Wait, maybe \( \angle u \) and \( \angle v \) are same - side interior angles, so they are supplementary? Wait, no, \( 104 + 76=180 \), so \( \angle u \) and \( \angle v \) are same - side interior angles (or maybe alternate interior? No, let's re - examine. The lines \( p \) and \( q \) are parallel, and the transversal is the vertical line. \( \angle u \) is on line \( p \), \( \angle v \) is on line \( q \). So \( \angle u \) and \( \angle v \) are same - side interior angles, so they are supplementary? Wait, no, \( \angle u = 104^{\circ} \), so \( \angle v=180 - 104 = 76^{\circ} \)? Wait, no, the diagram shows \( \angle w = 76^{\circ} \), maybe \( \angle v=\angle u \)'s supplement? Wait, let's do step by step.

Step1: Determine angle type

\( \angle u \) and \( \angle v \) are same - side interior angles (because \( p\parallel q \) and the transversal cuts them, forming same - side interior angles).

Step2: Use supplementary property

Same - side interior angles are supplementary, so \( m\angle u + m\angle v=180^{\circ} \). Given \( m\angle u = 104^{\circ} \), then \( m\angle v=180 - 104 = 76^{\circ} \). Wait, but \( \angle w = 76^{\circ} \), maybe \( \angle v=\angle w \)? Because \( \angle v \) and \( \angle w \) are vertical angles? Wait, no, the diagram: \( \angle v \) and \( \angle w \) are adjacent? Wait, maybe I made a mistake. Let's look again. The vertical line is the transversal. \( \angle u \) is on line \( p \), \( \angle v \) is on line \( q \). So \( \angle u \) and \( \angle v \) are same - side interior angles, so \( m\angle u + m\angle v = 180^{\circ} \). So \( m\angle v=180 - 104 = 76^{\circ} \).

To determine if \( \angle6 \) and \( \angle7 \) are corresponding angles when \( a\parallel b \) and \( c\parallel d \), we recall the definition of corresponding angles: Corresponding angles are in the same position relative to the parallel lines and the transversal. For \( \angle6 \) and \( \angle7 \), we check their positions. \( \angle6 \) is on line \( c \), \( \angle7 \) is on line \( a \). The transversals are the other lines. Since \( a\parallel b \) and \( c\parallel d \), but the position of \( \angle6 \) (on \( c \), between the two transversals) and \( \angle7 \) (on \( a \), at the intersection of two transversals) does not match the definition of corresponding angles (same relative position: e.g., both above the parallel lines, both below, both to the left of the transversal etc.). So \( \angle6 \) and \( \angle7 \) are not corresponding angles.

Step1: Identify angle relationship

Since \( p\parallel q \), \( \angle u \) and \( \angle v \) are same - side interior angles (or we can use the fact that \( \angle u + \angle v=180^{\circ} \) if they are same - side interior, but wait, \( m\angle u = 75.8^{\circ} \), \( m\angle w = 104.2^{\circ} \). Wait, \( \angle v \) and \( \angle w \): no, \( \angle u \) and \( \angle v \): since \( p\parallel q \), \( \angle u \) and \( \angle v \) are same - side interior angles, so \( m\angle u+m\angle v = 180^{\circ} \)? Wait, \( 75.8+104.2 = 180 \), so \( m\angle v=180 - 75.8=104.2^{\circ} \)? No, that can't be. Wait, maybe \( \angle v \) and \( \angle u \) are alternate interior angles? No, the diagram: \( \angle u \) is on line \( p \), \( \angle v \) is on line \( q \). Wait, maybe \( \angle v = \angle u \)'s supplement? Wait, \( m\angle u = 75.8^{\circ} \), so \( m\angle v=180 - 75.8 = 104.2^{\circ} \)? But \( m\angle w = 104.2^{\circ} \), so \( \angle v=\angle w \)? Maybe \( \angle v \) and \( \angle w \) are vertical angles? No, let's do it properly.

Step1: Angle relationship

Since \( p\parallel q \), \( \angle u \) and \( \angle v \) are same - side interior angles, so \( m\angle u + m\angle v=180^{\circ} \).

Step2: Calculate \( m\angle v \)

Given \( m\angle u = 75.8^{\circ} \), then \( m\angle v=180 - 75.8 = 104.2^{\circ} \).

Answer:

\( m\angle v = 76^{\circ} \) (assuming \( \angle u \) and \( \angle v \) are same - side interior angles and supplementary)

Problem 2: