Question

1 in this diagram, lines m and n are parallel.
diagram of lines m, n, and transversal l with angles 1-8
which statement about ∠4 and ∠8 is true?
a. they are corresponding angles, and they are congruent.
b. they are alternate exterior angles, and they are congruent.
c. they are corresponding angles, and they are not congruent.
d. they are alternate exterior angles, and they are not congruent.

2 this diagram shows lines m and n.
diagram of intersecting lines m and n with angles 1-4
the measure of ∠3 is 2x + 17,
and the measure of ∠4 is 5x + 23.
what is the measure of ∠1?
a. 20°
b. 57°
c. 123°
d. 180°

Explanation:

Problem 1

Step1: Identify angle types

Lines \( m \parallel n \), transversal \( l \). \( \angle 4 \) (interior, upper right of \( m \)) and \( \angle 8 \) (exterior, lower right of \( n \)): check corresponding angles. Corresponding angles are in same relative position. \( \angle 4 \) and \( \angle 8 \) are not in same relative position (one interior, one exterior). Wait, no—wait, \( \angle 4 \) is at \( m \), \( \angle 8 \) at \( n \), both below transversal? Wait, no, diagram: \( m \) top, \( n \) bottom, transversal \( l \) vertical. \( \angle 4 \): between \( m \) and \( n \), right of \( l \). \( \angle 8 \): below \( n \), right of \( l \). Wait, no, maybe I mislabel. Wait, \( \angle 4 \) is adjacent to \( \angle 3 \) (top, left of \( l \) on \( m \)): \( \angle 1, \angle 2 \) top of \( l \) on \( m \), \( \angle 3, \angle 4 \) bottom of \( l \) on \( m \); \( \angle 5, \angle 6 \) top of \( l \) on \( n \), \( \angle 7, \angle 8 \) bottom of \( l \) on \( n \). So \( \angle 4 \) (on \( m \), right of \( l \), interior) and \( \angle 8 \) (on \( n \), right of \( l \), exterior, below \( n \)). Wait, corresponding angles: same position relative to parallel lines and transversal. \( \angle 4 \) and \( \angle 8 \): both right of transversal, \( \angle 4 \) above \( n \), \( \angle 8 \) below \( n \)? No, \( m \parallel n \), so \( \angle 4 \) (interior, on \( m \)) and \( \angle 8 \) (exterior, on \( n \))—wait, no, \( \angle 4 \) and \( \angle 8 \): are they corresponding? Wait, \( \angle 4 \) and \( \angle 8 \): \( \angle 4 \) is at \( m \), \( \angle 8 \) at \( n \), both on the right side of transversal \( l \), and \( \angle 4 \) is above \( n \), \( \angle 8 \) is below \( n \)? No, \( m \) and \( n \) are parallel, horizontal. Transversal \( l \) is vertical. So \( \angle 4 \): (m, right of l, below m's top line? Wait, no, the diagram: \( m \) is a horizontal line, \( n \) is another horizontal line below \( m \), parallel. Transversal \( l \) is vertical, intersecting both. So \( \angle 1 \) (top left of l on m), \( \angle 2 \) (top right of l on m), \( \angle 3 \) (bottom left of l on m), \( \angle 4 \) (bottom right of l on m); \( \angle 5 \) (top left of l on n), \( \angle 6 \) (top right of l on n), \( \angle 7 \) (bottom left of l on n), \( \angle 8 \) (bottom right of l on n). So \( \angle 4 \) (m, bottom right of l) and \( \angle 8 \) (n, bottom right of l). Wait, no, \( \angle 4 \) is on \( m \), \( \angle 8 \) on \( n \), both bottom right of transversal. So they are corresponding angles? Wait, corresponding angles: when two parallel lines cut by transversal, corresponding angles are congruent. Wait, but \( \angle 4 \) is on \( m \), \( \angle 8 \) on \( n \), same position (right of transversal, below the horizontal line? Wait, \( m \) and \( n \) are horizontal, so \( \angle 4 \) is below \( m \)'s top? No, \( m \) is a single line, so \( \angle 1 \) and \( \angle 2 \) are above \( m \), \( \angle 3 \) and \( \angle 4 \) are below \( m \)? No, no—intersecting lines: when two lines intersect, they form vertical angles. So \( l \) intersects \( m \), forming \( \angle 1, \angle 2, \angle 3, \angle 4 \) (vertical angles: \( \angle 1 \) and \( \angle 3 \) are vertical? No, \( \angle 1 \) and \( \angle 3 \) are adjacent, forming linear pair. Wait, no, \( l \) is vertical, \( m \) is horizontal: so \( \angle 1 \) (top left, 90? No, not necessarily. Wait, the problem says \( m \parallel n \), so \( l \) is transversal. So \( \angle 4 \) and \( \angle 8 \): \( \angle 4 \) is on \( m \), between \( m \) and \( n \) (interior),…

Step1: Identify angle relationship

Lines \( m \) and \( n \) intersect, so \( \angle 3 \) and \( \angle 4 \) are supplementary (linear pair: they form a straight line, so \( \angle 3 + \angle 4 = 180^\circ \)).

Step2: Set up equation

Given \( \angle 3 = 2x + 17 \) and \( \angle 4 = 5x + 23 \), so:
\[
(2x + 17) + (5x + 23) = 180
\]

Step3: Solve for \( x \)

Combine like terms:
\[
7x + 40 = 180
\]
Subtract 40:
\[
7x = 140
\]
Divide by 7:
\[
x = 20
\]

Step4: Find \( \angle 3 \) or \( \angle 4 \), then \( \angle 1 \)

First, find \( \angle 3 \): \( 2(20) + 17 = 40 + 17 = 57^\circ \). \( \angle 1 \) and \( \angle 3 \) are vertical angles (opposite angles when two lines intersect), so vertical angles are congruent. Thus, \( \angle 1 = \angle 3 = 57^\circ \). Wait, but wait: \( \angle 1 \) and \( \angle 3 \) are vertical angles? Let's check the diagram: lines \( m \) and \( n \) intersect, forming \( \angle 1, \angle 2, \angle 3, \angle 4 \). So \( \angle 1 \) and \( \angle 3 \) are vertical angles (opposite), so they are congruent. Alternatively, \( \angle 4 = 5(20) + 23 = 100 + 23 = 123^\circ \), and \( \angle 1 \) and \( \angle 4 \) are supplementary? Wait, no—wait, \( \angle 1 \) and \( \angle 2 \) are supplementary, \( \angle 2 \) and \( \angle 3 \) are supplementary, etc. Wait, no, when two lines intersect, vertical angles are equal, and linear pairs are supplementary. So \( \angle 1 \) and \( \angle 3 \) are vertical angles (opposite), so \( \angle 1 = \angle 3 \). We found \( \angle 3 = 57^\circ \), so \( \angle 1 = 57^\circ \). Wait, but let's confirm: \( \angle 3 = 2x + 17 = 57 \), \( \angle 4 = 5x + 23 = 123 \), and \( 57 + 123 = 180 \), which checks out (linear pair). Then \( \angle 1 \) is vertical to \( \angle 3 \), so \( \angle 1 = 57^\circ \), which is option B.

Answer:

A. They are corresponding angles, and they are congruent.

Problem 2