Question

in the diagram below the horizontal lines are all parallel.
let
$m\angle 1 = 2x + 18$ and $m\angle 2 = 8x + 22$.

• create and use a model to determine $m\angle 1$ and $m\angle 2$. show your work or explain your answer.
• let $m\angle 5 = 104 - 6x$. determine $m\angle 3$ and explain how you determined this measure. show your work or explain your answer.

Explanation:

Step1: Identify Relationship Between ∠1 and ∠2

Since lines \( l \), \( m \), \( n \) are parallel, and the transversal creates alternate exterior (or corresponding) angles? Wait, actually, ∠1 and ∠2: let's see, the transversal cuts the parallel lines, so ∠1 and ∠2 should be equal? Wait, no, maybe same - side? Wait, no, looking at the diagram, ∠1 and ∠2: since the horizontal lines are parallel, and the transversal, ∠1 and ∠2 are equal? Wait, no, maybe they are supplementary? Wait, no, let's think again. Wait, the lines are parallel, so if we consider the transversal, ∠1 and ∠2: actually, ∠1 and ∠2 are equal because of the parallel lines (corresponding angles or alternate interior angles). Wait, no, maybe they are equal. So we set \( 2x + 18=8x + 22 \)? Wait, no, that would give negative x. Wait, maybe they are supplementary? So \( (2x + 18)+(8x + 22)=180 \). Let's check:

Step1: Set Up Equation for ∠1 and ∠2

Since the horizontal lines are parallel, and the transversal, ∠1 and ∠2 are same - side interior angles? Wait, no, maybe ∠1 and ∠2 are supplementary. So:
\( (2x + 18)+(8x + 22)=180 \)
Combine like terms: \( 10x+40 = 180 \)

Step2: Solve for x

Subtract 40 from both sides: \( 10x=180 - 40=140 \)
Divide by 10: \( x = 14 \)

Step3: Find m∠1 and m∠2

For \( m\angle1=2x + 18 \), substitute \( x = 14 \):
\( m\angle1=2(14)+18=28 + 18 = 46 \)
For \( m\angle2=8x + 22 \), substitute \( x = 14 \):
\( m\angle2=8(14)+22=112 + 22 = 134 \)
Check: \( 46+134 = 180 \), which is supplementary, so that makes sense.

Step4: Find m∠5 and then m∠3

First, find \( x = 14 \), so \( m\angle5=104-6x \). Substitute \( x = 14 \):
\( m\angle5=104-6(14)=104 - 84 = 20 \)
Now, looking at the diagram, ∠3, ∠5, and the angle at the intersection (let's say ∠4) form a triangle? Wait, or ∠3 and ∠5: since line \( m \) is parallel to line \( l \) or \( n \), maybe ∠3 is equal to \( 180-(m\angle4 + m\angle5) \)? Wait, no, maybe ∠3 and ∠5: let's think about the triangle. Wait, the angle at ∠4: since line \( m \) is parallel to line \( l \), ∠4 is equal to ∠1 (corresponding angles), so \( m\angle4 = 46 \). Then in the triangle, ∠3+∠4+∠5 = 180? Wait, no, maybe ∠3 is equal to \( 180-(m\angle4 + m\angle5) \)? Wait, no, let's see: ∠4 is 46, ∠5 is 20, so ∠3=180 - 46 - 20=114? Wait, no, maybe ∠3 and ∠5: wait, maybe ∠3 is equal to \( 180 - m\angle5 - m\angle4 \), but ∠4 is equal to ∠1 (46). So:
\( m\angle3=180 - m\angle4 - m\angle5 \)
\( m\angle4 = 46 \), \( m\angle5 = 20 \)
\( m\angle3=180 - 46 - 20 = 114 \)? Wait, no, maybe I made a mistake. Wait, alternatively, since line \( m \) is parallel to line \( l \), and the transversal, maybe ∠3 is equal to \( 180 - m\angle5 \)? Wait, no, let's re - examine.

Wait, first, we found \( x = 14 \), so \( m\angle5=104-6\times14=104 - 84 = 20 \). Now, looking at the diagram, ∠3, ∠4, and ∠5: ∠4 is equal to ∠1 (46) because of parallel lines (corresponding angles). Then, in the triangle (or the angle sum), ∠3+∠4+∠5 = 180? Wait, no, maybe ∠3 is equal to \( 180-(m\angle4 + m\angle5) \). So \( 180-(46 + 20)=114 \). Wait, but let's check again.

Wait, maybe another approach: ∠3 and ∠5: since line \( m \) is parallel to line \( l \), and the transversal, ∠3 is equal to \( 180 - m\angle5 - m\angle4 \), but ∠4 is 46, ∠5 is 20, so 180 - 46 - 20 = 114.

Answer:

For the first part: \( m\angle1 = 46^{\circ} \), \( m\angle2 = 134^{\circ} \)

For the second part: \( m\angle3=114^{\circ} \) (assuming the angle sum in the triangle - like figure, with ∠4 = 46 and ∠5 = 20, so ∠3=180 - 46 - 20 = 114)