Question

14 lines t and s are parallel. given that m∠5 = 125°, what is the measure of ∠8?

Explanation:

Step1: Identify Relationship (Supplementary Angles)

∠5 and ∠8 are supplementary (linear pair), so \( m\angle5 + m\angle8 = 180^\circ \).

Step2: Substitute and Solve

Given \( m\angle5 = 125^\circ \), substitute: \( 125^\circ + m\angle8 = 180^\circ \).
Solve for \( m\angle8 \): \( m\angle8 = 180^\circ - 125^\circ = 55^\circ \).
Also, ∠8 and ∠1 are corresponding angles (lines \( t \) and \( s \) parallel, transversal vertical line), so \( m\angle1 = m\angle8 = 55^\circ \). Wait, maybe typo: if "∠1" (assuming question is ∠1 or ∠8? Wait, original question: "What is the measure of ∠1?" (maybe typo in user, but based on diagram: ∠5 and ∠1? Wait, no: ∠5 and ∠1: ∠5 and ∠1 are same-side? Wait, no: lines \( t \) and \( s \) are parallel, transversal is the vertical line. ∠5 and ∠1: ∠5 is above line \( t \), right of vertical; ∠1 is above line \( t \), left of vertical. So ∠5 and ∠1 are supplementary? Wait, no: ∠5 and ∠1: ∠5 and ∠1 are adjacent? Wait, maybe the question is ∠1. Wait, let's recheck:

Wait, the diagram: vertical line (transversal) intersects two parallel lines \( t \) (top) and \( s \) (bottom). Angles: above \( t \): ∠1 (left), ∠5 (right); between \( t \) and \( s \): ∠2 (left), ∠6 (right); below \( s \): ∠3 (left), ∠7 (right); and ∠4 (left), ∠8 (right). So ∠5 and ∠1: they are adjacent? No, ∠5 and ∠1 are on a straight line? Wait, no, the vertical line is a transversal, so ∠5 and ∠1: ∠5 is adjacent to ∠6, ∠1 is adjacent to ∠2. Wait, maybe the question is ∠1. Wait, ∠5 and ∠1: are they supplementary? Wait, ∠5 and ∠1: if the transversal is vertical, then ∠5 and ∠1 are same-side interior? No, ∠5 is above \( t \), right; ∠1 is above \( t \), left. So they are a linear pair? Wait, no, the horizontal line is \( t \), vertical line is transversal. So ∠1 and ∠5: they form a linear pair? Wait, ∠1 and ∠5 are adjacent, forming a straight line (the top horizontal line and vertical line), so they are supplementary. Wait, no: ∠1 and ∠5 are on a straight line (the top horizontal line), so they are supplementary. Wait, no, the top horizontal line is a straight line, so ∠1 + ∠5 = 180°? Wait, no, ∠1 is left of vertical, ∠5 is right of vertical, on the top horizontal line. So yes, they are supplementary. Wait, but earlier I thought ∠5 and ∠8 are supplementary. Wait, ∠5 is on top horizontal (line \( t \)) right, ∠8 is on bottom horizontal (line \( s \)) right, below. So ∠5 and ∠8: are they related? Wait, line \( t \) and \( s \) are parallel, so ∠5 and ∠8: are they same-side? No, ∠5 is above \( t \), ∠8 is below \( s \). Wait, maybe ∠5 and ∠8 are vertical angles? No, ∠5 is above \( t \), ∠8 is below \( s \). Wait, I think I made a mistake earlier. Let's correct:

∠5 and ∠1: on line \( t \), vertical transversal: ∠1 (left, top), ∠5 (right, top) – they are a linear pair? No, a linear pair is adjacent and form 180°. Wait, ∠1 and ∠5: are they adjacent? The vertical line is between them, so ∠1 is left of vertical, ∠5 is right of vertical, on the same horizontal line (line \( t \)). So they are adjacent, forming a straight line (line \( t \) is straight), so ∠1 + ∠5 = 180°? Wait, no, line \( t \) is a straight line, so the sum of angles on a straight line is 180°. So ∠1 and ∠5 are adjacent, forming a linear pair, so \( m\angle1 + m\angle5 = 180^\circ \). Wait, that's different from before. So if \( m\angle5 = 125^\circ \), then \( m\angle1 = 180^\circ - 125^\circ = 55^\circ \). But earlier, I thought ∠5 and ∠8 are supplementary, but ∠8 is below line \( s \), right. So ∠5 and ∠8: are they related? ∠5 and ∠8: since lines \( t \) and \( s \) are parallel, ∠5 and ∠8: ∠5 is above \( t \), right; ∠8 is below \( s \), right. So they are same-side? No, ∠5 and ∠8: ∠5 and ∠8 are vertical angles? No, ∠5 and ∠8: let's see, ∠5 and ∠1 are linear pair (180°), ∠1 and ∠3 are vertical angles, ∠3 and ∠7 are corresponding, ∠7 and ∠8 are linear pair. Wait, maybe the question is ∠1. So let's re-express:

If the question is "What is the measure of ∠1?" (assuming typo, as original says "∠1?"):

Step1: Linear Pair (∠1 and ∠5)

∠1 and ∠5 are supplementary (linear pair on line \( t \)), so \( m\angle1 + m\angle5 = 180^\circ \).

Step2: Solve for \( m\angle1 \)

Substitute \( m\angle5 = 125^\circ \):
\( m\angle1 = 180^\circ - 125^\circ = 55^\circ \).

Answer:

\( 55^\circ \)