Question

look at this diagram:
(diagram with lines and points: t, q, p, r, u, w, v, s as shown)
if \\(\overleftrightarrow{qs}\\) and \\(\overleftrightarrow{tv}\\) are parallel lines and \\(m\angle qrp = 41^\circ\\), what is \\(m\angle sru\\)?
\\(\square^\circ\\)
submit

Explanation:

Step1: Identify angle relationships

Since \( \overleftrightarrow{QS} \parallel \overleftrightarrow{TV} \) and \( \overleftrightarrow{WP} \) is a transversal, \( \angle QRP \) and \( \angle SRU \) are vertical angles? Wait, no, actually, \( \angle QRP \) and \( \angle SRU \): Wait, first, \( \angle QRP \) and \( \angle TUV \) would be corresponding angles, but actually, \( \angle SRU \) and \( \angle QRP \): Wait, no, let's see. Wait, \( \angle QRP \) and \( \angle SRU \): Wait, \( \angle QRP \) is 41 degrees, and since \( QS \parallel TV \), and \( WP \) is a transversal, \( \angle QRP \) and \( \angle SRU \): Wait, actually, \( \angle QRP \) and \( \angle SRU \) are supplementary? No, wait, no. Wait, \( \angle QRP \) is 41 degrees, and \( \angle SRU \) and \( \angle QRP \): Wait, no, let's think again. Wait, \( \angle QRP \) and \( \angle SRU \): Wait, \( \angle QRP \) is 41°, and since \( QS \) and \( TV \) are parallel, and \( WP \) is a transversal, \( \angle QRP \) and \( \angle SRU \): Wait, no, actually, \( \angle SRU \) and \( \angle QRP \): Wait, \( \angle QRP \) is 41°, and \( \angle SRU \) is adjacent to a straight line? Wait, no, let's correct. Wait, \( \angle QRP = 41° \), and \( \angle SRU \) and \( \angle QRP \): Wait, \( \angle SRU \) and \( \angle QRP \) are same - side interior angles? No, wait, no. Wait, \( \angle QRP \) is 41°, and \( \angle SRU \) is vertical to some angle? Wait, no, let's look at the diagram. \( QS \) and \( TV \) are parallel, \( WP \) is a transversal. \( \angle QRP = 41° \), then \( \angle SRU \) is supplementary to \( \angle QRP \)? Wait, no, 180 - 41 = 139? Wait, no, that's not right. Wait, no, \( \angle QRP \) and \( \angle SRU \): Wait, \( \angle QRP \) is 41°, and \( \angle SRU \) is equal to 180° - 41°? Wait, no, let's think about linear pairs or supplementary angles. Wait, \( \angle QRP \) and \( \angle SRU \): Wait, \( \angle QRP \) is 41°, and \( \angle SRU \) is adjacent to a straight line, so \( m\angle SRU = 180° - 41° = 139° \)? Wait, no, that's not correct. Wait, no, let's re - examine. Wait, \( QS \) and \( TV \) are parallel, \( WP \) is a transversal. \( \angle QRP = 41° \), then \( \angle SRU \) and \( \angle QRP \): Wait, \( \angle SRU \) is supplementary to \( \angle QRP \) because they are same - side interior angles? No, same - side interior angles are supplementary. Wait, yes! Because \( QS \parallel TV \), and \( WP \) is a transversal, so \( \angle QRP \) and \( \angle SRU \) are same - side interior angles, so they are supplementary. So \( m\angle SRU = 180° - 41° = 139° \). Wait, but let's check again. Wait, \( \angle QRP = 41° \), and \( \angle SRU \): if \( QS \parallel TV \), then \( \angle QRP \) and \( \angle SRU \) are same - side interior angles, so they add up to 180°. So \( 180 - 41 = 139 \). So \( m\angle SRU = 139° \).

Step1: Determine the relationship between angles

Since \( \overleftrightarrow{QS}\parallel\overleftrightarrow{TV} \) and \( \overleftrightarrow{WP} \) is a transversal, \( \angle QRP \) and \( \angle SRU \) are same - side interior angles. Same - side interior angles are supplementary, which means \( m\angle QRP + m\angle SRU=180^{\circ} \).

Step2: Calculate \( m\angle SRU \)

We know that \( m\angle QRP = 41^{\circ} \). Substitute this value into the equation \( m\angle QRP + m\angle SRU = 180^{\circ} \).
\[

$$\begin{align*} 41^{\circ}+m\angle SRU&=180^{\circ}\\ m\angle SRU&=180^{\circ}- 41^{\circ}\\ m\angle SRU&=139^{\circ} \end{align*}$$

\]

Answer:

\( 139 \)