Question

in the image shown, $overleftrightarrow{pf} parallel overleftrightarrow{yk}$, where $mangle phm = (11z + 6)^circ$, $mangle mhx = 47^circ$, and $mangle hwr = 72^circ$. the value of $z$ is $6^circ$ $\boldsymbol{downarrow}$, with a measure of $angle fhr$ of $108^circ$ $\boldsymbol{downarrow}$ and a measure of $angle gwr$ of $\boldsymbol{square}$ $\boldsymbol{downarrow}$.

Explanation:

Step1: Identify Vertical Angles or Corresponding Angles

Since \( \overleftrightarrow{PF} \parallel \overleftrightarrow{YK} \), and considering the transversal, \( \angle GWR \) and \( \angle PHM \) might be related, but also, \( \angle GWR \) and \( \angle HWR \) form a linear pair? Wait, no, let's check. Wait, \( \angle HWR = 72^\circ \), and \( \angle GWR \) is supplementary to \( \angle HWR \)? Wait, no, actually, \( \angle GWR \) and \( \angle HWR \) are adjacent and form a linear pair? Wait, no, looking at the diagram, \( G \), \( W \), \( H \)? Wait, no, \( G \) is on the line through \( W \) and \( H \)? Wait, maybe \( \angle GWR \) is vertical to \( \angle FHR \)? Wait, no, \( \angle FHR \) is \( 108^\circ \), and \( \angle GWR \) should be equal to \( \angle PHM \) or related. Wait, first, we know \( z = 6 \), so \( m\angle PHM = 11z + 6 = 11(6) + 6 = 72^\circ \)? Wait, no, wait \( 11*6 +6 = 72 \)? Wait, no, 11*6 is 66, plus 6 is 72. Wait, but \( \angle HWR = 72^\circ \), and since \( PF \parallel YK \), \( \angle PHM \) and \( \angle HWR \) are corresponding angles? Wait, maybe \( \angle GWR \) is supplementary to \( \angle HWR \)? Wait, no, \( \angle GWR \) and \( \angle HWR \) are adjacent, forming a linear pair? Wait, \( \angle GWR + \angle HWR = 180^\circ \)? No, that can't be. Wait, maybe \( \angle GWR \) is equal to \( \angle FHR \)? Wait, \( \angle FHR \) is \( 108^\circ \), and \( \angle GWR \) should be equal to \( \angle FHR \) because they are corresponding angles or vertical angles. Wait, let's re-examine.

Wait, \( \overleftrightarrow{PF} \parallel \overleftrightarrow{YK} \), and the transversal is \( HR \) or \( HW \)? Wait, \( \angle FHR = 108^\circ \), and \( \angle GWR \) is vertical to \( \angle FHR \)? No, \( \angle GWR \) and \( \angle FHR \) – wait, maybe \( \angle GWR \) is equal to \( \angle FHR \) because of parallel lines. Wait, alternatively, \( \angle GWR \) and \( \angle HWR \) – \( \angle HWR = 72^\circ \), and \( \angle GWR = 180^\circ - 72^\circ = 108^\circ \)? Wait, no, that would be if they are supplementary. Wait, but \( \angle FHR \) is \( 108^\circ \), so maybe \( \angle GWR = 108^\circ \)? Wait, no, let's check the angles.

Wait, \( m\angle PHM = 11z + 6 \), \( z = 6 \), so \( 11*6 +6 = 72^\circ \). \( \angle MHX = 47^\circ \), but maybe that's a distractor. \( \angle HWR = 72^\circ \), which is equal to \( \angle PHM \) (corresponding angles, since \( PF \parallel YK \) and transversal \( HG \)). Then, \( \angle GWR \) and \( \angle FHR \) – \( \angle FHR \) is \( 108^\circ \), and \( \angle GWR \) should be equal to \( \angle FHR \) because they are vertical angles or corresponding angles. Wait, actually, \( \angle GWR \) and \( \angle FHR \) are corresponding angles, so they should be equal. Since \( \angle FHR = 108^\circ \), then \( \angle GWR = 108^\circ \)? Wait, no, maybe I made a mistake. Wait, \( \angle HWR = 72^\circ \), and \( \angle GWR \) is supplementary to \( \angle HWR \)? Wait, \( \angle GWR + \angle HWR = 180^\circ \)? No, that would be if they are adjacent and form a straight line. Wait, \( W \) is on \( YK \), and \( G \) is on the line through \( W \) and \( H \). So \( \angle GWR \) and \( \angle HWR \) are adjacent angles forming a linear pair, so their sum is \( 180^\circ \). Wait, \( \angle HWR = 72^\circ \), so \( \angle GWR = 180^\circ - 72^\circ = 108^\circ \). Yes, that makes sense. So \( \angle GWR = 108^\circ \).

Step1: Calculate \( \angle GWR \)

Since \( \angle GWR \) and \( \angle HWR \) form a linear pair, their sum is \( 180^\circ \).
\( m\angle GWR = 180^\circ - m\angle HWR \)
\( m\angle GWR = 180^\circ - 72^\circ \)
\( m\angle GWR = 108^\circ \)

Answer:

\( 108^\circ \)