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 blank, with a measure of $angle fhr$ of blank and a measure of $angle gwr$ of blank.

Explanation:

Step1: Identify Alternate Interior Angles

Since \( \overleftrightarrow{PF} \parallel \overleftrightarrow{YK} \), \( \angle PHM \) and \( \angle HWR \) are alternate interior angles? Wait, no, let's check the angles. Wait, \( \angle PHM \) and \( \angle HWR \)? Wait, actually, \( \angle PHM \) and \( \angle HWR \) – wait, maybe corresponding angles? Wait, \( \angle PHM \) and \( \angle HWR \): Wait, \( \angle PHM = (11z + 6)^\circ \), \( \angle HWR = 72^\circ \), and \( \angle MHX = 47^\circ \). Wait, maybe \( \angle PHM + \angle MHX = \angle PHX \), but since \( PF \parallel YK \), \( \angle PHX \) should be equal to \( \angle HWR \)? Wait, no, let's re-examine.

Wait, \( \overleftrightarrow{PF} \parallel \overleftrightarrow{YK} \), and \( \overleftrightarrow{HG} \) and \( \overleftrightarrow{HB} \) are transversals. Wait, \( \angle PHM \) and \( \angle HWR \): Wait, \( \angle PHM = (11z + 6)^\circ \), \( \angle HWR = 72^\circ \), and \( \angle MHX = 47^\circ \). Wait, maybe \( \angle PHM + \angle MHX = \angle PHX \), and \( \angle PHX \) is equal to \( \angle HWR \) because they are corresponding angles? Wait, no, \( \angle HWR = 72^\circ \), \( \angle MHX = 47^\circ \), so \( \angle PHM + \angle MHX = \angle PHX \), and since \( PF \parallel YK \), \( \angle PHX = \angle HWR \)? Wait, that doesn't make sense. Wait, maybe \( \angle PHM \) and \( \angle HWR \) are equal? Wait, no, let's check the diagram again.

Wait, the diagram: \( PF \) is horizontal, \( YK \) is horizontal, so they are parallel. \( H \) is on \( PF \), \( W \) and \( R \) are on \( YK \). \( HG \) and \( HB \) are lines through \( H \) and \( W, R \) respectively. \( M \) and \( X \) are on the lines through \( H \). So \( \angle PHM = (11z + 6)^\circ \), \( \angle MHX = 47^\circ \), so \( \angle PHX = \angle PHM + \angle MHX = (11z + 6) + 47 = (11z + 53)^\circ \). Now, since \( PF \parallel YK \), \( \angle PHX \) and \( \angle HWR \) are corresponding angles, so they should be equal? Wait, \( \angle HWR = 72^\circ \), so \( 11z + 53 = 72 \)? Wait, no, that would give \( 11z = 19 \), which is not integer. Wait, maybe I got the angles wrong.

Wait, maybe \( \angle PHM \) and \( \angle HWR \) are alternate interior angles. Wait, \( \angle PHM \) is at \( H \) on \( PF \), \( \angle HWR \) is at \( W \) on \( YK \). So if \( HG \) is the transversal, then \( \angle PHM \) and \( \angle HWR \) are alternate interior angles, so they should be equal. So \( 11z + 6 = 72 \)? Then \( 11z = 66 \), so \( z = 6 \). Ah, that makes sense. So \( \angle PHM = 11*6 + 6 = 72^\circ \), which is equal to \( \angle HWR = 72^\circ \), so that's alternate interior angles.

Step2: Find \( \angle FHR \)

Now, \( \angle FHR \): since \( PF \parallel YK \), and \( HB \) is a transversal, \( \angle FHR \) and \( \angle HWR \) – wait, no, \( \angle MHX = 47^\circ \), and \( \angle FHR \) is vertical to \( \angle PHM + \angle MHX \)? Wait, no, \( \angle PHM = 72^\circ \), \( \angle MHX = 47^\circ \), so \( \angle PHX = 72 + 47 = 119^\circ \), but \( \angle FHR \) is equal to \( \angle PHX \) because they are vertical angles? Wait, no, \( \angle PHX \) and \( \angle FHR \) are vertical angles? Wait, \( H \) is the intersection, so \( \angle PHX \) and \( \angle FHR \) are vertical angles, so they should be equal. Wait, but \( \angle HWR = 72^\circ \), \( \angle MHX = 47^\circ \), so maybe \( \angle FHR = 180^\circ - 72^\circ - 47^\circ \)? No, wait, let's think again.

Wait, \( \angle PHM = 72^\circ \) (since \( z = 6 \)), \( \angle MHX = 47^\circ \), so \( \angle PHX = 72 + 47 = 119^\circ \). Now, \( \angle FHR \) is vertical to \( \angle PHX \), so \( \angle FHR = 119^\circ \)? Wait, no, maybe \( \angle FHR \) is equal to \( \angle HWR + \angle MHX \)? Wait, \( \angle HWR = 72^\circ \), \( \angle MHX = 47^\circ \), so \( 72 + 47 = 119^\circ \), which matches. Alternatively, since \( PF \parallel YK \), \( \angle FHR \) and \( \angle HWR \) with the other angle: Wait, maybe \( \angle FHR = 180^\circ - \angle PHM = 180 - 72 = 108^\circ \)? No, that's not right. Wait, let's check the vertical angles. \( \angle PHM = 72^\circ \), so \( \angle FHM = 180 - 72 = 108^\circ \)? No, \( \angle PHM \) and \( \angle FHM \) are supplementary. Wait, maybe I made a mistake in Step1.

Wait, Step1: \( \overleftrightarrow{PF} \parallel \overleftrightarrow{YK} \), transversal \( HG \), so \( \angle PHM \) and \( \angle HWR \) are alternate interior angles, so \( \angle PHM = \angle HWR \). So \( 11z + 6 = 72 \), so \( 11z = 66 \), \( z = 6 \). That's correct. Then \( \angle PHM = 72^\circ \), \( \angle MHX = 47^\circ \), so \( \angle PHX = 72 + 47 = 119^\circ \). Now, \( \angle FHR \) is vertical to \( \angle PHX \), so \( \angle FHR = 119^\circ \). Alternatively, since \( PF \parallel YK \), \( \angle FHR \) and \( \angle HWR \) with \( \angle MHX \): Wait, \( \angle HWR = 72^\circ \), \( \angle MHX = 47^\circ \), so \( \angle FHR = 72 + 47 = 119^\circ \).

Step3: Find \( \angle GWR \)

\( \angle GWR \): since \( \angle HWR = 72^\circ \), and \( \angle GWR \) is supplementary to \( \angle HWR \) (because they are linear pair), so \( \angle GWR = 180 - 72 = 108^\circ \). Wait, no, \( \angle GWR \) and \( \angle HWR \) are adjacent angles on a straight line, so they are supplementary. So \( \angle GWR = 180 - 72 = 108^\circ \).

Answer:

The value of \( z \) is \( 6 \), with a measure of \( \angle FHR \) of \( 119^\circ \) and a measure of \( \angle GWR \) of \( 108^\circ \).