Question

pf || yk is shown where ( mangle phm = (8k + 11)^circ ), ( mangle whr = 47^circ ), and ( mangle wrb = 105^circ ) the value of ( k ) is dropdown with a measure of ( angle rhf ) of dropdown with ( 47^circ ), ( 58^circ ), ( 75^circ ), ( 105^circ ) and a measure of ( angle ywg ) of dropdown

Explanation:

Step1: Find \( m\angle HRF \) using supplementary angles

Since \( \angle WRB = 105^\circ \), and \( \angle WRB \) and \( \angle HRF \) are supplementary (they form a linear pair), we have:
\( m\angle HRF = 180^\circ - 105^\circ = 75^\circ \)? Wait, no, wait. Wait, \( PF \parallel YK \), so maybe we can use alternate interior angles or corresponding angles. Wait, first, let's look at \( \angle WHR = 47^\circ \), and \( \angle WRB = 105^\circ \). Wait, in triangle \( WHR \)? No, \( PF \parallel YK \), so the transversal \( HR \) and \( HW \)? Wait, maybe first find \( m\angle RHF \). Wait, \( \angle WRB = 105^\circ \), and since \( PF \parallel YK \), \( \angle RHF \) and \( \angle WRB \) are same - side interior angles? No, wait, \( \angle WHR = 47^\circ \), and \( \angle WRB = 105^\circ \), so in the triangle (if we consider triangle \( WHR \)), but actually, \( PF \parallel YK \), so the transversal \( HG \) and \( HD \)? Wait, maybe we can find \( m\angle PHM \) first. Wait, \( \angle WRB = 105^\circ \), and \( \angle WHR = 47^\circ \), so the angle at \( H \) for the transversal: Wait, \( PF \parallel YK \), so \( \angle PHM \) and \( \angle YWH \) are corresponding angles? Wait, no, let's think again.

Wait, \( \angle WRB = 105^\circ \), so its supplementary angle \( \angle HRK = 180 - 105 = 75^\circ \)? No, \( \angle WRB \) and \( \angle HRK \) are vertical angles? No, \( \angle WRB \) and \( \angle HRK \) are not. Wait, \( YK \) is a straight line, so \( \angle WRB + \angle HRK = 180^\circ \)? No, \( W - R - K \) is a straight line, so \( \angle WRB + \angle HRB = 180^\circ \)? Wait, maybe I made a mistake. Let's use the fact that \( PF \parallel YK \), so the sum of same - side interior angles is \( 180^\circ \). Wait, \( \angle WHR = 47^\circ \), and \( \angle WRB = 105^\circ \), so the angle \( \angle RHF \): Wait, \( \angle RHF \) and \( \angle WRB \) are same - side interior angles? No, \( PF \parallel YK \), transversal \( HR \), so \( \angle RHF + \angle WRB = 180^\circ \)? No, that's not right. Wait, \( \angle WHR = 47^\circ \), and \( \angle WRB = 105^\circ \), so in the triangle \( WHR \), the third angle \( \angle HWR = 180 - 47 - 105 = 28^\circ \)? No, that doesn't seem right. Wait, maybe the angle \( \angle PHM=(8k + 11)^\circ \) and \( \angle RHF \) are related. Wait, maybe \( \angle RHF = 180^\circ-\angle WRB = 75^\circ \)? No, \( \angle WRB = 105^\circ \), so \( 180 - 105 = 75 \), but then \( \angle PHM \) and \( \angle RHF \) are equal? Wait, no, \( \angle WHR = 47^\circ \), so \( \angle PHM + \angle WHR=\angle RHF \)? Wait, no, let's look at the diagram. \( H \) is the intersection of \( PF \) and the two lines \( HG \) and \( HD \). \( YK \) is parallel to \( PF \), with \( W \) and \( R \) on \( YK \).

Wait, maybe the correct approach is: Since \( PF\parallel YK \), \( \angle RHF + \angle WRB = 180^\circ \)? No, \( \angle WRB = 105^\circ \), so \( \angle RHF = 180 - 105 = 75^\circ \)? Wait, but then \( \angle PHM=(8k + 11)^\circ \), and \( \angle PHM \) and \( \angle RHF \) are related. Wait, \( \angle WHR = 47^\circ \), so \( \angle PHM + \angle WHR=\angle RHF \)? Wait, \( (8k + 11)+47 = 75 \)? No, that would be \( 8k+58 = 75 \), \( 8k = 17 \), which is not an integer. So I must have made a mistake.

Wait, maybe \( \angle WRB = 105^\circ \), so its alternate interior angle with \( \angle RHF \) is not, but \( \angle YWG \) and \( \angle WRB \) are supplementary? Wait, \( \angle WRB = 105^\circ \), so \( \angle YWG = 180 - 105 = 75^\circ \)? No, the options for \( \angle RHF \) are \( 47^\circ \), \( 58^\circ \), \( 75^\circ \), \( 105^\circ \). Wait, let's try another way. \( \angle WHR = 47^\circ \), and \( \angle WRB = 105^\circ \), so the angle \( \angle RHF \) is equal to \( 180 - 47 - 105 = 28^\circ \)? No, that's not an option. Wait, maybe \( PF\parallel YK \), so \( \angle PHM \) and \( \angle YWH \) are corresponding angles. \( \angle YWH = 180 - \angle WHR - \angle WRB \)? No, \( \angle WHR = 47^\circ \), \( \angle WRB = 105^\circ \), so \( \angle YWH = 180 - 47 - 105 = 28^\circ \), which is not helpful.

Wait, maybe the angle \( \angle RHF \) is equal to \( 180 - 105 = 75^\circ \), and \( \angle PHM=(8k + 11)^\circ \), and \( \angle PHM + \angle WHR=\angle RHF \)? No, \( 8k + 11+47 = 75 \), \( 8k=17 \), no. Wait, maybe \( \angle PHM \) and \( \angle RHF \) are equal because they are alternate interior angles. So \( 8k + 11=75 \), then \( 8k = 64 \), \( k = 8 \). Wait, but then \( \angle RHF = 75^\circ \), which is an option. Then \( \angle YWG \): since \( \angle WHR = 47^\circ \), and \( PF\parallel YK \), \( \angle YWG \) and \( \angle PHM \) are corresponding angles? Wait, \( \angle PHM=(8\times8 + 11)=75^\circ \), and \( \angle YWG \) is equal to \( 180 - 105 = 75^\circ \)? Wait, no, \( \angle WRB = 105^\circ \), so \( \angle YWG = 180 - 105 = 75^\circ \), and \( \angle RHF = 75^\circ \), \( k = 8 \).

Wait, let's re - do:

1. Since \( PF\parallel YK \), \( \angle RHF \) and \( \angle WRB \) are same - side interior angles? No, \( \angle WRB = 105^\circ \), so \( \angle RHF = 180 - 105 = 75^\circ \) (because they are same - side interior angles and \( PF\parallel YK \)).

2. Also, \( \angle PHM=(8k + 11)^\circ \), and \( \angle PHM \) and \( \angle RHF \) are equal (alternate interior angles? Wait, maybe \( \angle PHM \) and \( \angle RHF \) are equal because of the transversal. Wait, if \( \angle PHM=(8k + 11)^\circ \) and \( \angle RHF = 75^\circ \), then \( 8k+11 = 75 \).

Step1: Solve for \( k \)

Set \( 8k + 11=75 \)
Subtract 11 from both sides: \( 8k=75 - 11=64 \)
Divide both sides by 8: \( k = \frac{64}{8}=8 \)

Step2: Find \( m\angle RHF \)

From the above, \( m\angle RHF = 75^\circ \)

Step3: Find \( m\angle YWG \)

Since \( \angle WRB = 105^\circ \), and \( \angle YWG \) and \( \angle WRB \) are supplementary (linear pair? Wait, no, \( \angle YWG \) and \( \angle WRB \) are same - side interior angles? Wait, \( PF\parallel YK \), transversal \( WG \), so \( \angle YWG + \angle WRB = 180^\circ \)? No, \( \angle WRB = 105^\circ \), so \( \angle YWG = 180 - 105 = 75^\circ \)? Wait, but also, \( \angle YWG \) and \( \angle PHM \) are corresponding angles, and \( \angle PHM = 75^\circ \), so \( \angle YWG = 75^\circ \)

Answer:

The value of \( k \) is \( 8 \), the measure of \( \angle RHF \) is \( 75^\circ \), and the measure of \( \angle YWG \) is \( 75^\circ \)