in the image shown, $overleftrightarrow{pf} parallel overleftrightarrow{yk}$, where $mangle phm = (11z + 6)^circ$, $mangle mhx = 47^circ$, and $mangle hwr = 72^circ$.

Question

Accepted Answer

To determine $ m\angle GWR $, we analyze the relationships between the angles:

### Step 1: Identify Supplementary Angles
Since $ \angle HWR = 72^\circ $, and $ \angle GWR $ and $ \angle HWR $ are supplementary (they form a linear pair), their sum is $ 180^\circ $.

### Step 2: Calculate $ m\angle GWR $
Using the supplementary angle property:
$$
m\angle GWR + m\angle HWR = 180^\circ
$$
Substitute $ m\angle HWR = 72^\circ $:
$$
m\angle GWR + 72^\circ = 180^\circ
$$
Solve for $ m\angle GWR $:
$$
m\angle GWR = 180^\circ - 72^\circ = 108^\circ
$$

Wait, but let's check again. Wait, maybe I made a mistake. Wait, $ \angle GWR $ and $ \angle HWR $: Wait, actually, looking at the diagram, $ \angle GWR $ and $ \angle HWR $ – no, wait, $ \angle GWR $ and $ \angle HWR $ are adjacent and form a linear pair? Wait, no, maybe $ \angle GWR $ is vertical or corresponding? Wait, no, let's re-examine.

Wait, $ PF \parallel YK $, and $ HG $ is a transversal. Wait, $ \angle FHR $ is $ 108^\circ $, and since $ PF \parallel YK $, $ \angle FHR $ and $ \angle HRK $ are supplementary? No, wait, $ \angle HWR = 72^\circ $, and $ \angle GWR $: Wait, maybe $ \angle GWR $ is equal to $ \angle FHR $ because of corresponding angles? Wait, $ PF \parallel YK $, and $ HG $ is a transversal. Wait, $ \angle FHG $ and $ \angle GWR $ – no, maybe I messed up.

Wait, the given $ m\angle HWR = 72^\circ $, and $ \angle GWR $ is supplementary to $ \angle HWR $? Wait, no, $ \angle GWR + \angle HWR = 180^\circ $? Wait, $ \angle HWR = 72^\circ $, so $ \angle GWR = 180 - 72 = 108^\circ $? But let's check the options. The options are $ 47^\circ $, $ 61^\circ $, $ 72^\circ $, $ 108^\circ $. Wait, but maybe I made a mistake. Wait, $ \angle GWR $ and $ \angle FHR $ are corresponding angles? Since $ PF \parallel YK $, and $ HR $ is a transversal? No, $ HG $ and $ HB $ are transversals. Wait, $ \angle FHR = 108^\circ $, and $ \angle GWR $ should be equal to $ \angle FHR $ because of corresponding angles? Wait, $ PF \parallel YK $, and $ HG $ is a transversal. Wait, $ \angle PHG $ and $ \angle GWR $ – no, maybe $ \angle GWR $ is $ 108^\circ $, but let's check again.

Wait, the problem says "a measure of $ \angle GWR $ of". The options are $ 47^\circ $, $ 61^\circ $, $ 72^\circ $, $ 108^\circ $. Wait, maybe $ \angle GWR $ is equal to $ \angle FHR $, which is $ 108^\circ $, so $ m\angle GWR = 108^\circ $.

But let's re-express:

### Step 1: Recognize Corresponding Angles
Since $ PF \parallel YK $, and $ HR $ is a transversal, $ \angle FHR $ and $ \angle HRK $ are supplementary? No, $ \angle FHR + \angle HRK = 180^\circ $, but $ \angle HRK $ is equal to $ \angle HWR $? No, $ \angle HWR = 72^\circ $, and $ \angle HRK $ is also $ 72^\circ $ (vertical angles)? Wait, no, $ \angle HWR $ and $ \angle HRK $ are alternate interior angles? Since $ PF \parallel YK $, $ \angle PH R = \angle HRK $. Wait, $ \angle PHM = (11z + 6)^\circ $, $ z = 6 $, so $ \angle PHM = 11(6) + 6 = 72^\circ $. Then $ \angle PHM = 72^\circ $, and $ \angle M H X = 47^\circ $, so $ \angle PHX = 72 + 47 = 119^\circ $? No, that doesn't make sense. Wait, maybe $ \angle PHM $ and $ \angle M H X $ are adjacent angles on a straight line? Wait, $ PF $ is a straight line, so $ \angle PHM + \angle M H X + \angle XHF = 180^\circ $? No, the diagram shows $ H $ with lines $ PF $, $ YK $, $ HG $, $ HB $, and $ MX $.

Wait, the value of $ z $ is given as $ 6^\circ $, so $ \angle PHM = 11(6) + 6 = 72^\circ $. Then $ \angle PHM = 72^\circ $, and $ \angle M H X = 47^\circ $, so $ \angle PHX = 72 + 47 = 119^\circ $? No, that's not right. Wait, maybe $ \angle PHM $ and $ \angle M H X $ are vertical angles? No, $ MX $ is a line, so $ \angle PHM $ and $ \angle XHF $ are vertical angles? Wait, maybe I'm overcomplicating. The problem already gives that $ z = 6^\circ $, $ \angle FHR = 108^\circ $, and we need $ \angle GWR $.

Since $ PF \parallel YK $, and $ HG $ is a transversal, $ \angle FHG $ (which is $ \angle FHR $) and $ \angle GWR $ are corresponding angles, so they are equal. Therefore, $ m\angle GWR = 108^\circ $.

# Answer:
$ 108^\circ $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step 1: Identify Supplementary Angles

Step 2: Calculate \( m\angle GWR \)

Step 1: Recognize Corresponding Angles

Answer: