Question

in the figure, ( overline{ab} ), ( overline{de} ), and ( overline{cf} ) are parallel. if ( 204 < x + y + z < 214 ), which of the following could be a value of ( w - y )?
i. 24
ii. 33
a i only
b ii only
c i and ii
d there is not enough information to determine which could be a value of ( w - y )

Explanation:

Step1: Use the property of parallel lines

When two parallel lines are cut by a transversal, the alternate - interior angles are equal.
Since \(AB\parallel CF\), \(w=x + y\) (by the property of alternate - interior angles).
Since \(DE\parallel CF\), \(z=y\) (alternate - interior angles).
We know that \(204\lt x + y+z\lt214\). Substitute \(z = y\) into the inequality, we get \(204\lt x + 2y\lt214\).
Also, \(w - y=(x + y)-y=x\).

Step2: Analyze the inequality for \(x\)

From \(204\lt x + 2y\lt214\), we can rewrite it as \(204-2y\lt x\lt214 - 2y\).
Since \(x\gt0\) and \(y\gt0\), we consider the following:
If \(y = 85\), then \(204-2\times85=204 - 170 = 34\) and \(214-2\times85=214 - 170 = 44\).
If \(y = 80\), then \(204-2\times80=204 - 160 = 44\) and \(214-2\times80=214 - 160 = 54\).
If \(y=70\), then \(204 - 2\times70=204-140 = 64\) and \(214-2\times70=214 - 140 = 74\).
If \(y = 60\), then \(204-2\times60=204 - 120 = 84\) and \(214-2\times60=214 - 120 = 94\).
If \(y=55\), then \(204-2\times55=204 - 110 = 94\) and \(214-2\times55=214 - 110 = 104\).
If \(y = 50\), then \(204-2\times50=204 - 100 = 104\) and \(214-2\times50=214 - 100 = 114\).
If \(y = 40\), then \(204-2\times40=204 - 80 = 124\) and \(214-2\times40=214 - 80 = 134\).
If \(y=35\), then \(204-2\times35=204 - 70 = 134\) and \(214-2\times35=214 - 70 = 144\).
If \(y = 30\), then \(204-2\times30=204 - 60 = 144\) and \(214-2\times30=214 - 60 = 154\).
If \(y=25\), then \(204-2\times25=204 - 50 = 154\) and \(214-2\times25=214 - 50 = 164\).
If \(y = 20\), then \(204-2\times20=204 - 40 = 164\) and \(214-2\times20=214 - 40 = 174\).
If \(y=10\), then \(204-2\times10=204 - 20 = 184\) and \(214-2\times10=214 - 20 = 194\).
If \(y = 5\), then \(204-2\times5=204 - 10 = 194\) and \(214-2\times5=214 - 10 = 204\).

We can also use another approach.
Since \(AB\parallel DE\parallel CF\), by the property of parallel lines and transversals:
\(w=x + y\) (alternate - interior angles for \(AB\parallel CF\) with transversal \(BC\)) and \(z = y\) (alternate - interior angles for \(DE\parallel CF\) with transversal \(CD\)).
We know that \(204\lt x + y+z\lt214\), substituting \(z = y\) gives \(204\lt x + 2y\lt214\).
Let's assume \(x + 2y=204 + k\) where \(0\lt k\lt10\).
\(w - y=x\).
If \(x = 24\), then \(204-2y\lt24\lt214 - 2y\).
From \(204-2y\lt24\), we get \(2y\gt204 - 24=180\), \(y > 90\). From \(24\lt214 - 2y\), we get \(2y\lt214 - 24 = 190\), \(y\lt95\).
If \(x = 33\), then \(204-2y\lt33\lt214 - 2y\).
From \(204-2y\lt33\), \(2y\gt204 - 33 = 171\), \(y>85.5\). From \(33\lt214 - 2y\), \(2y\lt214 - 33=181\), \(y\lt90.5\).

Answer:

C. I and II